TENSOR TOMOGRAPHY FOR SURFACES · X-ray transform on negatively curved surfaces and spectral...

TENSOR TOMOGRAPHY FOR SURFACES

by Thibault LEFEUVRE (*)

Résumé. Ce mémoire se penche sur de récents résultats obtenus dansla théorie des problèmes inverses sur les surfaces. En particulier, nousdémontrons l’injectivité de la transformée en rayon-X pour les 2-tenseurs symétriques sur les surfaces de type Anosov (surfaces surlesquelles le flot géodésique est de type Anosov) et l’injectivité dela transformée en rayon-X pour les m-tenseurs symétriques (m > 0quelconque) pour les surfaces simples à bord strictement convexe.Nous montrons comment ces résultats permettent de prouver despropriétés de rigidité spectrales ou de bord rigide.

Résumé. This memoire surveys some recent results in inverse pro-blem theory for surfaces. More precisely, we prove the injectivityof the X-ray transform of symmetric 2-tensors for Anosov surfaces(surfaces on which the geodesic flow is Anosov) and the injectivity ofthe X-ray transform of symmetric m-tensors (m > 0) for simple sur-faces with strictly convex boundary. We show how these results canbe applied to prove spectral rigidity properties or boundary rigidityproperties.

(*) under the supervision of Colin Guillarmou.

2 THIBAULT LEFEUVRE

Table des matières

1. Introduction 51.1. Historical background 51.1.1. The Radon transform 51.1.2. Generalization 71.2. Content of this memoire 81.3. Acknowledgement 92. Preliminaries of Riemannian geometry 102.1. Notations 102.2. The geodesic flow 122.2.1. The standard point of view 122.2.2. The dual point of view 122.3. Structure of SM 152.3.1. The horizontal distribution 152.3.2. The Sasaki metric 162.4. Surface theory 182.4.1. Isothermal coordinates 182.4.2. The moving frame 182.4.3. Canonical volume form on SM 213. Functional spaces on SM 233.1. The space L2(SM) 233.1.1. An orthogonal decomposition of L2(SM) 233.1.2. The Hilbert transform 243.2. Symmetric tensors on a manifold 253.2.1. Symmetric tensors 253.2.2. Functions on SM and symmetric tensors 273.2.3. The inner derivation 283.2.4. Actions on 1-forms 293.3. The Riccati equation 303.3.1. The Jacobi vector fields 303.3.2. The Riccati equation 313.3.3. α-controlled surfaces 323.4. A technical toolbox 333.4.1. The raising and lowering operators η+ and η− 343.4.2. Max Noether’s theorem 353.4.3. The Pestov identity 363.4.4. Proof of Lemma 3.7 374. The X-ray transform 384.1. Definition of the X-ray transform 38

TENSOR TOMOGRAPHY FOR SURFACES 3

4.1.1. A first definition 384.1.2. The notion of s-injectivity 384.1.3. The Livcic property 394.2. Definition on a compact manifold with boundary 404.2.1. Another definition of the X-ray transform 404.2.2. The map I 414.2.3. The adjoint map I∗0 424.2.4. The operator I∗0 I0 444.3. Definition on a manifold without boundary 484.3.1. The point of view of distributions 484.4. A few words about the hypothesis 494.4.1. Strict convexity for manifolds with boundary 494.4.2. The hypothesis of simplicity and the injectivity of I0, I1 555. X-ray transform on negatively curved surfaces and spectral

rigidity 575.1. Introduction 575.1.1. The spectrum of the Laplacian 575.1.2. What about a converse ? 585.1.3. Spectrally rigid manifolds 585.2. Injectivity of the ray transform 585.3. Proof of Theorem 5.4 605.4. A second result 636. Spectral rigidity on Anosov surfaces 646.1. Introduction 646.2. Surjectivity of I∗0 646.3. Surjectivity of I∗1 666.4. Injectivity of I2 686.4.1. The mixed Sobolev norm 686.4.2. Attempt for a general proof 706.4.3. Injectivity of I2 717. Injectivity of the X-ray transform on simple surfaces 737.1. Introduction 737.2. Pestov identity in presence of an attenuation 737.3. End of the proof 757.4. Surjectivity of I∗0 787.5. Application to the deformation boundary rigidity problem 818. The boundary rigidity property 848.1. Notations 848.2. A first reduction 84

4 THIBAULT LEFEUVRE

8.3. The scattering relation determines the Dirichlet-to-Neumannmap 86

8.4. The DN map determines the conformal class 888.5. Conclusion of the proof 889. Conclusion 89Annexe A. (Pseudo)differential operators 90A.1. Differential operators 90A.2. Pseudodifferential operator 91Annexe B. Existence of isothermal coordinates on a surface 93Annexe C. On Anosov flows 95C.1. Definition 95C.2. The Anosov theorem 97C.3. Livcic’s periodic theorem 98Annexe D. Decomposition of symmetric tensors 101D.1. Decomposition in potential and solenoidal parts 101D.2. Decomposition in a vicinity of the boundary 104Annexe E. The Riemann-Roch theorem 106E.1. Holomorphic line bundles 106E.2. The first Chern class 107E.3. The Riemann-Roch formula 108BIBLIOGRAPHY 110


1. Introduction

1.1. Historical background

1.1.1. The Radon transform

Historically, one of the first example of inverse problem was given by J.Radon in 1917, in his celebrated article Über die Bestimmung von Funk-tionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten (On thedetermination of functions from their integral values along certain mani-folds). He introduced a transform — which we now refer to as the Radontransform — integrating functions of R2 along straight lines. In other words,given a function f , the Radon transform Rf is the application :

Rf : L 7→∫L

f(x)dx,

where L denotes a line of the plane and dx the Lebesgue measure on L. Byparametrizing a line by its slope p and its intercept τ , the Radon transformcan be written :

Rf(p, τ) =∫ +∞

−∞f(x, px+ τ)dx

But the usual form is that given in polar coordinates (r, θ), where θ denotesthe angle of the slope and r the offset of the line. We have :

Rf(r, θ) =∫R2f(x, y)δ0(x cos θ + y sin θ − r)dxdy

r

θ

Figure 1.1. The polar coordinates

6 THIBAULT LEFEUVRE

This expression is rather interesting and its computation shows a cha-racteristic sinusoid shape. It is referred to as a sinogram. Given a black-and-white picture, the function f that one can take is the intensity of thegrey in a given point (x, y) of the picture :

Figure 1.2. The initial picture ...

Figure 1.3. ... and its sinogram.

In his article, Radon proved an inversion formula, that is, he pointed outthe possibility to reconstruct the initial function f when given its transform,and gave the explicit formula :

f(x, y) = 1(2π)2

∫ π

0(Rf(·, θ) ∗ h) (x cos θ + y sin θ)dθ,

where h is the function such that h(ξ) = |ξ|.Medical imaging techniques such as CT (Computerized Tomography)

actually compute Radon transforms. The idea is the following : given acertain structure (like lambs, tissues or cells) with unknown density, onesends X-ray through the shape. Given an initial beam, with known intensity,


it is possible to gather its attenuated intensity when coming out of the shape— this is precisely the Radon transform. Then, using an inversion formulafor the Radon transform, one recovers the unknown density.

1.1.2. Generalization

The Radon transform was just the start of a long theory of inverse pro-blems. It gave birth to many applications in a large number of fields. Letus just quote a few of them, in order to motivate the reader’s interest forit. We refer to [24] for more details.

One of the most astonishing application of the inverse problem theoryarises in geophysical imaging, in order to determine the inner structure ofthe Earth. The seismic waves going through the Earth’s crust increase theirspeed with depth, which curves the ray back to the surface. In some sense,they actually follow geodesics, where the metric is given by the Earth’sdensity. The question raised is therefore : to what extent is it possible toreconstruct the Earth’s density when knowing how seismic waves propagateinto the Earth’s crust ?

Such a problem is also studied in ultrasound tomography, when trying todetect tumors using blood flow measurements or in non-invasive industrialmeasurements for reconstructing the velocity of a fluid in motion. In thiscase, the problem involves integration formulas of 1-tensors along geodesics.The integration of 4-tensors can describe the perturbation of travel timesof compressional waves propagating in anisotropic elastic media.From a more theoretical point of view, given a Riemannian manifold

(M, g), the tomography is the idea to reconstruct a tensor T , given its inte-gral along some geodesics of the manifold. Indeed, just like in the previouscase of R2, it is possible to define a ray transform, similar to the Radontransform. The questions which naturally rise are therefore : can one find anexplicit inversion formula ? Is the ray transform an injective application ?Or what are the obstructions for the ray transform to become injective ?The manifolds considered are usually compact with or without boundary.

In the case with boundary, one of the most studied example is that ofsimple manifolds (that is, free of conjugate points) and with strictly convexboundary. As to manifolds without boundary, a very interesting class ofmanifolds is provided by the set of Anosov manifolds, that is, the set ofmanifolds for which the geodesic flow is hyperbolic (see Appendix C for adefinition of Anosov flows).

8 THIBAULT LEFEUVRE

1.2. Content of this memoire

All the results exposed in this memoire are related to inverse problemsfor surfaces. We relate recent theorems proved by G. P. Paternain, M. Saloand G. Uhlmann, mostly in [23] and [25]. Some of the theorems detailedbelow have been extended to greater dimensions. When it is the case, itwill be mentioned, but all our proofs will be done in the two-dimensionalcase. Note that some of the results exposed here are still open in dimensiongreater than 2.

In a preliminary part, we will recall some elements of Riemannian geo-metry which will be used throughout the rest of the memoire. We detail thebasic tools of the geometry of surfaces and their unit tangent bundle, intro-ducing the standard moving frame X,H, V . Then, in Section 3, we seehow these geometric properties can be related to some specific propertiesof the functional space L2(SM). In particular, we introduce a decompo-sition in Fourier elements in the circle bundles and the lower and raisingoperators η+, η−. Section 5, is devoted to the proof of the injectivity ofthe ray transform on negatively curved surfaces. This is based on a cele-brated result of V. Guillemin and D. Kazhdan [14], published in 1979. Inparticular, the injectivity of I2 will allow us to show that such a surface isspectrally rigid, that is an isospectral continuous deformation of the metric(a deformation preserving the spectrum of the Laplace-Beltrami operator)is always an isometry.It is known that the geodesic flow on a negatively curved closed ma-

nifold is Anosov (see Appendix C for a definition). In Section 6, we willtry to extend our previous result to Anosov surfaces, that are Riemanniansurfaces for which the geodesic flow is Anosov (thus including negativelycurved surfaces). We will show that I2 is injective, thus proving that sucha surface is also spectrally rigid, but the proof does not extend to obtainthe injectivity of Im for m > 3.

In a third part, we will forget boundaryless surfaces and turn to compactsurfaces which are simple with strictly convex boundary, asking the samequestion. We will show that Im is injective, for any m > 0. This will implythat the manifold is deformation boundary rigid.

The appendix contains a few elements of introduction to different theo-ries referred to throughout the memoire. It is far from being exhaustiveand detailed. In particular, most of the proofs are omitted but some refe-rences are given. We detail some more subtle results like the existence oflocal isothermal coordinates for surfaces, or the decomposition of symme-tric tensors.


1.3. Acknowledgement

This memoire validates the "Master de Mathématiques Fondamentales"of University Paris VI, as well as the fourth year of the Ecole Polytechnique.It was carried out under the supervision of Colin Guillarmou. I deeply thankhim for his advice and suggesting me this subject.

10 THIBAULT LEFEUVRE

2. Preliminaries of Riemannian geometry

The main reference for this section is [8]. In the following, we consi-der (M, g) an n-dimensional smooth manifold endowed with a Riemannianmetric g.

2.1. Notations

Given a local chart (U, φ) on M , we will denote by (xi)16i6n the localcoordinates and we write in these coordinates

g =n∑

i,j=1gijdx

idxj ,

where gij(x) = g

(∂

∂xi,∂

∂xj

). We denote by (gij) the coefficient of the

inverse matrix. Given x ∈ M , the norm of v ∈ TxM is given by |v|x =gx(v, v)1/2, which will sometimes be denoted 〈v, v〉1/2x . In the following, wewill often drop the index x. Note that if we are given other coordinates(yj), then one can check that in these new coordinates :

(2.1) g =n∑

i,j,k,l=1gij

∂xi∂yk

∂xj∂yl

dykdyl

We define the musical isomorphism at x ∈M by

[ :∣∣∣∣ TxM → T ∗xM

v 7→ v[ = g(v, ·)This is an isomorphism between the two vector bundles TM and T ∗M

since they are equidimensional and g is symmetric definite positive, thusnon-degenerate. Given an orthonormal basis (ei) of TxM , we will denoteby (ei) the dual basis of T ∗xM which is, in other words, the image of thebasis (ei) by the musical isomorphism.

If E is a vector bundle over M , then the projection will be denoted π :E →M . We denote by Γ(M,E) the set of smooth sections of E. Γ(M) de-notes Γ(M,TM), the set of vector fields. We will denote by Γ(M,⊗mS T ∗M)the set of smooth symmetric covariant m-tensors on M . On the cotangentbundle T ∗M , we denote by ω the canonical symplectic form, which wewrite in coordinates ω =

∑ni=1 dp

i ∧ dxi. We recall that it is obtained asthe differential of the canonical 1-form λ ∈ Ω1(T ∗M), defined intrinsicallyas

λ(x,p)(ξ) = p(dπ(x,p)(ξ)

),


for a point (x, p) ∈ T ∗M, ξ ∈ T(x,p)T∗M and where π : T ∗M →M denotes

the projection.Given a point x ∈ M , if (ei) is an orthonormal basis of TxM , we define

dvol = e1 ∧ ... ∧ en. In local coordinates, it is given by the formula :

(2.2) dvol =√

det(gij)dx1 ∧ ... ∧ dxn

We recall that the Laplacian in local coordinates is given by :

(2.3) ∆f =n∑

i,j=1

1√det g

∂i

(√det ggij∂jf

)We denote by ∇ the Levi-Civita connection on TM . In local coordinates,

the Christoffel symbols are defined such that :

(2.4) ∇ ∂∂xi

∂

∂xj=

n∑k=1

Γkij∂

∂xk

They are given by the Koszul formula :

(2.5) Γkij = 12

n∑l=1

gkl(∂gil∂xj

+ ∂gjl∂xi− ∂gij∂xl

)We denote the torsion tensor T∇, which is defined as :

T∇(X,Y ) = ∇XY −∇YX − [X,Y ]

The curvature tensor is denoted by F∇ and defined as :

(2.6) F∇(X,Y )Z = ∇X∇Y Z −∇Y∇XZ −∇[X,Y ]Z

In particular, we clearly have F∇(X,Y ) = −F∇(Y,X). We recall that theLevi-Civita connection is the unique torsion-free and g-metric connection,namely it satisfies for any X,Y, Z ∈ Γ(M) :

(2.7) T∇(X,Y ) = ∇XY −∇YX − [X,Y ] = 0

(2.8) Z · (g(X,Y )) = g(∇ZX,Y ) + g(X,∇ZY )

The sectional curvature K is given at x by

Kx(e1, e2) = 〈F∇(e1, e2)e1, e2〉,

where e1, e2 ∈ TxM are orthogonal. In particular, in the case of a surface,which is what we will be mostly interested in, the sectional curvature is areal number referred to as the Gaussian curvature (or simply the curvatureif the context is not ambiguous).


2.2. The geodesic flow2.2.1. The standard point of viewGiven a curve γ : I →M , we recall that it is a geodesic if it satisfies the

geodesic equation :

(2.9) ∇γ γ = 0

In local coordinates, this equation becomes :

(2.10) γk +n∑

i,j=1Γkij γiγj = 0

For any initial data (γ(0), γ(0)) = (x, v) ∈ TM , the Cauchy theorem en-sures the local existence of a solution (γ, γ) defined for t small. If M iscomplete (for instance if M is compact), then the existence is global. For apoint (x, v) ∈ TM , we denote by ϕt(x, v) = (γ(t), γ(t)) the geodesic flow,when it is defined. Thanks to the geodesic equation written in coordinates,we know that the infinitesimal generator X of the geodesic flow (which wewill sometimes call the geodesic vector field) is given in coordinates by

(2.11) X(x, v) = vk∂

∂xk− Γkijvivj

∂

∂vk,

where we used the Einstein convention of summation. This will be done inthe following in order to simplify the notations.

2.2.2. The dual point of viewChanging our point of view, geodesics can also be seen as critical points

of a certain Lagrangian. Indeed, consider the Lagrangian density L definedon TM by L(x, v) = 1

2 |v|2x. Then, the critical points of the energy

E(γ) =∫ b

a

L(γ, γ) dt =∫ b

a

|γ(t)|2γ(t) dt,

sought among the piece-wise C1 curves γ joining γ(a) to γ(b), are the geo-desics. In this perspective, one can check that the geodesic equation (2.9)is nothing less but a reformulation of the Euler-Lagrange equations :

d

dt

(∂L

∂v(γ, γ)

)= ∂L

∂x(γ, γ)

In a more general frame, the Lagrange transform is the application whichis given in coordinates by :

(2.12) L :

∣∣∣∣∣∣TM → T ∗M

(x, v) 7→ L(x, v) = (x, ∂L∂vi

(x, v)dxi)


It transports a variational problem from the Lagrangian perspective to theHamiltonian perspective. In our particular case, one can check that theLagrange transform is simply the musical isomorphism introduced in theprevious subsection, namely L(x, v) = (x, v[). It can be expressed in localcoordinates by :

(2.13) L(x, v) =(x, gijv

idxj)

And its inverse is given by L−1(x, p) = (x, p]), where ] is the standardnotation for the inverse of [. In coordinates, we have :

(2.14) L−1(x, p) =(x, gijpi

∂

∂xj

)

The natural metric for covectors is given by g−1, which therefore makes themusical isomorphism an isometry between TM and T ∗M . For a covectorξ, we will indistinctly write |ξ|2g or |ξ|2g−1 but these two notations refer tothe same quantity, namely gijξiξj .We now show how to recover the geodesic flow on T ∗M from the Hamil-

tonian point of view. We consider the Hamiltonian H(x, p) = 12 |p|

2x which

is given in local coordinates by :

H(x, p) = 12g

ijpipj

We denote by XH its flow on T ∗M which is such that ω(XH , ·) + dH = 0.Let us consider an integral curve (x(t), p(t)) for the vector field XH . Weknow that it satisfies the Hamilton equations :

dxidt

= ∂H

∂pi,dpidt

= −∂H∂xi

The n first equations can be written in coordinates :

dxidt

=n∑j=1

gijpj ,


that is dxdt

[

= p(t). The n next equations yield to :

dpidt

= −12

n∑k,l=1

∂igklpkpl

= 12

n∑k,l,u,v=1

gku∂iguvgvlpkpl

= 12

n∑u,v=1

∂iguvdxqdt

dxvdt

,

where we used the formula giving the derivative of the coefficients of theinverse matrix according in the second last line. But since dx

dt= p(t)[, we

have :

dpidt

= d

dt

(n∑k=1

gikdxkdt

)

=n∑

k,l=1∂lgik

dxldt

dxkdt

+n∑k=1

gikd2xkdt2

From the two previous formulas, we get :n∑k=1

gikd2xkdt2

= 12

n∑k,l=1

(∂igkl − ∂kgil − ∂lgik) dxkdt

dxldt

By inverting this relation, it is possible to obtain the expression of dx2/dt2 :

d2xsdt2

= −n∑

k,l=1

12

n∑i=1

gsi (∂kgil + ∂lgik − ∂igkl)dxldt

dxkdt

Then, applying Koszul formula (2.5), we recover the Christoffel symbols,and eventually the geodesic equation (2.10) in coordinates. Given (γ(0), γ(0)) =(x0, v0) and (x0, p0) its image by the Lagrange transform, (x(t), p(t)) is no-thing but the image by the Lagrange transform of the geodesic (γ(t), γ(t)).In other words, we have proved the

Proposition 2.1. — The Lagrange transform conjugates the geodesicflow ϕt on TM , generated by X, and the geodesic flow ϕ∗t on T ∗M , gene-rated by XH .

Actually, this is the dual point of view : instead of looking at the tangentvector γ to the geodesic, one looks at the hyperplanes distribution whichare orthogonal to γ.


_γ(t)

ker(p(t))

γ(t)

Figure 2.1. The geodesic flow on TM and T ∗M

2.3. Structure of SM

2.3.1. The horizontal distribution

Let us first state some very general results. We consider a vector bundleπ : E → M of rank r with a connection ∇. Given a path γ : I → M suchthat γ(0) = x, γ(0) = X ∈ TxM and an initial value s0 ∈ Ex = π−1(x),there exists a unique lift of γ to a path s : I → E — it is the paralleltransport of s0 along γ — such that s(0) = s0, π(s) = γ and

(2.15) ∇γs = 0

Now, we can associate to X the vector

(2.16) X = d

dt

∣∣∣∣t=0

s(t) ∈ Ts0E

One can check that X is well defined and only depends on the choice of(γ(0), γ(0)). It is a linear application TxM → Ts0E which we denote byθ : X 7→ X. In local coordinates, if we write X = Xi ∂

∂xi, s0 = sjej , then

the parallel transport equation (2.15) yield to :

(2.17) θ(X) = Xi ∂

∂xi− ΓkijXisj

∂

∂sk

We define the horizontal distribution on (E,∇) at s0 ∈ E as :

(2.18) H∇s0= Span(θ(X), X ∈ Tπ(s0)) ⊂ Ts0E

In the following, we will drop the notation ∇ for Hs0 , but we insist onthe fact that Hs0 is entirely determined by the choice of ∇. Actually, onecan check that choosing a connection ∇ is strictly equivalent to choosinga smooth distribution of vector spaces H ⊂ TE. Note that we have thefollowing theorem :

Theorem 2.2. — The following assertions are equivalent :— The distribution s0 7→ Hs0 is integrable.


— The connection ∇ is flat, i.e. T∇ vanishes.— The holonomy i.e. the parallel transport of a section along a closed

loop is invariant by homotopy of this loop.

Since θ is clearly injective by definition, it defines an isomorphism bet-ween Tπ(s0)

∼−→ Hs0 . We define the vertical distribution on (E,∇) at s0 asVs0 = ker dπs0 = Eπ(s0) ⊂ Ts0E. Thanks to (2.15), it is easy to check thatθ is exactly the inverse of the differential dπs0 restricted to the subspaceHs0 of Ts0E. We therefore have the splitting :

(2.19) Ts0E = Hs0 ⊕ Vs0

Given a vector ξ ∈ Ts0E, which we write in local coordinates ξ =Xi ∂

∂xi+ Y k

∂

∂sk, (2.17) shows that it is in H∇s0

if and only if

(2.20) Y k + ΓkijXisj = 0,

for all 1 6 k 6 r, and it is in Vs0 if and only if Xi = 0 for all 1 6 i 6 n.

2.3.2. The Sasaki metric

We now apply the previous formalism to the particular case when E =TM and construct a canonical metric 〈〈·, ·〉〉 on TM , called the Sasakimetric. If ∇ is the Levi-Civita connection on TM , then it automaticallydetermines a splitting like in (2.19).In this case, there also exists a canonical application K : T(x,v)TM →

TxM whose kernel is exactly H(x,v). Somehow, it can be seen as a com-plementary application to dπ. Given ξ ∈ T(x,v)TM , we can consider alocal curve c in TM such that c(0) = (x, v) and c(0) = ξ. We can writec(t) = (γ(t), Z(t)), where Z is a vector field along the curve γ. We define :

(2.21) K(x,v)(ξ) = ∇γZ(0)

One can easily check that the application K now defines an isomorphismbetween V(x,v) and TxM and its kernel is precisely H(x,v), the horizontaldistribution.

Definition 2.3. — Given a point (x, v) ∈ TM and ξ, η ∈ T(x,v)TM ,we set :— If ξ, η are vertical, namely ξ, η ∈ V(x,v) = TxM , then 〈〈ξ, η〉〉 :=〈K(ξ),K(η)〉,

— If ξ, η are horizontal, then 〈〈ξ, η〉〉 := 〈dπ(x,v)(ξ), dπ(x,v)η〉,— V(x,v) and H(x,v) are orthogonal for 〈〈·, ·〉〉.


Note that the last line induces in particular that the decomposition (2.19)is orthogonal for the Sasaki metric.

In the rest of this paragraph, we explain how to recover the Sasaki metricfrom the symplectic point of view. We recall that λ is the canonical 1-formdefined on T ∗M and introduced in the previous section. It can be pulledback via the musical isomorphism to get α = [∗λ ∈ Ω1(TM). Then

α(x,v)(ξ) = λ(x,v[)(d [(x,v)(ξ))

= v[(dπ(x,v[) d [(x,v)(ξ)

)= v[

(d (π [)(x,v) (ξ)

)= 〈v, dπ(x,v)(ξ)〉= 〈〈Z, ξ〉〉,

for some vector field Z, according to Riesz representation theorem. Weclaim that Z is actually the geodesic vector field X. Indeed, first observethat X is a horizontal vector field (i.e. X(x, v) ∈ H(x,v)), which is animmediate consequence of the expressions in local coordinates (2.11) and(2.20). In particular, in local coordinates, expression (2.17) yield to :

(2.22) dπ(x,v)(X(x, v)) = vi∂

∂xi

In order to prove that Z = X, we just have to check that the expressions〈v, dπ(x,v)(ξ)〉 and 〈〈X, ξ〉〉 agree when ξ runs through a basis of T(x,v)TM .If ξ is in V(x,v) (that is ξ is vertical), this is immediate since dπ(ξ) = 0 and

X is horizontal. If ξ = ξi = ∂

∂xi− Γkijvj

∂

∂vk, then dπ(ξi) = ∂

∂xiby (2.17)

and 〈v, dπ(x,v)(ξ)〉 = vi. But by definition of the Sasaki metric, since bothX and ξi are horizontal :

〈〈X, ξi〉〉 = 〈dπ(X), dπ(ξi)〉 = vi

We therefore obtain the equality :

(2.23) α(x,v)(ξ) = 〈〈X, ξ〉〉

In other words, α = X [, where [ : TTM → T ∗TM is the musical isomor-phism given by the Sasaki metric.

Remark 2.4. — Behind this is hidden the fact that SM (the unit tangentbundle), and therefore S∗M , both have the structure of a contact manifold,but we will not give further details about this.


2.4. Surface theory

2.4.1. Isothermal coordinates

In this paragraph, we introduce isothermal coordinates, which will bewidely used is the following.

Definition 2.5. — Let (M, g) be an n-dimensional Riemannian mani-fold. Isothermal coordinates are local coordinates such that the metric canbe written g = e2λ(dx2

1 + ...+ dx2n), where λ is a smooth function.

In dimension n > 3, in a neighborhood of a point, isothermal coordinatesmay not exist. However, in dimension n = 2, we have the

Theorem 2.6. — Let (M, g) be a Riemannian surface and x ∈ M .There exists isothermal coordinates in a neighborhood of x.

We provide a proof of this theorem in Appendix B based on the resolutionof a Dirichlet problem. The existence of isothermal coordinates on a surfaceis closely link to the existence of a Riemann structure (or holomorphicstructure) on the surface, that is a covering by charts U,ϕ such thatthe transition maps are all holomorphic, as explained in the Appendix. Acomplex structure on M is an endomorphism J ∈ End(TM) such thatJ2 = −id. In particular, the data of a given conformal class together withan orientation of the manifold is equivalent to that of a complex structure.The Koszul formula allows to compute the Christoffel symbols in the

isothermal coordinates. We obtain :

Γ1ij j = 1 j = 2

i = 1 ∂1λ ∂2λ

i = 2 ∂2λ −∂1λ

Γ2ij j = 1 j = 2

i = 1 −∂2λ ∂1λ

i = 2 ∂1λ ∂2λ

Note that in this case, the curvature ofM has a rather simple expression :

(2.24) K = −e−2λ ∆λ

2.4.2. The moving frame

We now assume that M is a surface and ∇ is the Levi-Civita connectionon M , associated to the Riemannian metric g. Instead of studying thetangent bundle TM , we will study the unit tangent bundle SM . Note,


that since we will mostly be interested in studying properties of the geodesicflow, this will not restrict our considerations as far as geodesics are coveredat constant speed (and one can always assume it is arc-length parametrized,up to a preliminary reparametrization).SM can be seen as a subriemannian manifold of TM . In the following,

we may identify the tangent space TxM at a point with the complex planeC and therefore work with complex coordinates. In other words, this simplymeans that we see M as a Riemannian surface, endowed with a complexstructure i, and i simply acts on tangent vectors by π/2-rotating them. Sxwill denote the restriction of TxM to the unit circle. Given a point (x, v) ∈SM , the rotation rθ : Sx → Sx defined in coordinates by rθ(x, v) = (x, eiθv)is a flow on SM which provides an infinitesimal generator V (x, v), spanninga direction of T(x,v)SM . The horizontal distribution H(x,v) is unchangedand since SM is submanifold of TM of dimension 3, one gets the orthogonalsplitting :

T(x,v)SM = Span(V (x, v))⊕H(x,v)

Moreover, thanks to the expression (2.22), one clearly sees that the vectorfield X is unitary on SM . This is also the case for the vector field V . Inorder to obtain a orthonormal basis of T(x,v)SM at a point (x, v) ∈ SM , weneed to provide a third vector field H ∈ H(x,v), unitary and orthogonal toX. It is chosen so that X,H, V is positively oriented. We call this basisthe moving frame on SM .

We can give explicit formulas for this vector fields in isothermal coor-dinates. In these coordinates, in a neighborhood U of a point x ∈ M , wewrite g = e2λ(x1,x2)(dx2

1 + dx22). The local coordinates induced on TM |U

are denoted by (x1, x2, v1, v2). Thus, we can consider local coordinates onSM |U , given by :

(x1, x2, θ) 7→ (x1, x2, e−λ cos θ, e−λ sin θ)

Thanks to the Koszul formula (2.5), the Christoffel symbols can be com-puted explicitly. The expression (2.11) then yield to :(2.25)

X(x, θ) = e−λ(

cos θ ∂

∂x1+ sin θ ∂

∂x2+(− ∂λ

∂x1sin θ + ∂λ

∂x2cos θ

)∂

∂θ

)V (x, θ) is simply given by :

(2.26) V (x, θ) = ∂

∂θ

Eventually,H(x, θ) can be obtained as a normal vector to the plane spannedby the vectors X,V , or as a positive rotation of X in the plane Hx,v of


angle π/2. We get :(2.27)

H(x, θ) = e−λ(− sin θ ∂

∂x1+ cos θ ∂

∂x1−(∂λ

∂x1sin θ + ∂λ

∂x2cos θ

)∂

∂θ

)From these formulas, we can derive the Cartan structural equations. Inparticular, using the expression (2.24) for the curvature, one gets :

(2.28) [X,H] = K · V

(2.29) [V,X] = H

(2.30) [H,V ] = X

X(x; v)

H(x; v)V (x; v)

π−1(fxg)

TxM

M

x v

iv

π

K−1(v)

T(x;v)SM

H(x;v)

0

Figure 2.2. The unit tangent bundle

We can study this basis from the dual point of view. Via the musicalisomorphism defined thanks to the Sasaki metric, we obtain a dual basisα, β, ψ on T ∗SM (note that the 1-form α has already been introducedin the previous section). Let us give a short description of this basis. Givenξ ∈ T(x,v)SM , since dπ(x,v)(X) = v, dπ(x,v)(H) = iv,K(x,v)(V ) = iv, wehave :


— α(x,v)(ξ) = 〈〈X, ξ〉〉 = 〈v, dπ(x,v)(ξ)〉,— β(x,v)(ξ) = 〈〈H, ξ〉〉 = 〈iv, dπ(x,v)(ξ)〉,— ψ(x,v)(ξ) = 〈〈V, ξ〉〉 = 〈iv,K(x,v)(ξ)〉.

The kernel of ψ is precisely the horizontal distribution H. The dual formu-lation (1) of the Cartan structural equations is :

(2.31) α ∧ β = K · dψ

(2.32) ψ ∧ α = dβ

(2.33) β ∧ ψ = dα

Remark 2.7. — For the reader’s convenience, let us just quote here theinversion formulas :

∂

∂x1= eλ (cos θ ·X − sin θ ·H)− ∂λ

∂x2V

∂

∂x2= eλ (sin θ ·X + cos θ ·H) + ∂λ

∂x1V

2.4.3. Canonical volume form on SM

As we mentioned before, the metric g induces a canonical volume formon the surface M which can be written in isothermal coordinates, thanksto formula (2.2) :

(2.34) dvol = e2λdx1 ∧ dx2

We can also define a canonical volume form on SM , denoted by Θ, asthe canonical volume form induced by the Sasaki metric. It is called theLiouville measure on SM . Since the dual basis α, β, ψ is orthonormal forthe Sasaki metric, then one simply has :

(2.35) Θ := α ∧ β ∧ ψ

From the definition (2.35), and the Cartan structural equations, one canprove that the fields X,H and V preserve the volume form Θ. Indeed, letus prove it for X. If LX denotes the Lie derivative with respect to X, thenone has :(LXΘ) (X,H, V ) = LX (Θ(X,H, V ))−Θ([X,X], H, V )

−Θ(X, [X,H], V )−Θ(X,H, [X,V ])

1. Dual in the sense that we apply d [ to the Cartan structural equations on thevector fields.


Since Θ(X,H, V ) = 1, the first term is zero and using Cartan structu-ral equations, the three other terms are zero. So LXΘ is zero, that is, Xpreserves the volume-form Θ.We also have Θ = π∗(dvol)∧dθ. Indeed, since Θ and π∗(dvol)∧dθ are both

volume forms, we know that they may differ by a multiplicative function.But Θ(X,H, V ) = 1 and π∗(dvol)∧dθ(X,H, V ) = dvol(dπ(X), dπ(H)) = 1.Given u ∈ C∞(SM), this yields to the integration formula over the fibers :

(2.36)∫SM

u ·Θ =∫M

(∫SMπ(p)

u|SMπ(p)(p) dθ(p))dvol(π(p))

In particular :

(2.37) vol(SM) = 2π · vol(M)

Remark 2.8. — The volume form Θ can also be seen as the contact formα ∧ (dα)n−1. Here, since n = 2, we simply obtain α ∧ dα = α ∧ β ∧ ψ = Θby the dual formulation of the Cartan structural equation (3.6).

Remark 2.9. — The construction of the Sasaki metric on the tangentbundle and the canonical volume form on the unit tangent bundle is notspecific to dimension 2 and can easily be generalized to greater dimensions.Actually, the unit tangent bundle is locally a product space U × Sn−1 andthe Liouville measure on SM is nothing but the product measure dvol×dSwhere dS denotes the euclidean volume form on Sn−1.


3. Functional spaces on SM

3.1. The space L2(SM)As an introduction, we first recall the

Theorem 3.1 (Stone). — Let H be a Hilbert space and (Ut)t∈R be agroup of unitary operators defined on H and strongly continuous at t = 0.Then, there exists a self-adjoint operator A, densely defined on H calledthe infinitesimal generator of the group, such that :

Ut = eitA

The flows generated respectively by the vector fields X,H and V arevolume-preserving as we saw in the previous paragraph. Therefore, theyact on L2(SM) as unitary operators and by Stone’s theorem, their infini-tesimal generators — the operators −iX,−iH and −iV — are self-adjointoperators densely defined on L2(SM). In particular, their domain of defi-nition contains C∞(SM).

3.1.1. An orthogonal decomposition of L2(SM)We begin with a first proposition :

Proposition 3.2. — The space L2(SM) breaks up into an orthogonalsum of subspaces :

L2(SM) =⊕n∈Z

Hn,

where Hn is the eigenspace of the operator −iV corresponding to the ei-genvalue n.

Démonstration. — Since M is compact and smooth, we can always tri-angulate M into (Mi)16i6N where each Mi is small enough so that it iscontained in a local chart for which there exists isothermal coordinates. Wehave :

L2(SM) =N⊕i=1

L2(SMi)

Now, given aMi and u ∈ L2(SMi), one may use the isothermal coordinatesin order to decompose the function in a Fourier series

u =∑n∈Z

un,

where un ∈ Hn(SMi) is given in the coordinates by :

un(x, θ) =(

12π

∫ 2π

0u(x, t)e−int dt

)einθ = un(x)einθ


And :

Hn =N⊕i=1

Hn(SMi)

Let us insist here on the fact that the function un is defined regardless ofthe system of isothermal coordinates that is chosen. Indeed, consider twosets of coordinates (x, θ) and (y, α) for which the metrics can be respectivelywritten e2λ(dx2

1 + dx22) and e2µ(dy2

1 + dy22) and a transition map ϕ. We

fix x and y = ϕ(x) and compare un(x) with vn(y). Note that, since ϕ isconformal (see Appendix B), dϕx preserves the angles so it is the compositeof a homothetic transformation (of ratio e2(λ−µ)) with a rotation of angle ω.Therefore, the θ coordinate in the first system is sent to α = ψ(θ) = θ+ω.Thus, replacing this in the definition with the integral, we obtain the soughtresult.

Remark 3.3. — In particular, one gets from this decomposition that||u||2L2(SM) =

∑k∈Z ||uk||2L2(SM). We will denote by πk : L2(SM) → Hk

the orthogonal projection and by πk : L2(SM) → L2(M) the map suchthat πk(u) = uk.

Definition 3.4. — We define u+ and u− the respective even and oddparts of u : SM → C by :

u+ :=∑k∈2Z

uk, u− :=∑

k∈1+2Zuk

3.1.2. The Hilbert transform

Definition 3.5. — We say that u : SM → C is holomorphic if uk = 0for all k < 0 and antiholomorphic if uk = 0 for k > 0.

Definition 3.6. — For uk ∈ Hk, we define its Hilbert transform byHuk := −isgn(k)uk.

Observe in particular that u is holomorphic if and only if (Id−iH)u = u0and antiholomorphic if and only if (Id+ iH)u = u0. We have the followingcommutant formula :

Lemma 3.7. —[H, X]u = H · u0 + (H · u)0

We postpone the proof of this Lemma to the paragraph 3.4.4, where theproper tools will be introduced.


The previous relation can even be refined. Indeed, we can define the oddand even parts H− and H+ of the Hilbert transform by considering for asmooth u : SM → C :

H−u = Hu−, H+u = Hu+

In particular, an even (resp. odd) function is transformed into an even(resp. odd) function by the Hilbert transform. The equality of the previouslemma can thus be written :

(3.1) H+X · u−X · H−u = (H · u)0, H−X · u−X · H+u = H · u0

3.2. Symmetric tensors on a manifold

The references for this paragraph are [16], [28] and to a lesser extent [3].Note that throughout this paragraph, the Einstein convention will be usedin order to avoid the summation symbol on i1, ..., im.

3.2.1. Symmetric tensors

A tensor is a section of the fiber bundle Γ(M,⊗mT ∗M). In local coordi-nates (xi), its covariant coordinates are given by :

Ti1,...,im = T

(∂

∂xi1, ...,

∂

∂xim

)By "raising" the indices, one gets the contravariant coordinates, namely :

T i1,...,im = gi1j1 ...gimjmTj1,...,jm

We recall that Γ(M,⊗mS T ∗M) denotes the set of symmetricm-tensors onM , that arem-covariant tensors symmetric in each coordinate. Given anm-tensor T ∈ Γ(M,⊗mT ∗M), there is a natural operation of symmetrizationmaking this tensor a symmetric one :

σ = 1m!

∑π∈Sm

ρπ,

where ρπ acts on an m-tensor by :

ρπT = Ti1...imdxπ(i1) ⊗ ...⊗ dxπ(im)

Γ(M,⊗∗ST ∗M) =⊕

m>0 Γ(M,⊗mS T ∗M) is a Z-graded algebra endowedwith the symmetric product σ(u⊗ v).The set Γ(M,⊗mS T ∗M) is naturally endowed with an L2 scalar product

q. Indeed, given x ∈ M , T ∗xM is endowed with the metric g−1x as we have

seen (which makes the musical isomorphism an isometry between TxM and


T ∗xM) and this extends to ⊗mS T ∗xM by setting for two symmetric m-tensorsu = σ(ξ1 ⊗ ...⊗ ξm), v = σ(η1 ⊗ ...⊗ ηm) (with ξi, ηj ∈ T ∗xM) :

qx(u, v) =∑π∈Sm

g−1x (ξ1, ηπ(1))...g−1

x (ξm, ηπ(m))

Eventually, we define for u, v ∈ Γ(M,⊗mS T ∗M) :

q(u, v) =∫M

qx(u, v)dvol

This scalar product can be extend to the algebra Γ(M,⊗∗ST ∗M) by de-claring the sets Γ(M,⊗mS T ∗M) and Γ(M,⊗m′S T ∗M) to be orthogonal form 6= m′.We define the inner derivative of a symmetric tensor by :

dT := σ∇T,

where σ is the operator of symmetrization. It is a symmetric (m+1)-tensor.Its divergence is the operator defined by :

δT := −tr12(∇T ),

where the trace is taken over the two first coordinates. In local coordinates,one can write :

(δT )i1...im−1 = −∂Ti1...im−1j

∂xkgjk

One can prove that the formal adjoint of d is δ for the L2-product intro-duced previously.We can now state the main theorem of this paragraph :

Theorem 3.8. — Let (M, g) be a compact Riemannian manifold withor without boundary (in this case, we will use the convention ∂M = ∅).Let T be a smooth symmetric m-tensor field. Then there exists a uniquesmooth symmetric m-tensor field fs and a unique smooth (m − 1)-tensorfield v such that :

f = fs + dv, δfs = 0, v|∂M = 0

The last condition is empty if M is boundaryless. We call fs the sole-noidal part of the tensor f and v its potential part. We provide a proof ofthis theorem in the Appendix D, where this result is even proved in theSobolev regularity Hk.


3.2.2. Functions on SM and symmetric tensors

There exists a canonical map

Φm : Γ(M,⊗mS T ∗M)→ C∞(TM),

defined by Φm(T )(x, v) = Tx(v, ..., v). If T is written in coordinates

T = Ti1...imdxi1 ⊗ ...⊗ dxim

then Φm(T ) the m-homogenous polynomial in v given in the same coordi-nates by :

Φm(T )(x, v) = Ti1...im(x)vi1 ...vim ,where vik denotes the ik-th coordinate of the vector v. Most of the time,we will only consider the restriction of this function to the unit tangentbundle SM . Note in particular that Φ0 is just the pull-back of a functionf ∈ C∞(M) via the projection π : SM →M . In the isothermal coordinates,for (x, v) = (x, e−λeiθ) ∈ SM , we have

Φm(T )(x, v) = Ti1...im(x)e−mλ (cos θ)p (sin θ)m−p ,

where p = # k, ik = 1 which, by using Euler formulas, yields to

Φm(T )(x, v) = Ti1...im(x) e−mλ

2mim−p∑06k6p,06l6m−p

(p

k

)(m− pl

)(−1)leiθ(m+2(k−p−l))

We therefore obtain a decomposition T =∑mk=−m Tk, where each Tk ∈ Hk.

Note in particular, according to the previous formula, that ifm is even, thenT = T+ and ifm is odd, then T = T−. Moreover, if T is real, then Tk = T−k.

The orthogonal projection uk of a function u ∈ L2(SM) on Hk canbe identified with a section of the k-th tensor power of the canonicalline bundle κ i.e. κ⊗k (see Appendix E for further details). Namely, ifk > 0, then we consider uk 7→ uke

kλ(dz)⊗k (if k 6 0, we consider uk 7→uke

kλ(dz)⊗(−k)). Then in coordinates if (x, v) = (x, e−λ(cos θ + i sin θ)) ∈SM , we get :

ukekλ(dz)⊗k

((x, e−λ(cos θ + i sin θ))

)= uke

kλe−kλeikθ = uk(x, θ)

Now, we show how to do "the way back", that is how to recover a m-symmetric tensor, given a certain class of smooth function on SM . Let usdenote by Rm the set of smooth, real-valued functions u on SM such thatuk = 0 for |k| > m + 1 and u = u+ (resp. u = u−) if m is even (resp.odd). Consider u ∈ Rm and assume m is even (the same arguments apply


for m odd). In the isothermal coordinates (x, θ), we have uk = ukeikθ and

we define

Uk = 2 · Re(uke

kλ(dz)⊗k)

It is a k-symmetric tensor such that if (x, v) = (x, e−λeiθ) ∈ SM , then :

Uk(x, v, ..., v) = f−k(x, v) + fk(x, v)

We can raise the degree of this tensor by two, by tensoring with the metricg and symmetrizing, namely we define αUk = σ (Uk ⊗ g). Moreover, therestriction of the metric to SM is the constant function 1SM defined onSM , namely Φ2(g) = 1SM . Thus, the restriction of αUk to SM is stilluk + u−k. Now we set

U :=m/2∑k=0

αkUm−2k,

and we have by construction Φm(U) = u. In other words, we have provedthat :

Proposition 3.9. — For each m > 0, Φm : Γ(M,⊗mS T ∗M) → Rmis bijective. In particular, R0 can be identified with the set of smoothfunctions C∞(M).

3.2.3. The inner derivation

The main idea which will be used throughout the rest of this memoireis to transfer the analysis of tensor fields on functions defined on the unittangent bundle via the previous identification. Thus, we need to understandfrom this functional point of view what is the inner derivative of a tensor.We have the

Proposition 3.10. —

(3.2) Φk+1(dT ) = X · Φk(T )


Démonstration. — Let T be a smooth symmetric m-tensor. In local co-ordinates (x1, ..., xn), we can express the coordinates of ∇T :

(∇T )i0...im = ∇T(

∂

∂xi0, ...,

∂

∂xim

)=(∇ ∂

∂xi0T

)(∂

∂xi1, ...,

∂

∂xim

)= ∂

∂xi0·(T

(∂

∂xi1, ...,

∂

∂xim

))−

m∑j=1

T

(∂

∂xi1, ...,∇ ∂

∂xi0

∂

∂xij, ...,

∂

∂xim

)

= Ti1...im∂xi0

−m∑j=1

n∑k=1

Γki0ijTi1...ij−1kij+1...im

Thus, ∇T can be written in coordinates :

∇T = (∇T )i0...imdxi0 ⊗ ...⊗ dxim ,

where the (∇T )i0...im have just been given and the sum is taken over 1 6i0, ..., im 6 n. As a consequence, σ∇T , that is the symmetrization of ∇Tcan be seen via the application Φm+1 as the function (the sum is stillimplicit) :

∂Ti1...im∂xi0

vi0vi1 ...vim −m∑j=1

n∑k=1

Γki0ijTi1...ij−1kij+1...imvi0 ...vim

Now, the first term can easily be factored as(vi0

∂

∂xi0

)·(Ti1...imv

i1 ...vim),

while the second term involves some more efforts in the computation (onehas to rearrange the summation) and can also be factored as(

Γi0ijvivj

∂

∂vi0

)·(Ti1...imv

i1 ...vim)

We recognize thanks to the expression (2.25) the vector field X acting onTi1...imv

i1 ...vim .

3.2.4. Actions on 1-formsEventually, let us end this paragraph with two short computations which

will be used in the sequel. Let A be a smooth real-valued 1-form on M .According to the previous paragraph, we can identify A with the functionΦ1(A) ∈ Ω−1 ⊕ Ω1. Let us denote a = Φ1(A). Then we have the :


Lemma 3.11. —Φ0(δA) = η+a−1 + η−a1

In particular, a 1-form is solenoidal if and only if η+a−1 + η−a1 = 0.Démonstration. — In local coordinates, using the Christoffel symbols for

the isothermal coordinates, one can see that (the other terms vanish) :

δA = −(∂A1

∂x1+ ∂A2

∂x2

)Applying the expressions given in Remark 2.7 linking the ∂/∂xi to themoving frame (and thus to η±) and the fact that

a1 = e−2λA1 − iA2

2

a−1 = e−2λ A1 + iA2

2 ,

we obtain the sought result.

Lemma 3.12. —Φ1(?db) = H · Φ0(b)

Démonstration. — A computation in local coordinates using expression(2.27) for H provides the result.

3.3. The Riccati equation

The reference for this paragraph is [22].

3.3.1. The Jacobi vector fields

Let γ : [a, b]→M be a geodesic joining γ(a) and γ(b). The Jacobi vectorfields are the vector fields along γ for which the hessian of the energyfunctional is degenerate. The normal Jacobi vector fields are those amongthe Jacobi vector fields which are orthogonal to γ. They are the vectorfields along γ for which the hessian of the length functional is degenerate.Standard computations allow to prove that Jacobi vector fields satisfy theJacobi equation along γ

Y ′′ +R(Y, γ)Y = 0,

where R denotes the curvature tensor and with initial conditions Y (a) =v, Y ′(a) = w. The normal Jacobi vector fields are obtained for the sameequations but with initial values v, w⊥γ(a). The vector space of Jacobi(resp. normal Jacobi) vector fields along γ is a 2n-dimensional (resp. 2(n−


1)-dimensional) vector space : this comes from the Cauchy theorem, andthe fact that a Jacobi field is completely determined by its initial valuesv and w. Note that the most interesting Jacobi fields are those for whichv = 0 (which geometrically means that the ends in the variation formulaof the hessian of the energy functional are fixed).The Jacobi vector fields can be used in order to compute the differen-

tial of the geodesic flow ϕt. Indeed, given (x, v) ∈ TM , if ξ ∈ T(x,v)TM ,and Y denotes the Jacobi vector field with initial conditions Jξ(0) =dπ(x,v)(ξ), Jξ(0) = K(x,v)(ξ), then :

(3.3) d(ϕt)(x,v)(ξ) = (dπϕt(x,v))−1(Y (t)) + (Kϕt(x,v))−1(Y (t))

In the two-dimensional case, the normal Jacobi equation reduces to ascalar equation, namely :

f ′′ +Kf = 0,with f(a) = λ, f ′(a) = µ. We say that a point q = γ(t0) is conjugate top = γ(a) if there exists a normal Jacobi field along γ such that f(t0) = 0and f(a) = 0. In particular, if (x1, v1) = ϕt(x0, v0) are conjugate, ta-king ξ = V (x0, v0) in the previous equation (3.3) (which satisfies Y (0) =dπ(x,v)(V ) = 0), we obtain

d(ϕt)(x0,v0)(V ) = (Kϕt(x,v))−1(Y (t)) ∈ V(x1,v1) = Span(V (x1, v1))

3.3.2. The Riccati equationSetting u = f ′/f , we obtain the equivalent equation along γ,

u′ + u2 +K = 0

which can be rewritten as the Riccati equation on SM :

(3.4) X · u+ u2 +K = 0

We will admit the following theorems. They will be used in the next para-graph in order to provide an α-control.

Theorem 3.13 (Hopf). — If (M, g) is free of conjugate points, thenthere exists a bounded measurable function u : SM → R, differentiable inthe direction X, solution to the Riccati equation (3.4).

For a proof of Hopf’s theorem, we refer to his short original article [17].It is possible to better Hopf’s theorem for Anosov manifolds :

Theorem 3.14. — If (M, g) is Anosov, then there exists two boundedsolutions r+, r− of class C1 to the Ricatti equation (3.4), which are everyw-here distinct (r+− r− > 0 actually). Moreover, −H(x, v) + r+(x, v)V (x, v)(resp. −H + r−V ) spans the bundle Es(x, v) (resp. Eu(x, v)).


Remark 3.15. — Actually, the regularity can be bettered up to C2−ε

for any ε > 0, as established by Katok-Hurder in [18] because we are indimension 2.

We will also omit the proof of this theorem. We refer to [20], whereit is proved for an even wider class of flows, that of Anosov λ-geodesicflows. There are various implications of this theorem. Let us just quote twoof them. The first one is that there cannot exist conjugate points on anAnosov manifold because there exists bounded (measureble and C1(SM))solutions to the Riccati equation (and therefore non-vanishing solutions tothe initial Jacobi equations). The second implication is rather surprising.This last theorem actually allows to prove that the sphere and the toruscannot carry an Anosov flow, in an elegant fashion. Indeed, assume this isthe case, then we have the existence of such functions r±. Take r = r+ orr− and integrate the Riccati equation over the unit tangent bundle SM :∫

SM

X · r +∫SM

r2 +∫SM

K = 0

The first term vanish since the Liouville form on SM is preserved by thegeodesic flow X. Since K is constant over the fibers, the formula of inte-gration over the fibers (C.1) gives :∫

SM

K = 2π∫M

K = 4π2χ(M),

by the Gauss-Bonnet formula. Thus :∫SM

r2 = −4π2χ(M)

Since χ(S2) = 2, we obtain a contradiction. For the torus, χ(T2) = 0, sowe obtain r = 0, that is r+ = r− = 0, which is absurd too.

3.3.3. α-controlled surfaces

Let us conclude this section by a less common result on surfaces. It will beused in Section 6. Given α ∈ [0, 1], we will say that a surface is α-controlledif for any ψ ∈ C∞(SM) such that ψ|∂(SM) = 0, one has :

||X · ψ||2 − (Kψ,ψ) > α||X · ψ||2

Theorem 3.16. — A surface (M, g) which is free of conjugate points is0-controlled. In other words, for any ψ ∈ C∞(SM), such that ψ|∂(SM) = 0,

(3.5) ||X · ψ||2 − (Kψ,ψ) > 0


Démonstration. — Consider a solution u : SM → R to the Riccati equa-tion given by Hopf’s Theorem 3.13. For ψ ∈ C∞(SM), we obtain :

|X·ψ−uψ|2 = |X·ψ|2−2Re(uψX·ψ)+u2|ψ|2 = |X·ψ|2+|ψ|2(X·u+u2)−X·(u|ψ|2)

Using the invariance of the Liouville volume form Θ on SM by the geodesicflow X and the fact that ψ vanishes on ∂(SM) = 0, we obtain :

||X · ψ − uψ||2 = ||X · ψ||2 + (X · u+ u2, |ψ|2) = ||X · ψ||2 − (Kψ,ψ),

where we used the Riccati equation in the last equality.

Remark 3.17. — Actually, one can prove that the equality in (3.5) holdsif and only if ψ = 0. Indeed, in this case, using the previous equalities, weobtain X · ψ = uψ for u = r±, so (r+ − r−)ψ = 0 and ψ = 0.

For an Anosov surface, one can even prove a better result, namely :

Theorem 3.18. — Assume (M, g) is Anosov. Then, there exists α > 0,such that for all ψ ∈ C∞(SM),

(3.6) ||X · ψ||2 − (Kψ,ψ) > α(||X · ψ||2 + ||ψ||2

)Démonstration. — Consider the solutions r± to the Riccati equation

(3.4). We know that they are bounded and C1, r+−r− > 0 and the manifoldis compact, so there exists constants C,D > 0, such that C 6 r+−r− 6 D.We define A = X ·ψ− r−ψ,B = X ·ψ− r+ψ. Thanks to our previous com-putations, we know that ||A||2 = ||B||2 = ||X · ψ||2 − (Kψ,ψ). Moreover,we have :

ψ = 1r+ − r−

(A−B)

X · ψ = λA+ (1− λ)B,

with λ = r+/(r+ − r−). From these equations, we obtain the constant αsuch that :

2α||ψ||2 6 ||A||2

2α||X · ψ||2 6 ||A||2,

which gives the sought result.

3.4. A technical toolbox

The following results can be found in [14] and [25].


3.4.1. The raising and lowering operators η+ and η−

Definition 3.19. — We define the following operators on C∞(SM) :

(3.7) η+ = 12(X − iH)

(3.8) η− = 12(X + iH)

An immediate computation shows that the Cartan structural equationscan be rewritten as :

(3.9) [−iV, η+] = η+

(3.10) [−iV, η−] = −η−

(3.11) [η+, η−] = iK

2 V

We set Ωk = Hk ∩C∞(SM). In the rest of this paragraph, we detail someuseful results concerning these operators.

Proposition 3.20. — η± : Ωk → Ωk±1 and the formal adjoint of η+ is−η−.

Démonstration. — Let u ∈ Ωk. Then, we have by (3.9) :

(−iV )η+u = [−iV, η+]u+ η+(−iV )u = η+u+ kη+u = (k + 1)η+u

The same computation can be applied to η−, using (3.10). The computationof the adjoint comes from the fact that −iX and −iH are self-adjointoperators on a dense set of L2(SM) containing C∞(SM).

We now assume that the surface (M, g) is topologically equivalent to atorus with g > 2 holes. In particular, this is the case if (M, g) has negativecurvature, or if (M, g) is Anosov, that is the geodesic flow is Anosov (seeTheorem C.4 in the Appendix). Then we have the following result :

Proposition 3.21. — — For k 6 −2, η+ is surjective and dim ker (η+) =(−2k − 1)(g − 1).

— For k = −1, dim ker (η+) = g.— For k = 0, dim ker (η+) = 1.— For k > 1, η+ : Ωk → Ωk+1 is injective.

Proposition 3.22. — — For k 6 −1, η− : Ωk → Ωk−1 is injective.— For k = 0, dim ker (η−) = 1.— For k = 1, dim ker (η−) = g.— For k > 2, η− is surjective and dim ker (η−) = (2k − 1)(g − 1).


Démonstration. — Thanks to the explicit expression of the vector fieldsX and H in isothermal coordinates (x, θ), it is possible to compute expli-citly η±u for u ∈ Ωk. Indeed, if u(x, y, θ) = h(x, y)eikθ in local isothermalcoordinates, then one has

(3.12) η−(u) = e−(k+1)λ∂(hekλ)ei(k−1)θ,

(3.13) η+(u) = e(k−1)λ∂(he−kλ)ei(k+1)θ,

where ∂ = 12( ∂∂x− i ∂

∂y) and ∂ = 1

2( ∂∂x

+ i∂

∂y).

Therefore, the operators η± are almost the operators ∂ and ∂, whichhave been well-studied in literature so far. For instance, as far as η− isconcerned, its surjectivity and/or injectivity is equivalent to that of ∂. Thisstems from the Riemann-Roch theorem (see Appendix E or [29] (Chapter10) for instance).

Proposition 3.23. — η+ : Ωk → Ωk+1 and η− : Ωk → Ωk−1 are firstorder elliptic operators.

Démonstration. — Let us use the expression (3.12). Ωk can be seen as aline bundle over M and we can write :

η−(heikθ) = (e−λ∂h+ ah)ei(k−1)θ,

for some smooth functions a. Now, the symbol of ∂ is given by −i(ξx+ iξy)so the principal symbol of η− is e−λ(−iξx + ξy) 6= 0, as long as ξ 6= 0.

Corollary 3.24. — Ωk = η+(Ωk−1)⊕ker (η−), and the decompositionis orthogonal.

Corollary 3.25. — If f is smooth and η+u = f (resp. η−u = f), thenu is smooth.

3.4.2. Max Noether’s theorem

Let us end this section by a result of Max Noether which appears inthe proof of the injectivity of the ray transform on Anosov manifold. Wewill not give a proof of this theorem and refer to [6] for further details. If(M, g) is a smooth oriented surface, we know that it admits an underlyingholomorphic structure, making it a Riemann surface (see Appendix B forinstance). We say that M is hyperelliptic if there exists a holomorphic mapf : M → S2 of degree two.


Theorem 3.26 (Noether). — On a non-hyperelliptic surface of genusg > 2, the kernel of ∂ : κ⊗m → κ⊗m⊗ κ is spanned by the m-fold productsof the abelian differentials of the first kind.

In the caseM is hyperelliptic, one can use the fact the a closed Riemannsurface of genus g > 2 admits Galois covers of arbitrary degree. Given aninteger n > 1, there exists a Galois cover π : N →M of degree n, where Nis a Riemann surface with genus n(g − 1) + 1. If M is hyperelliptic, thenone can show that N will be hyperelliptic only if n = 2 or 4 (see [19]), soby taking n > 5, we can ensure that N is not hyperelliptic. This trick isused once in the memoire.A consequence of Max Noether’s theorem is the following. Assume that

M is a non-hyperelliptic surface. Then, according to Theorem 3.26, weknow that the k-fold products of abelian differentials of the first kind spanthe space of holomorphic k-differentials. In other words, writing this interms of smooth functions on SM , if η−u = 0 and u ∈ Ωk, then thereexists a(j)

i ∈ Ω1 such that η−a(j)i = 0 and :

u =N∑i=1

a(1)i ...a

(k)i

This fact will be crucial in Section 6.

3.4.3. The Pestov identity

In this paragraph, the scalar product considered is the canonical scalarproduct in L2(SM).

Proposition 3.27. — Let u ∈ C∞(SM). Then :

(3.14) ||XV u||2 − (KV u, V u) + ||Xu||2 − ||V Xu||2 = 0

Démonstration. — We denote by P the partial differential operator onSM defined as :

P := V X

The formal adjoint P ∗ of P is XV . We can compute the bracket [P ∗, P ]thanks to the Cartan structural equations :

[P ∗, P ] = XV V X − V XXV= ([X,V ]− V X)V X − V X ([X,V ]− V X)= V XV X +HVX − V XV X − V XH

= V [H,X]−X2

= −X2 + V KV


Now, we have for u ∈ C∞(SM)

||Pu||2 = (Pu, Pu) = (u, P ∗Pu) = (u, [P ∗, P ]u) + ||P ∗u||2,

which is the identity sought when replacing the operator P by the operatorsV and X.

Remark 3.28. — In particular, if u ∈ Hk, then :

(k2 + 1)||Xu||2 − k2(Ku, u)− ||V Xu||2 = 0

Using X = η+ + η−, we obtain :

(3.15) − k

2 (Ku, u) + ||η−u||2 = ||η+u||2

3.4.4. Proof of Lemma 3.7

Démonstration. — Note that it is sufficient to check the formula for u ∈Ωk. If |k| > 1, then the right member clearly vanishes and it is easy tocheck that X and H commute. If k = 1 (the case k = −1 being analogous),H · u0 vanishes, X · u ∈ Ω0 ⊕ Ω2 and HX · u = −i(X · u − (X · u)0). Onthe other hand, XHu = −iX ·u. Thus [H, X]u = i(X ·u)0, which coincideswith (H · u)0 since X = η+ + η−, H = −i(η+ − η−). The case k = 0 is alsoimmediate.


4. The X-ray transform

4.1. Definition of the X-ray transform

4.1.1. A first definition

Let us first give the general idea behind the X-ray transform, beforedetailing it according to the nature of the manifold involved. Let (M, g)be a n-dimensional Riemannian manifold with or without boundary. Wedenote by G the set of its closed unit-speed geodesics. In the boundarylesscase, this simply means periodic geodesic while in the case with boundary,this means geodesics which exit the manifold in finite time. Let T be asmooth symmetric m-tensor. In the following, we will still denote by T

the smooth function Φm(T ) ∈ C∞(SM), canonically associated to T , asexplained in Section 3.2.2. We define the ray transform Im for γ ∈ G by :

ImT (γ) =∫ T

0Tγ(t)(γ(t))dt,

where L denotes the length of the geodesic. Of course, this definition canbe extended to the case of infinite time geodesics like trapped geodesic inthe case with boundary (geodesics which do not exit the manifold) as longas the tensor integrated satisfies some good integration properties, but wewill not go that much into this issue.

4.1.2. The notion of s-injectivity

The fundamental question we want to answer is : what properties onf (or T ) can we recover from the knowledge of its ray transform If (orImT ) ? For instance, if ImT = 0, that is ImT (γ) = 0 for any γ ∈ G, canwe prove that T = 0 ? The answer to this question is usually negative.If T is a symmetric (m − 1)-tensor, we know, according to (3.2), that itsinner derivative dT = σ∇T is given, in terms of functions on SM by X ·T .Moreover, on a manifold with boundary, if we assume that T |∂M = 0, thenit becomes clear that Im(dT ) = 0 by the fundamental theorem of Analysis.Recall that, according to Theorem 3.8, one can decompose any smooth

symmetric m-tensor T in T = T s + dh, where T s is a symmetric m-tensorwith zero divergence and h is an (m − 1)-tensor which is zero on ∂M (inthe case M admits a boundary). We refer to Appendix D for a proof ofthis result. The part T s is called the solenoidal part of the tensor T , whilethe part dh is called the potential part. We have just seen that we cannotexpect to recover the potential part of a tensor from the knowledge of itsray transform.


Definition 4.1. — We will say that the ray transform Im is s-injective,if it is injective on the set of solenoidal m-tensors. In other words, it its-injective if for any symmetric m-tensor T , ImT = 0 implies T = dh,for some (m − 1)-tensor h such that h|∂M = 0. In the case of functions,that is m = 0, the s-injectivity reduces to the injectivity, that is for anyf ∈ C∞(M), I0f = 0 implies f = 0.

Remark 4.2. — In the rest of this memoire, I will denote the ray trans-form acting on smooth functions in C∞(SM) (or, when explicitly detailedin L2(SM)). We will use the index Im to insist on the fact that we referto the ray transform acting on symmetric m-tensors or, equivalently, onsmooth functions in the space Rm.

4.1.3. The Livcic property

Given x ∈ ∂M , we define the second fundamental form :

Sx :∣∣∣∣ Tx(∂M)× Tx(∂M)→ R

(v, w) 7→ g(∇vν, w)We say that the manifold is strictly convex if Sx is definite positive for anyx ∈ ∂M . A manifold is said to be simple if it is simply connected, doesnot possess any conjugate points and is strictly convex. One can prove thatsuch a manifold is diffeomorphic to a ball of Rn and the exponential mapis a diffeomorphism in each point of the manifold. Indeed, the absence ofconjugate points imply that the exponential map is a local diffeomorphism(this can be seen using equation (3.3) where the differential of the expo-nential is computed explicitly in terms of the Jacobi vector fields). SinceMis complete (by the Hopf-Rinow theorem, because it is compact), the ex-ponential map is surjective and becomes in each point a covering map andsince M is simply connected, it is a diffeomorphism. Thus, given x ∈

M ,

there exists a closed compact set Kx ⊂ TxM such that expx : Kx →M is adiffeomorphism. NowKx is diffeomorphic to a ball because it is star-shapedin 0 (the geodesics in Kx are sent on "rays" or straight lines) and ∂Kx isa smooth graph over a sphere εSx for some ε > 0 (where Sx denotes theunit tangent sphere in x) because of the strictly convex boundary condi-tion (see Section 4.4.1). In particular, one obtains that such a manifold isnon-trapping because any geodesic starting from x will hit the boundaryin finite time. Note that the converse is also true : if the surface is non-trapping, then the π1(M) has to be trivial, otherwise there would exist aperiodic geodesic (see [8], Section 2.98 for instance).The two cases that we shall consider are :


— compact closed Anosov manifolds (manifolds for which the geodesicflow is Anosov on the unit tangent bundle),

— compact and simple manifolds.The interest of these two sets of manifolds lies in the fact that they both sa-tisfy the Livcic property : if the integral of a smooth function f ∈ C∞(SM)is zero along every closed integral curve of the geodesic field X, then thenthere exists a function u ∈ C∞(SM) such that f = X · u. In the case of acompact and simple manifold, this means that the integral of f is zero alongevery geodesic (in particular, all the geodesics have their extremal pointson the boundary of M) and the function u has to satisfy u|∂M = 0 by thefundamental theorem of Analysis. In the case of a compact closed Anosovmanifold, this means that the integral of f is zero along every periodicgeodesic.Let us end this paragraph with an important remark. We assume that

the manifold (M, g) satisfies the smooth Livcic property. Thus, we knowthat if ImT = 0, there exists a smooth function h such that T = X · h.Now, assume we can prove that u ∈ Rm−1, then going back to the tensorsvia the application Φ, this exactly means that there exists a smooth tensorh such that T = dh, and Im is s-injective. As a consequence, on a manifoldsatisfying the Livcic property, the injectivity of Im actually "reduces" toproving that if ImT = 0, then there exists a smooth function u ∈ Rm−1such that T = X · u.

4.2. Definition on a compact manifold with boundary

4.2.1. Another definition of the X-ray transform

Let (M, g) be a n-dimensional manifold with boundary. We denote byν the unit outer normal to ∂M . The unit tangent bundle SM of M is a(2n− 1)-dimensional manifold with boundary. We denote by π : SM →M

the projection. ∂(SM) is therefore a (2n− 2)-dimensional manifold whichwe can write ∂(SM) = ∂+(SM) ∪ ∂−(SM) where :

∂±(SM) = (x, v) ∈ ∂(SM),∓g(ν(x), v) > 0

We also define :

∂0(SM) = (x, v) ∈ ∂(SM), g(ν(x), v) = 0

For (x, v) ∈ SM , we denote by ϕt(x, v) the geodesic flow starting from thepoint x in the direction v.


It is always possible to endow the manifold (M, g) in a boundarylessmanifold (N,h) such that if i denotes the inclusion, then i∗h = g. Wedefine the travel time τ : SM → [0,∞] by :

τ(x, v) = inft > 0, π(φt(x, v)) ∈ N \M

Of course, this definition does not depend on the embedding chosen. Clearly,this time can be infinite. We say that the manifold M is non-trapping ifτ(x, v) < ∞ for any point (x, v) ∈ SM . In the following, we will assumeM to be non-trapping.Given a smooth function f defined on M , we define its ray transform

by :

(4.1) I0 :

∣∣∣∣∣ C∞(M)→ C∞(∂+(SM))f 7→

((x, v) 7→ I0f(x, v) =

∫ τ(x,v)0 f(π(ϕt(x, v))) dt

)It is the integral of f taken along the geodesic curves. Just like in theprevious paragraph, it can be easily generalized to any symmetric m-tensorT on M . Indeed, let us still denote by T the smooth function induced bythe tensor on SM (it is Φm(T ) in our previous notations). Then we definethe ray transform Im of the tensor by :

∀(x, v) ∈ ∂+(SM), ImT (x, v) =∫ τ(x,v)

0T (ϕt(x, v)) dt

4.2.2. The map IFrom now on, we assume that the manifold is simple. We will discuss

these assumptions in a next paragraph. In particular, we insist on the factthat such a manifold is necessarily non-trapping.

The natural measure Θ on ∂(SM) is the restriction of the Liouville mea-sure Θ on SM to the boundary. On ∂+(SM), we consider the measuredµ(x, v) = 〈ν(x), v〉Θ(x,v). The ray transform can naturally be extended toa bounded operator I : L2(SM)→ L2(∂+(SM), µ).

Proposition 4.3. — I : L2(SM)→ L2(∂+(SM), µ) is bounded

The proof of this proposition relies on Santaló’s formula, which traduces,in terms of measure, a disintegration of the Liouville measure Θ on SM

along the flow lines of the geodesic vector field X :

Lemma 4.4. — Let (M, g) be a manifold with boundary such that theset of trapped points has zero measure and h ∈ L1(SM). Then :∫

SM

h ·Θ =∫∂+(SM)

(∫ τ(x,v)

0h(ϕt(x, v))dt

)dµ


Proof of Lemma 4.4. — It is sufficient to prove the result for h ∈C∞c (

SM) and a density argument allows to conclude. Given (x, v) ∈ SM ,

we have∫ τ(ϕt(x,v))

0h(ϕs(ϕt(x, v)))ds =

∫ τ(x,v)−t

0h(ϕs+t(x, v))ds =

∫ τ

t

(x, v)h(ϕs(x, v))ds,

so we obtainX·∫ τ(x,v)

0 h(ϕs(x, v))ds = −h(ϕt(x, v)). Thus, applying Green’sformula, using the fact that X preserves the Liouville measure :∫

SM

h ·Θ = −∫SM

X ·

(∫ τ(x,v)

0h(ϕs(x, v))ds

)·Θ

=∫∂SM

(∫ τ(x,v)

0h(ϕs(x, v))ds

)〈X, ν〉 · Θ

It is immediate that the integral vanishes on (x, v) ∈ ∂(SM), 〈v, ν(x)〉 > 0,which gives the sought result.

Proof of Proposition 4.3. — Since M is strictly convex, we know (seeParagraph 4.4.1) that the travel time τ : ∂+(SM) → [0,∞) is continuousand the manifold is non-trapping, so there exists a maximum travel timeT = sup τ .

||If ||2L2(∂+(SM),µ) =∫∂+(SM)

(∫ τ(x,v)

0f(ϕt(x, v))dt

)2

dµ

6∫∂+(SM)

T

(∫ τ(x,v)

0f(ϕt(x, v))2dt

)dµ

= T

∫SM

f2 ·Θ,

where we used Santalo’s formula in the last equality. Note in particularthat I is bounded by the square root of the maximum travel time.

4.2.3. The adjoint map I∗0

The natural adjoint of I0 : L2(M) → L2(∂+(SM), µ) is an operatorI∗0 : L2(∂+(SM), µ)→ L2(M). For h ∈ C∞(∂+(SM)), we define hψ(x, v) =h(ϕ−τ(x,−v)(x, v)).


(x; v)

'τ(x;v)(x; v)

'−τ(x;−v)(x; v)

Figure 4.1. The travel time

Proposition 4.5. — The adjoint of I0 for the natural L2 inner productsis the operator

I∗0 :

∣∣∣∣∣ L2(∂+(SM), µ)→ L2(M)

h 7→(x 7→ I∗0h(x) =

∫Sxhψ(x, v)dv

),

where Sx is the sphere bundle in x and dv is the volume form on Sx, i.e. therestriction of the Liouville form on Sx (which is also the euclidean volumeform on the sphere).

Démonstration. — Consider f ∈ C∞(M), w ∈ C∞(∂+(SM)). Note thatfor (x, v) ∈ ∂+(SM), we have w(x, v) = wψ(ϕt(x, v)), for any t ∈ [0, τ(x, v)].Then, we have :

(I0f, w)L2(∂+(SM),µ) =∫∂+(SM)

(∫ τ(x,v)

0f(ϕt(x, v))dt

)w(x, v)dµ

=∫∂+(SM)

(∫ τ(x,v)

0f(ϕt(x, v))wψ(ϕt(x, v))dt

)dµ

=∫SM

fwψ ·Θ

=∫M

f(x)(∫

Sx

wψ(x, v)dv)dvolg,


where we applied Santalo’s formula in the second last equality and theformula of integration over the fibers (C.1) in the last one.

Remark 4.6. — Note that for h ∈ C∞(SM), h0(x) = 12π∫Sxh(x, v)dv.

Here, we obtain :I∗0h(x) = 2π(hψ)0(x)

We are mostly interested in smooth functions on the unit sphere bundle(which will be seen for us as the associated function to a smooth tensor). Aproblem may come from the fact that it is not clear that I0 : C∞(SM) →C∞(∂+(SM)), because of the non-smoothness of τ (see below). Moreover, itis also rather uncertain that I∗0 : C∞(∂+(SM))→ C∞(SM), all the more sosince this is actually wrong. Indeed, for w ∈ C∞(∂+(SM)), the smoothnessof wψ is only guaranteed on SM \ ∂0(SM) and thus the smoothness of

I∗0w(x) =∫Sx

wψ(x)dv

is not guaranteed on ∂M . In order to avoid problems of regularity, weconsider C∞α (∂+(SM)), the set of functions h ∈ C∞(∂+(SM)) such that hψis smooth on SM . We will only study the restriction of I∗0 : C∞α (∂+(SM))→C∞(M). In the following, we will see that some proofs of injectivity for theX-ray transform rely on the surjectivity of this operator.

4.2.4. The operator I∗0 I0

The reference for this paragraph is [26], where it was proved for the firsttime. We also refer to Appendix A for a short review on some elementaryfacts about pseudodifferential operators. Note that this paragraph is notspecific to dimension 2, and we consider a dimension n > 2, even thoughit will only be applied in the two-dimensional case in the following. In thisparagraph, we are going to prove the

Theorem 4.7. — The operator I∗0 I0 is an elliptic pseudodifferentialoperator of order −1. Its principal symbol is given by σ(I∗0 I0) = c|ξ|−1

g−1 .

It will be used in order to obtain the surjectivity of I∗0 in a further section.Let us discuss a bit about this operator. We have :

(I∗0 I0f)(x) =∫Sx

(∫ τ(x,v)

−τ(x,−v)f(ϕt(x, v))dt

)dv

= 2∫Sx

(∫ τ(x,v)

0f(ϕt(x, v))dt

)dv


Note that for a geodesic, we have ϕt(x, v) = ϕ1(x, tv) = expx(tv). Sincethe manifold is simple, expx is a diffeomorphism onto M , i.e. there existsa closed ball Bx such that expx : Bx →M is a diffeomorphism. We changevariables in the previous formula. Setting Φ : (t, v) 7→ expx(tv) = y, wehave |det dΦ(t,v)| = tn−1|det d(expx)tv|. Note that t = d(x, y) (where thedistance is computed with the Riemannian metric). The change of variablein the integral gives :

(I∗0 I0f)(x) =∫M

K(x, y)f(y)dvolg(y),

with the kernel :K(x, y) = 2 |det d(exp−1

x )y|dn−1(x, y)

This is clearly a smooth function outside the diagonal. Since d(expx)0 = id,we have |det d(exp−1

x )0| = 1 so the singularity of the kernel around thediagonal is contained in the term 1/dn−1(x, y).We recall that the manifold is smooth, as well as the metric. We have

the following

Lemma 4.8. — In coordinates, there exists smooth functions Gij(x, y)such that Gij(x, x) = gij(x) and :

d2(x, y) = Gij(x, y)(x− y)i(x− y)j

Démonstration. — We fix x and define f(y) = d2(x, y). We can alwaysassume that the neighborhood considered is smaller than the radius ofinjectivity of the manifold. Given y, we can thus write y = expx(v(y)),where v(y) = (expx)−1(y) ∈ TxM and f(y) = ||v(y)||2x. We denote by ϕitthe flow generated by the ∂/∂xi. Thus :

∂f

∂xi(x) = d

dt

(f(ϕit(x))

)∣∣∣∣t=0

= 2〈dv0(∂/∂xi), v(x)〉x = 0,

since v(x) = 0. So x is a critical point and we can compute the hessian,using that dv0 = id :

Hessxf(

∂

∂xi,∂

∂xi

)= 2||∂/∂xi||2x = 2gii(x)

By polarizing the previous identity and developing f up to the second order,we obtain :

f(y) = gij(x)(x−y)i(x−y)j+O(||x−y||3) = gij(x)(1+O(||x−y||))(x−y)i(x−y)j

This provides the sought result with Gij(x, y) = gij(x)(1+O(||x−y||)). This will allow to compute the principal symbol of I∗0 I0 :


Démonstration. — Thanks to the previous lemma, we can therefore writein coordinates, in a vicinity of a point x :

K(x, y) = 2 |det d(exp−1x )y|

(Gij(x, y)(x− y)i(x− y)j)(n−1)/2

Consider a covering of M by a finite number of charts (Uα, ϕα) and apartition of unity

∑χα = 1 related to this covering. Each χα has compact

support in Uα. We take χ′α with compact support in Uα such that χ′α = 1on supp(χα). Consider a smooth function u on M . We write everything incoordinates in ϕα(Uα) ⊂ Rn (and we drop the index α) :

I∗0 I0(χu)(x) =∫K(x, y)χ′(y)χ(y)u(y)dvolg(y)

=∫U

K(x, y)χ′(y)(∫

Rneiy·ξuχ(ξ)dξ

)√det g(y)dy

=∫Rneix·ξ

(∫U

ei(y−x)·ξK(x, y)χ′(y)√

det g(y)dy)uχ(ξ)dξ

=∫Rneix·ξp(x, ξ)uχ(ξ)dξ,

where · denotes the usual scalar product on Rn, with

p(x, ξ) =∫Rne−iz·ξK(x, x−z)χ′(x−z)

√det g(x− z)dz =

∫Rne−iz·ξF (x, z)dz,

which can also be written

F (x, z) = 1(2π)n

∫Rnp(x, ξ)eiz·ξdξ

Now, F is a smooth (outside the diagonal) function and F (x, ·) ∈ L1(Rn)since it is compactly supported and we have established that F (x, z) ∼z→02|z|−(n−1). Let us prove that F ∈ S−(n−1)(Rnx×Rnz ) in the sense that (here,we are interested in the behavior as z ∼ 0)

(4.2) ∀α, β, ∀z 6= 0, |∂βx∂γzF (x, z)| 6 Cβγ |z|−n+1−|γ|,

and then by [29] (Proposition 2.7, page 9), we will be able to conclude thatp = F2(F ) ∈ S−1(Rnx ×Rnξ ) (where the Fourier transform is taken over thesecond variable) (2) .

2. Let us just recall that this stems from the fact that if H]k

(Rn) denotes the set ofsmooth homogeneous functions on Rn \ 0, it is well known that the Fourier transformacts as F : H]

m(Rn)→ H]−m−n(Rn). This result can actually be extended somehow to

symbols.


We can write :

F (x, z) =2|det d(exp−1

x )x−z|χ′(x− z)√

det g(x− z)(Gij(x, x− z)zizj)(n−1)/2

=2|det d(exp−1

x )x−z|χ(x− z)(n+1)/2√

det g(x− z)(χ′(x− z)Gij(x, x− z)zizj)(n−1)/2

= λ(x, z)/θ(x, z)

Since λ is smooth and compactly supported, all its derivatives are boundedso the problem reduces to proving (4.2) for 1/θ instead of F , that is :

∀α, β, ∀z 6= 0,∣∣∣∣∂βx∂γz ( 1

θ(x, z)

)∣∣∣∣ 6 Cβγ |z|−n+1−|γ|

One way of proving this estimate is to use the Leibnitz formula in aninduction. Actually, the whole proof starts from the observation that θsatisfies the estimates

∀α, β, ∀z 6= 0,∣∣∂βx∂γz (θ(x, z))

∣∣ 6 Cβγ |z|n−1−|γ|

Thus, using ∂βx∂γz (θ/θ) = 0 :

θ · ∂βx∂γz (1/θ) =∑

β′+β′′=β,γ′+γ′′=γ,|β′|+|γ′|<|β|+|γ|

c(β′, γ′)∂β′

x ∂γ′

z (1/θ) ∂β′′

x ∂γ′′

z θ

And :∣∣θ · ∂βx∂γz (1/θ)∣∣ 6 C ∑

β′+β′′=β|z|−n+1−|β′| · |z|n−1−|β′′| 6 C|z|−|β|

Since θ is compactly supported and satisfies θ(x, z) ∼z→0 |z|n−1, we canwrite |θ(x, z)| > 1

2 |z|n−1 around zero, and thus :∣∣∂βx∂γz (1/θ)

∣∣ 6 C|z|−n+1−|β|

This proves the estimate (4.2).Let us end the computation. Since g(x) (the matrix of the bilinear form

g) is symmetric definite positive, it can be written as g(x) = s2(x) for somesmooth symmetric definite positive s. Using the change of variable u = sz,and denoting | · | the euclidean norm on Rn, we obtain :∫

Rne−iξ·z

2√

det g(x)gij(x)zizj dz =

∫Rne−i(s

−1ξ)·u 2|u|n−1 du

= cn|s−1ξ|−1

= cn|ξ|−1g−1 ,


where cn is a constant depending on the dimension, coming from the Fouriertransform F : H]−(n−1)(R

n)→ H]−1(Rn), and |·|g−1 denotes the norm givenby g−1, which is the canonical norm induced by g on the cotangent bundle.Thus, the principal symbol is

σ(I∗0 I0) = cn|ξ|−1g−1

4.3. Definition on a manifold without boundary

4.3.1. The point of view of distributions

Let (M, g) be a compact manifold without boundary. Given γ ∈ G, a clo-sed unit speed geodesic onM , we can define the distribution δγ ∈ D′(SM),which corresponds to the integration along (γ, γ) in SM . In other words,we define for f ∈ C∞(SM),

〈δγ , f〉 = If(γ)

This corresponds exactly with the definition of the X-ray transform onclosed geodesics γ ∈ G.D′(SM) (endowed with the weak-* topology) is the topological dual of

C∞(SM), so X acts on D′(SM) by duality (since it acts smoothly onC∞(SM)) that is, given T ∈ D′(SM), u ∈ C∞(SM),

〈X · T, u〉 = −〈T,X · u〉

From this, we can define the set of invariant distributions by the flow of X,that is :

D′inv(SM) = T ∈ D′(SM), X · T = 0We can already note that this set of invariant distributions does not containany L2 functions (not even L1 actually) but the constants. This comes fromthe fact that the geodesic flow is ergodic : as a consequence, any functionwhich is invariant by the flow is constant.

Proposition 4.9. — If (M, g) is Anosov, then the set δγ , γ ∈ G isdense in D′inv(SM).

Démonstration. — Assume f ∈ C∞(SM) satisfies 〈δγ , f〉 = 0 for allγ ∈ G. By Livcic’s theorem C.10, we know that f = X · u, for someu ∈ C∞(SM). Thus, for any T ∈ D′inv(SM), we obtain that 〈T, f〉 =〈T,X · u〉 = −〈X · T, u〉 = 0.


From now on, we assume that the manifold (M, g) is Anosov. Therefore,we can extend the definition of the X-ray transform to :

I :∣∣∣∣ C∞(SM)→ L(D′inv(SM),R)f 7→ (If : ν 7→ If(ν) = 〈ν, f〉) ,

where L(D′inv(SM),R) denotes the space of continuous linear forms onD′inv(SM), endowed with the weak-* topology. The application I is a conti-nuous application between the Fréchet space C∞(SM) (endowed with thecanonical norms) and the space L(D′inv(SM),R). Since D′inv(SM) is re-flexive (as a closed subspace of a reflexive space), the dual of L(D′inv(SM),R)is D′inv(SM). The adjoint of I is the map

I∗ :∣∣∣∣ D′inv(SM)→ D′(SM)ν 7→ (I∗ν : ϕ 7→ 〈ν, Iϕ〉)

In the sequel, we will mostly be interested in proving the s-injectivity ofIm : Rm ⊂ C∞(SM)→ L(D′inv(SM),R), that is showing that if Im(f) = 0,then there exists u ∈ Rm−1 such that f = X · u. It will often be obtainedthanks to the surjectivity of its adjoint I∗m : D′inv(SM) → D′(SM). Theorthogonal projector πk : L2(SM) → Hk acts by duality on D′(SM),that is any distribution ν ∈ D′(SM) can be decomposed into a Fourierseries where its coefficient νk = πk(ν) is given by 〈νk, ϕ〉 = 〈ν, πk(ϕ)〉.Thus, formally, the adjoint I∗m : D′inv(SM) → R∗m is given by I∗m(ν) =πm(ν) + πm−2(ν) + ...+ π−m(ν) (where R∗m denotes the dual of Rm). Thesurjectivity of this operator will be studied on Rm, that is for f ∈ Rm(seen as a subspace of R∗m by the natural identification of a function withthe linear form associated to it via the L2 scalar product), we will try tofind h ∈ D′inv(SM) such that I∗m(h) = f .

4.4. A few words about the hypothesis

4.4.1. Strict convexity for manifolds with boundary

For manifolds with boundary, we mentioned in the introduction to thissection that we will usually assume that they are strictly convex. Actually,this is not always the optimal hypothesis, depending on the problem weconsider. First, let us state the

Lemma 4.10. — Assume (M, g) is a compact surface with strictly convexboundary. Then geodesics in

M intersect ∂M transversally, i.e. a geodesic

coming fromM and touching a point x ∈ ∂M cannot be tangent to ∂M .


Démonstration. — We can always assume that M is embedded in a lar-ger manifold (N,h) such that g = i∗h, where h denotes the inclusion.Consider x0 ∈ ∂M . It is always possible to choose centered isothermal co-ordinates (x1, x2) in a neighborhood of x0 such that x0 is sent on x = 0, aneighborhood of x0 in M corresponds to a neighborhood of 0 in the upperhalf-plane x2 > 0 and ν(x0) is sent on the vector −∂/∂x2(0). Indeed, thiscan be done using the same argument as the one involved in the proof ofthe existence of isothermal coordinates with the Dirichlet condition thatg = 0 on x2 = 0 (see Appendix B). Therefore, thanks to the Koszul formulafor the Christoffel symbols, one can check that in 0 :

∇vν =(−∂2λ ∂1λ

−∂1λ −∂2λ

)v

The bilinear form (v, w) 7→ g(∇vν, w) is positive definite sinceM is strictlyconvex. This immediately implies that ∂2λ(0) < 0.Now recall that according to the expression (2.25), we can write in these

coordinates :

X(x1, x2, θ) = e−λ(

cos θ ∂

∂x1+ sin θ ∂

∂x2+(− ∂λ

∂x1sin θ + ∂λ

∂x2cos θ

)∂

∂θ

)Consider a smooth geodesic x(t) such that x(t0) = 0, x2(t) > 0 for t < t0

and x2(t0) = 0 (it is a geodesic contained inM that intersects tangentially

∂M at t = t0). The equation (x, θ) = X(x, θ) tells us (when projected on∂/∂x2(0)) that θ(t0) = 0 (or π, but this case being similar, we forget it)and θ(t0) = e−λ(0) cos θ(t0)∂x2λ(0) = e−λ(0)∂x2λ(0) < 0. So θ decreases ast is close to t0. Moreover, the equation x2(t) = e−λ(x(t)) sin θ(t) providesx2(t0) = eλ(0)θ(t0) < 0. Therefore, a Taylor expansion in a vicinity of t0yields to :

x2(t) = 12(t− t0)2x2(t0) +O((t− t0)3)

This is negative around t0 so we obtain a contradiction.

Remark 4.11. — Note that the arguments applied in the previous proofimmediately shows that in the strictly convex case, τ = 0 on ∂−(SM). If(x, v) ∈ ∂(SM) satisfies 〈v, ν(x)〉 < 0, then τ(x, v) > 0. The prototype forsuch a surface is the unit disk, embedded in R2, endowed with the euclideanmetric. Locally, we have the following picture :

Proposition 4.12. — Assume (M, g) has a strictly convex boundary.Then τ is zero on ∂−(SM), smooth on SM \ ∂0(SM), continuous on SM .


M

x0

0

Figure 4.2. The local isothermal coordinates and the form of thegeodesics in a neighborhood of the boundary

We see in particular that in the case where the manifold is non-trapping,there exists a maximum exit time for geodesics starting in ∂+(SM). Thiswill imply the boundedness of the X-ray transform I as an L2 operator.

Démonstration. — As mentioned before, it is clear that τ is zero on∂−(SM). The smoothness of τ on SM \ ∂0(SM) relies on the implicitfunction theorem. Indeed, consider a point (x0, v0) ∈ SM \ ∂−(SM) anddenote by x∗ = π(ϕτ(x0,v0)(x0, v0)) ∈ ∂M . In particular, t0 = τ(x0, v0) > 0as we have seen. We can always assume that (M, g) is embedded in (N,h).Consider a smooth function p defined on a vicinity of x∗ in N such that∂M = p = 0 in this neighborhood, and dp 6= 0. It can be lifted in thefibers to obtain an application, still denoted by p such that in a vicinity ofπ−1(x∗), we have ∂(SM) = p = 0. We look at the application

Φ : (x, v, t) 7→ p(ϕt(x, v)),

defined in a neighborhood of (x0, v0, t0). By definition, τ satisfies the im-plicit equation

Φ(x, v, τ(x, v)) = p(ϕτ(x,v)(x, v)) = 0

But, we have∂Φ∂t

(x0, v0, t0) = dpϕt0 (x0,v0) (X(ϕt0(x0, v0))) 6= 0,

since the geodesic intersects transversally the boundary at x∗ accordingto Lemma 4.10. Thus, the implicit function theorem allows us to concludethat τ is smooth on SM \∂−(SM). It is also smooth on ∂−(SM)\∂0(SM)


since τ is zero on it. The continuity can be obtained using local coordinatesfor instance.

Remark 4.13. — It can be easily seen that the convexity assumption isnecessary to ensure the continuity of τ . Indeed in the following figure, aslight modification of the initial speed can deeply affect τ .

(x; v)

M

Figure 4.3. The non-continuity of τ without the convexity assumption

Remark 4.14. — The problem of smoothness can be seen on a simpleexample. If one considers the disk of radius one, with center at (1, 0) ∈ R2,that is D =

(x− 1)2 + y2 6 1

, for the euclidean metric, then the exit

time of a geodesic starting at (0, 0) with direction (cos θ, sin θ) is given bythe function

τ(θ) =

2| cos θ|, θ ∈ [−π/2, π/2]0, elsewhere

0

1

−π=2 π=2

τ

θ

Figure 4.4. The non-smoothness of τ on a disk

Theorem 4.15. — Assume (M, g) is non-trapping and has strictly convexboundary. Then, the geodesics in

M intersect ∂M transversally and (M, g)

satisfies the smooth Livcic property.


If we consider a smooth function f such that If = 0, then it is clearthat the function u defined on SM such that f = X · u and u|∂−(SM) = 0(and thus u|∂+(SM) = 0 by If = 0) will be smooth on SM \ ∂0(SM). Theproblem comes from the regularity around ∂0(SM) because of τ .

Lemma 4.16. — Consider a smooth symmetricm-tensor f ∈ C∞(M,⊗mS T ∗M)such that Imf = 0. Then there exists a decomposition f = dh + p, wereh ∈ C∞(M,⊗m−1

S T ∗M), h|∂M=0, p ∈ C∞(M,⊗mS T ∗M), with

∀k > 0, ∂k(f − dh)∂νk

= 0

where for x ∈ ∂M , ν(x) denotes the unit outward vector.

Démonstration. — We consider the coordinates (r, θ) on (0, ε) × ∂M '(0, ε) × S1, like in Appendix D. The tensor f can be decomposed in f =dh + p, where h ∈ C∞(M,⊗m−1

S T ∗M), p ∈ C∞(M,⊗mS T ∗M) and h|∂M =0, i ∂

∂rp = 0.

First, assume p(0, θ0) = a(0, θ0)dθm 6= 0 for some θ0. Then, by smooth-ness, it is also true in a vicinity of (0, θ0) and we may assume (the other casebeing symmetric) that a(r, θ) > 0 in a neighborhood U of (0, θ0). Considera short geodesic starting from (0, θ0) (with initial direction v such thatvr 6= 0, vθ > 0) and staying within U , with exit time τ . We denote it by(γt, γt). Then

Imf(γ) = 0 =∫ τ

0a(γt)(γt)mθ dt,

which is absurd since a(γt)(γt)mθ > 0.

Now, we reason by induction. Assume that we have proven that ∂kp

∂νk= 0

for any k < l and that there exists a θ0 such that ∂lp

∂νl(0, θ0) 6= 0. As before,

we can assume, without loss of generality, that ∂la

∂νl(0, θ0) > 0, which is still

true in a neighborhood U of this point. Using a Taylor expansion, we get :

p(r, θ) = 1l!∂la

∂νl(0, θ)rldθm +O(|r|m+1)

In particular, there is a vicinity of (0, θ0) such that 1l!∂la

∂νl(0, θ)rl > 0 as

long as r > 0. Thus, considering a small geodesic as we did before, weobtain a contradiction.

Remark 4.17. — Actually, the proof only works for this derivative be-cause the geodesics that we consider are confined in a conical neighborhood(the value of (γ)θ does not change much) and p is of the form adθm. Had


we considered a tensor with a term dr⊗dθm−1 for instance, this would nothave worked, because the short geodesics would have involved a term (γ)rwhose sign is not constant along the geodesic.

We can now prove Livcic’s theorem in the case M is non-trapping andhas strictly convex boundary. What we are going to use is the following

Lemma 4.18. — Assume f ∈ C∞(M,⊗mS T ∗M) satisfies If = 0 and hascompact support in

M . Then f = X · u, for some smooth u ∈ C∞(SM).

Démonstration. — We set on SM :

u(x, v) = −∫ τ(x,v)

0f(ϕt(x, v))dt

We clearly haveX ·u = f and u is smooth since τ is smooth on SM\∂0(SM)(and the function u is zero in a neighborhood of ∂0(SM).

Démonstration. — Consider the annulus [0, ε) × ∂M = U like in theprevious proof and a cutoff function χ that is 1 in a vicinity of ∂M and withcompact support in U . Assume f ∈ C∞(M,⊗mS T ∗M) satisfies Imf = 0.We want to prove that there exists a smooth function u on SM such thatf = X · u (where f is seen as a smooth function on SM , that is Φm(f)).We will use the two decompositions detailed in Appendix D, namely f =dh1 + fs = dh2 + p, where fs is divergence-free and i ∂

∂rp = 0. Note that

the second decomposition is only valid in U .

Me

M

χ

~χ

M n U

0

1

Figure 4.5. The cut-off functions


We can embed (M, g) into a closed manifold (N,h) such that h is asmooth extension of g. We can consider Me, a tubular neighborhood ofM is N which is still strictly convex and non-trapping. Let us denote byV = (Me \M) ∪ U . Since p and all its radial derivative vanish, we canextend it smoothly on Me by the zero section. Consider a cutoff functionχ that is 1 on Me \M and on the support of χ, with compact support inV . Let us denote by J the X-ray transform on Me and by τe the exit timeof the geodesics. We now define :

u1(x, v) =∫ τe(x,−v)

0χp(ϕs(x,−v))ds

u2(x, v) = −∫ τ(x,v)

0fs(ϕs(x, v))ds

Let us make a few comments. Both functions satisfy by constructionX·u1 =χp,X · u2 = fs. u1 is defined on SMe and u2 is only defined on SM . u1 issmooth by the previous lemma, whereas u2 may not be smooth on ∂0(SM).We now consider :

λ = χ(h1 + u1) + (1− χ)(h2 + u2)

This is a smooth function defined on SM . Moreover :X · λ = χX · (h1 + u1) + (1− χ)X · (h2 + u2) +X · χ ((h1 + u1)− (h2 + u2))

= f +X · χ ((h1 + u1)− (h2 + u2))

Let us call µ = X · χ ((h1 + u1)− (h2 + u2)). This is a function supportedin an annulus strictly contained in

M and it satisfies Iµ = 0 by the previous

equality. Thus, by the previous lemma, it can be written µ = X · u3 forsome smooth u3. Therefore X ·(λ−µ) = f and λ−µ is smooth on SM .

4.4.2. The hypothesis of simplicity and the injectivity of I0, I1

We recall that a manifold is simple if it is free of conjugate points, sim-ply connected and has strictly convex boundary. It is possible to substitutethe hypothesis "simply connected" by "non-trapping" in the definition of asimple manifold because they become equivalent with the other assump-tions, as mentioned before. As we have seen in Section 3, the hypothesisthat the surface is free of conjugate points provides the crucial identity(3.5)

||X · ψ||2 − (Kψ,ψ) > 0,for any ψ ∈ C∞(SM). Note that in the case of an Anosov surface (andtherefore, in the case of a surface with negative curvature), this hypothesis


is unnecessary since we have seen that an Anosov manifold cannot carryconjugate points (this comes from the existence of the two distinct solu-tions r+ and r− to the Riccati equation). Coupled with the Pestov identity(7.1), the previous identity immediately provides the injectivity of the X-ray transform for 0- and 1-tensors. Thus, we can already state our firsttheorem of injectivity :

Theorem 4.19. — Let (M, g) be a simple surface satisfying the smoothLivcic property. Then I0 and I1 are s-injective.

In particular, since a surface with strictly convex boundary is smoothlyLivcic, we deduce from this theorem that I0 and I1 are injective on such asurface as long as it does not carry conjugate points.Démonstration. — Let us write once again the Pestov identity for u ∈

C∞(SM) :

||XV u||2 − (KV u, V u) + ||Xu||2 − ||V Xu||2 = 0

We already know that the sum of the first two terms is positive. And thus,for any u ∈ C∞(SM) :

(4.3) ||Xu||2 − ||V Xu||2 =∑k

(1− k2)||(Xu)k||2 = 0

Let f be a smooth function in C∞(M), that is a 0-tensor, such that I0f =0. Since M satisfies the smooth Livcic property, we know that f = X · u,for a smooth function u defined on M . Since f is a 0-tensor, it is clear that(Xu)k = 0, as long as k 6= 0. Thus the previous equality (4.3) immediatelyimplies that f = 0.Let A be a 1-tensor such that I1f = 0. We still denote by A the smooth

function Φ1(A) defined on SM . By the Livcic property, we can write A =X ·u, for some smooth function u. Since A is a 1-form, a simple computationshows that ||Xu||2−||V Xu||2 = ||A||2−||V A||2 = 0. Thus, using the Pestovidentity, we obtain the equality in (3.5), i.e. ||XV u||2 − (KV u, V u) = 0.This immediately implies that V u = 0 by Remark 3.17, so u ∈ C∞(M) isactually constant in the fibers. In other words, u ∈ R0.


5. X-ray transform on negatively curved surfaces and spec-tral rigidity

5.1. Introduction

5.1.1. The spectrum of the Laplacian

Since Weyl’s formula in 1911, — which provides an asymptotic develop-ment of the cumulative distribution function of the spectrum of the Lapla-cian with respect to the volume of the manifold — the link between thespectrum of the Laplacian and the geometric properties of a manifold hasbeen investigated. Recall that on a compact Riemannian manifold (M, g),there exists in any non trivial free homotopy class at least one smooth andclosed geodesic whose length is minimal in the class (see [8], Theorem 2.98for instance). We will call length spectrum the collection of lengths of theperiodic geodesics counted without multiplicities and marked length spec-trum the collection of lengths counted with multiplicites. Among the mainresults concerned with the link between length spectrum and spectrum ofthe Laplacian, we have the (see [5] and [12]) :

Theorem 5.1. — The spectrum of the Laplacian determines the lengthspectrum, that is the lengths of the periodic geodesics without multiplici-ties.

Theorem 5.2. — Let q be a smooth real-valued function. Then, thespectrum of ∆+q determines the integral of q over every periodic geodesics.

These two theorems will be used in this form in order to obtain thespectral rigidity of manifolds with negative curvature. Let us briefly explainwhere these two results originate. In the caseM is a compact manifold, theLaplace operator −∆ has compact resolvent so one can find a sequenceof positive eigenvalues λi → ∞ (with associated eigenfunctions fi). Notethat they form an orthonormal basis of L2(M) and any f ∈ L2(M) canbe decomposed in its "Fourier" mode f =

∑i αifi. We define the unitary

operator U(t) = eit√−∆. Then, its trace is a distribution on the real line

given by :χ(t) =

∑i

eit√λi

We then have the following result (see [13] for instance) :

Theorem 5.3. — T 6= 0 is in the singular support of χ if and only ifthere exists a periodic geodesic on M of period T . Thus, the spectrum ofthe Laplace operator determines the length of the periodic geodesics.


5.1.2. What about a converse ?

In particular, a long-standing question — now solved — has been to seeif two isospectral manifolds (manifolds with same spectrum of the Lapla-cian counted with multiplicities) were necessarily isometric. This is actuallywrong and it is now known that there exists Riemann surfaces with samegenus g > 4 and constant negative curvature −1 that are isospectral butnot isometric (see [1] for instance). However, J-P. Otal proved in 1990 (see[21] for a reference) a sort of converse : in negative curvature, two manifoldssharing the same marked length spectrum are isometric. Note that it is stillignored whether this result still holds in greater dimensions.

5.1.3. Spectrally rigid manifolds

This section is devoted to the proof of a theorem of V. Guillemin andD. Kazhdan (see [14] for the original paper). Let us first introduce somepreliminary notions before stating their result. We fix a Riemannian two-dimensional manifold (M, g). We say that a family of Riemannian metrics(gt)t∈(0,ε) is a deformation if gt depends smoothly on t and g0 = g. Thedeformation is said to be trivial if there exists an isotopy (φt)t∈(0,ε), that is afamily of diffeomorphisms depending smoothly on t, such that g = (φt)∗gt.

For each metric gt, we can consider the Laplace-Beltrami operator ∆t

and its spectrum. We say that the deformation is isospectral if the spectrumof the operators ∆t coincide, counting the multiplicities of the eigenvalues.A manifold is said to be spectrally rigid if any isospectral deformation istrivial. We can now state the theorem :

Theorem 5.4 (Guillemin-Kazhdan, 78). — Let (M, g) be a two-dimensionalRiemannian manifold with negative curvature. Then (M, g) is spectrally ri-gid.

This result can actually be seen as a linearization of a stronger result onecould expect, which is in the spirit of what we obtain in the context of theboundary rigidity problem. Namely, it could be reasonable to try to showthat two isospectral Riemannian manifolds are isometric (in the sense thatthere exists a diffeomorphism ψ between the two such that ψ∗g = g′), butwe have already explained that this is not true.

5.2. Injectivity of the ray transform

In this paragraph, we are going to prove the


Theorem 5.5. — Let (M, g) be a surface with negative curvature. Thenfor any m > 0, the X-ray transform Im is s-injective.

Since we assume the curvature to be negative, the geodesic flow is Anosovand the surface satisfies the Livcic property (see Appendix C).

Lemma 5.6. — Assume f : SM → R is a smooth function such thatf ∈ ⊕kk=−nΩk and f is zero over every periodic integral curve of X. Then,there exists a smooth function u ∈ ⊕n−1

k=−(n−1)Ωk such that f = X · u.

Proof of the lemma. — Since M is negatively curved and f is zeroover every periodic integral curve of X, we know by Livcic’s theorem (seeAppendix C) that there exists a smooth function u : SM → R such thatX ·u = −f . We decompose the functions as u =

∑k uk, f =

∑k fk. Writing

X = η+ + η− and projecting on each Hk, we obtain :

η+uk−1 + η−uk+1 = −fkMoreover, since we are in negative curvature, the Pestov identity (3.15)gives :

(5.1) ||η+uk|| > ||η−uk||

We know that fk = 0 for k > n+ 1 so :

(5.2) ||η+uk−1|| = ||η−uk+1||

Combining (5.1) and (5.2), we obtain :

||η+un|| = ||η−un+2|| 6 ||η+un+2|| = ||η−un+4|| 6 ||η+un+4|| = ...

In other words, (||η+un+2k||)k>0 is a non-decreasing sequence and it convergesto 0 since u ∈ L2(SM). It is therefore constantly zero. We obtain thesame result for odd numbers. The same arguments apply for k 6 −n so||η+uk|| = 0 for k > n and k 6 −n. Applying once again Pestov identity(3.15), we obtain for k > n and k 6 −n that uk is zero.

Proof of Theorem 5.5. — Consider T a symmetric m-tensor such thatImT = 0, that it T is zero over every integral closed curve of the geodesicvector field X. Then, by Lemma 5.6, we know that there exists a smoothfunction u ∈ Rm−1 such that T = X · u. As mentioned in Section 3.2.2, ugives rise to a symmetric (m− 1) tensor such that T = du which concludesthe proof.

Remark 5.7. — Actually, the same arguments apply in any dimension,up to a generalization of the technical tools introduced. In [3], it is provedthat Im is s-injective for all m on non-positively curved manifolds of anydimension.


Remark 5.8. — In particular, we see where the proof breaks down whenno assumption is made on the curvature. In Pestov’s identity, it is no longerpossible to obtain the fundamental inequality (5.1) which is at the core ofthe proof of Lemma 5.6. Therefore, one needs to make more assumptionson the manifold M and/or introduce new tools to tackle this obstacle.

5.3. Proof of Theorem 5.4

Let us first give a heuristical approach. Consider an isospectral defor-mation (gt)t∈(−ε,+ε). As mentioned in Section 3.2.2, since a metric is asymmetric 2-tensor, we can associate to each metric a smooth real-valuedfunction ft : TM → R. The restriction of ft to the unit tangent bundleSMt (which depends on t since it is given by the metric gt) is the constantfunction 1. Since ∂gt

∂tis still a symmetric 2-tensor, the restriction to the

unit tangent bundle SMt of the function ft = ∂ft∂t∈ C∞(TM) canonically

associated to it lies in Ht,−2 ⊕Ht,0 ⊕Ht,2, and we can write

ft = ft,−2 + ft,0 + ft,2,

with ft,−2 = ¯ft,2. In the following, Xt will denote the geodesic vector fieldon SMt with respect to the metric gt and ∇t the Levi-Civita connexion.The idea is to prove that for each t, the ft is zero over every integral curveof the geodesic field Xt. Since the surface is negatively curved, we knowin particular, according to Anosov’s theorem C.8, that the geodesic flowis Anosov. By Livcic’s theorem C.10, this will give a smooth real-valuedfunction ut : SM → R such that ft = −Xt ·ut. We will actually prove thatthis function is in Ht,−1 ⊕Ht,1 and is therefore associated to a 1-form θt.By the musical isomorphism, this provides a vector field Yt on M , whosefamily of isotopy is exactly the one we seek.

Consider a periodic geodesic γ for the metric g parametrized by arc-length. Then, we can find a family of closed geodesics (γt)t∈(0,ε) dependingsmoothly on t such that γ0 = γ. Since the deformation is isospectral andthe spectrum of the Laplace-Beltrami operators determines the length ofthe closed geodesics, according to Theorem 5.1, then we know that thegeodesics γt have all the same length.

Lemma 5.9. — ∫γt

ft = 0,


Démonstration. — It is sufficient to prove the lemma in the case t = 0since the result can be deduced from it by reparametrizing the family as(gt+s), with s ∈ (0, ε − t). The common length of the geodesic is denotedby L. We have :

1 = gt(∂γt∂s

(s), ∂γt∂s

(s)) = ft(γt(s),∂γt∂s

(s)),

for any 0 6 s 6 x, 0 6 t 6 ε. We see Γt : s 7→ (γt(s),∂γt∂s

(s)) as a path

in TM (and more precisely in SMt). By definition, we have ∂Γt∂s

= Xt(Γt)and the geodesic field Xt is unitary on the unit tangent bundle SMt. Thus :

∀t ∈ (0, ε),∫

Γft = L =

∫ L

0ft(Γt(s)) ds

Differentiating with respect to t at 0, we obtain :

0 =∫

Γf0 +

∫Γd(f0)

Now, the second term is clearly zero since d(f0) is exact and Γ is a loop.

Lemma 5.10. — For each t ∈ (0, ε), there exists a smooth real-valuedfunction ut : SMt → R such that −Xt · ut = ft and ut ∈ Ht,−1 ⊕Ht,1 withut,−1 = ut,1.

Démonstration. — According to Lemma 5.6, we know that the integralof ft over every periodic integral curve of the geodesic field Xt is zero.Therefore, by the previous lemma, we know that there exists a smooth real-valued function ut : SMt → R such that −Xt ·ut = ft, ut ∈ Ht,−1⊕Ht,1 andut,−1 = ut,1. Now, these functions depend smoothly on t. Indeed, we havefor instance ft,2 = η+

t ut,1 and f depends smoothly on t and the operatorη+t is an injective elliptic operator depending smoothly on t.

We can now complete the proof of the theorem.Proof of Theorem 5.4. — Since ut lies in Ht,−1 ⊕ Ht,1, it can be seen

as a symmetric real-valued 1-tensor, which is nothing but a 1-form θt ∈Ω1(M). The musical isomorphism [t corresponding to the metric gt providesa vector field Yt onM such that (2Yt)[t = θt (the 2 comes from the operatorof symmetrization σ). We now denote by (φt)t∈(0,ε) the isotopy induced bythe family of vector fields (Yt)t∈(0,ε). We claim that (φt)∗gt = g. In orderto prove this, all we have to do is to check that for any x ∈M, ξ, η ∈ TxM ,we have :

gt(d(φt)x(ξ), d(φt)x(η)) = g(ξ, η)


Since this equality is obviously true for t = 0, it is sufficient to prove thatthe derivative of the left-hand side with respect to t is zero.

By definition of φt, we have ∂φt∂t

(x) = Yt(φt(x)). Moreover, according to(3.2), the equality ft = −Xt · ut can be rewritten in terms of tensors :

∂gt∂t

(·, ·) = −σ (∇tθt) (·, ·) = −12 (gt((∇t)·(2Yt), ·) + gt(·, (∇t)·(2Yt)))

= − (gt((∇t)·Yt, ·) + gt(·, (∇t)·Yt))

By the Leibnitz rule, we have :∂

∂t′

∣∣∣∣t′=t

gt (dφt′(ξ), dφt′(η)) = ∂

∂t′

∣∣∣∣t′=t

gt (dφt′(ξ), dφt(η))

+ ∂

∂t′

∣∣∣∣t′=t

gt (dφt(ξ), dφt′(η))

In order to compute these terms, let us consider a path γ : (a, b) → M ,

parametrized by s such that γ(0) = φt(x), ∂γ∂s

∣∣∣∣s=0

= dφt(ξ). We thus have

a map c : (t− ε, t+ ε)× (a, b)→M such that c(t′, s) = φt′(γ(s)). Since ∇is torsion-free, ∇ = c∗∇ is torsion-free too and :

T ∇(∂

∂s,∂

∂t

)= 0 = ∇ ∂

∂s

∂

∂t− ∇ ∂

∂t

∂

∂s−[∂

∂s,∂

∂t

]︸︷︷︸

=0

In other words, ∇ ∂∂s

∂

∂t= ∇ ∂

∂t

∂

∂s(this can also be seen as a Schwarz lemma

when written in coordinates, using a symmetry of the Christoffel symbols).Using the fact that ∇ is the Levi-Civita connection (it is g-adapted), weget :∂

∂t′

∣∣∣∣t′=t

gt (dφt′(ξ), dφt(η)) = ∂

∂t′

∣∣∣∣t′=t

gt

(∂

∂s

∣∣∣∣s=0

φt′ γ(s), dφt(η))

= gt

(∇ ∂

∂t′

∂

∂s· (φt′ γ(s)) , dφt(η)

)∣∣∣∣t′=t,s=0

= gt

(∇ ∂

∂s

∂

∂t′· (φt′ γ(s)) , dφt(η)

)∣∣∣∣t′=t,s=0

= gt(∇d(φt)x(ξ)Yt(φt(x)), dφt(η)

)And thus, combining all the previous equalities :

∂

∂t′(gt′(d(φt′)x(ξ), d(φt′)x(η))) = 0

This concludes the proof of the theorem.


5.4. A second result

A Riemannian manifold is said to have simple length spectrum if itsperiodic geodesics are isolated and non-degenerated, and all the periodicgeodesics have different periods. By non-degenerate, we mean that Hessianof the energy functional along a periodic geodesic is itself non-degenerate. Itis known that the property of having a simple length spectrum is a genericproperty of Riemannian manifold.

Theorem 5.11. — LetM be a compact negatively Riemannian surfacewith simple length spectrum. Let q1 and q2 be two smooth real-valuedfunctions on M and assume ∆ + q1 and ∆ + q2 have the same spectrum.Then q1 = q2.

Démonstration. — Let us denote by f the pull-back on SM of the func-tion q1 − q2. Since the operators ∆ + q1, ∆ + q2 have same spectrum,we know that the integrals of q1, q2 over periodic geodesics are equal ac-cording to Theorem 5.2. As a consequence, the integral of f over everyperiodic geodesics if zero. But clearly f ∈ H0 since f is constant in thefibers. By injectivity of the ray transform I0, we obtain f = 0.


6. Spectral rigidity on Anosov surfaces6.1. IntroductionIn this part, we are going to prove the injectivity of the X-ray transform

on symmetric 2-tensors.

Theorem 6.1 (Paternain, Salo, Uhlmann, 2014). — If (M, g) is Anosov,then I2 is s-injective.

The proof is rather pedestrian and consists in different steps. Recall thatwe have already established the injectivity of I0 and I1 in Section 4.4.2,using the Pestov identity and the inequality provided by the 0-control of asimple surface. In the case of an Anosov surface, we know that we have theinequality (3.6) which betters the α-control for some 0 < α 6 1. This willbe used in order to prove the surjectivity of I∗0 . Then, using the surjectivityof I∗0 we will obtain the surjectivity of I∗1 . Eventually, the surjectivity ofthis operator will provide the injectivity of I2. The trick involved in thesecond proof of surjectivity relies on a successful definition of the productof two invariant (by the geodesic flow) distributions, as long as they are insuitable "mixed" Sobolev spaces (which will be introduced later).The proof exposed in this part mainly follows [25]. Throughout the first

paragraph, it will be assumed that the surface is non-hyperelliptic (seeTheorem 3.26), thus providing an explicit description of the kernel of η−.In the last paragraph, we show how the general case can be recovereed. Asa corollary, we will obtain the spectral rigidity of Anosov manifolds, thanksto the same proof as the one provided in Section 5.3.

Corollary 6.2. — An Anosov surface is spectrally rigid.

6.2. Surjectivity of I∗0Let us first prove the surjectivity of I∗0 on R0 = C∞(M). We recall that

for ν ∈ D′inv(SM), I∗0 (ν) = 2πν0, so we are going to prove that for anyh ∈ C∞(M), there exists a invariant distribution ν such that 2πν0 = h.More precisely, we are going to show the

Theorem 6.3. — If (M, g) is Anosov, then I∗0 : H−1inv(SM) → C∞(M)

is surjective. Moreover, given f ∈ C∞(M) and w ∈ H−1inv(SM) such that

I∗0w = h, we have πk(w) ∈ C∞(SM) for all even k.

It mainly relies on the following lemma, which looks like an energy iden-tity, rather common in PDEs. Throughout the rest of this section, thesymbol ] in index of a functional set E] will mean that we consider in Ethe orthogonal to constant functions for the natural L2-product on SM .


Lemma 6.4. — There exists a constant C such that ||u||H1 6 C||Pu||L2 ,for all u ∈ C∞] (SM).

Démonstration. — We recall that for an Anosov surface, there exists a0 < α 6 1 such that it is α-controlled and even better, that is, for allψ ∈ C∞(SM), we have :

||Xψ||2 − (Kψ,ψ) > α(||Xψ||2 + ||ψ||2

)Applying this inequality with ψ = V u, and using Pestov identity 7.1, weobtain :

||Pu||2 > ||Xu||2 + α(||V u||2 + ||XV u||2

)By using Cartan’s structural equation [V,X] = H, we obtain that Hu =V Xu−XV u = Pu−XV u, thus :

||Hu||2 6 2(||Pu||2 + ||XV u||2)

As a consequence, there exists a constant C such that :

C||Pu||2 > ||V u||2 + ||Xu||2 + ||Hu||2

Eventually, the Poincaré inequality on a closed Riemannian manifold tellsus that ||u||2H1 ≈ ||∇u||2L2 = ||V u||2 + ||Xu||2 + ||Hu||2, thus leading to :

||u||H1 6 C||Pu||L2

Now we use a classical argument involving the Hahn-Banach theorem inorder to prove the

Lemma 6.5. — For any f ∈ H−1] (SM), there exists a solution h ∈

L2(SM) to the equation P ∗h = f in SM . Moreover, we have the inequality||h||L2 6 C||f ||H−1 .

Démonstration. — Consider l : P (C∞] (SM)) → C defined by l(Pu) =〈u, f〉. By the previous lemma, we have :

|l(Pu)| 6 ||f ||H−1 ||u||H1 6 C||f ||H−1 ||Pu||L2

Thus, l : P (C∞] (SM))→ C is continuous so by the Hahn-Banach theorem,it admits a continuous extension which we still denote by l : L2(SM) →C, and satisfies |l(u)| 6 C||f ||H−1 ||u||L2 , for any u ∈ L2(SM). By theRiesz representation theorem, there exists h ∈ L2(SM), such that l(u) =(u, h)L2(SM), for all u ∈ L2(SM) and ||h||L2 6 C||f ||H−1 . As a conse-quence, for any u ∈ C∞] (SM), we have :

〈u, P ∗h〉 = 〈Pu, h〉 = l(Pu) = 〈u, f〉

Now, f is orthogonal to constant functions so P ∗h = f .


Proof of the Theorem 6.3. — Let f ∈ C∞(M). By the previous lemma,we know that there exists h ∈ L2(SM) such that P ∗h = XV h = −Xf .Now set w = V h + f ∈ H−1(SM). It is clear that X · w = 0, that isw ∈ H−1

inv(SM), and π0(w) = w0 = f . To obtain the smoothness of the w2k,we make use of the ellipticity of the operators η±, along with a bootstrapargument. Indeed, we know that X ·w = 0, that is for all k ∈ Z, η+wk−1 +η−wk+1 = 0. Therefore, η−w2 = −η+w0 = −η+f . Since f is smooth, η+f

is smooth and w2 is smooth by ellipticity of η−. Inductively, we concludethat all the w2k are smooth.

6.3. Surjectivity of I∗1

We recall that a smooth 1-form a = a−1 + a1 ∈ Ω−1 ⊕ Ω1 is said to besolenoidal if its divergence is zero. As we have seen, this is equivalent toη+a−1 +η−a1 = 0. We still confuse the 1-form and the function in C∞(SM)canonically associated to it.

Theorem 6.6 (Surjectivity of I∗1 ). — Let a be a smooth solenoidal 1-form. Then there exists w ∈ H−1

inv(SM) such that I∗1 (w) = w−1 + w1 = a.In particular, such a w can be taken such that w0 = 0.

Let T : C∞(SM)→⊕|k| >2 Ωk be the orthogonal projection such that

Tu = u−(π−1 + π0 + π1) (u). We also defineQ := TV X = TP and we haveQ∗ = XV T since T∗ is self-adjoint and P ∗ = XV . Just like in the previousparagraph, the surjectivity of Q∗ is obtained thanks to the following

Lemma 6.7. — There exists a constant C > 0, such that for any u ∈⊕|k| >1 Ωk, we have :

||u||H1 6 C||Qu||L2

Démonstration. — First note that

||Pu||2 = ||(Xu)1||2 + ||(Xu)−1||2 + ||Qu||2 6 ||Xu||2 + ||Qu||2

We already know by Lemma 6.4 that ||u||H1 6 C||Pu||. Therefore, aninequality of the form ||Xu|| 6 C||Qu|| would allow us to conclude. Werecall the Pestov identity (7.1) :

||Pu||2 = ||V Xu||2 = ||XV u||2 − (KV u, V u) + ||Xu||2

Since (M, g) is Anosov, we have for some 0 < α 6 1 :

||XV u||2 − (KV u, V u) > α(||XV u||2 + ||V u||2

),


which gives :

||Pu||2 > α||XV u||2 + ||Xu||2 > (1 + α)||Xu||2,

since u has no component in Ω0. Thus :

||Qu||2 = ||Pu||2 − (||(Xu)1||2 + ||(Xu)−1||2) > α||Xu||2

Remark 6.8. — Actually, this lemma proves the injectivity of I1. Indeed,assume f is a 1-form such that I1(f) = 0 and define u such that X ·u = f .We set v = u− u0. X · v has degree 1 and we want to show it is zero. ButQu = TV Xu = 0 by definition of T . The inequality implies immediatelythat u = 0.

The same trick involving the Hahn-Banach theorem applies here :

Lemma 6.9. — Consider f ∈ H−1(SM) such that f0 = 0. Then, thereexists h ∈ L2(SM) such that Q∗h = f . Moreover, we have the inequality||h||L2 6 C||f ||H−1 .

Démonstration. — In this proof, we will denote by Λ1 :=⊕|k| >1 Ωk.

Consider the linear form

l :∣∣∣∣ Q(Λ1)→ CQu 7→ 〈u, f〉

By the previous lemma, we have :

l(Qu) 6 ||f ||H−1 ||u||H1 6 C||f ||H−1 ||Qu||L2

Therefore, l is continuous on Q(Λ1), so it admits a continuous extension bythe Hahn-Banach theorem (which we still denote l) such that l : L2(SM)→C with |l(v)| 6 C||f ||H−1 ||v||L2 . By the Riesz representation theorem, weknow that there exists a h ∈ L2(SM), such that l = (·, h)L2 , with ||h||L2 6C||f ||H−1 . Then, for u ∈ C∞(SM) :

〈u,Q∗h〉 = 〈Qu, h〉 = 〈Q(u− u0), h〉 = l (Q(u− u0)) = 〈u− u0, f〉 = 〈u, f〉,

where the last equality holds because f0 = 0.

Proof of Theorem 6.6. — Consider a smooth solenoidal 1-form a anddefine f := −X · (a−1 + a1). Since a is solenoidal, η+a−1 + η−a1 = f0 = 0.By the previous lemma, we know that there exists h ∈ L2(SM), such thatQ∗h = XV Th = f . We define w = V Th + a−1 + a1 ∈ H−1(SM) whichsatisfies by construction X ·w = 0. And w−1 +w1 = a−1 + a1, w0 = 0.


6.4. Injectivity of I2As mentioned before, the main idea relies on the fact that it is possible to

multiply invariant distributions, as long as they are in some "good" Sobolevspaces, in order to obtain another invariant distribution.

6.4.1. The mixed Sobolev normIn this paragraph, we introduce the space

L2xH

sθ (SM) = u ∈ D′(SM), ∀k ∈ Z, uk ∈ L2(SM),

||u||L2xH

sθ

:=( +∞∑k=−∞

〈k〉2s||uk||2L2

)1/2

<∞

,

(6.1)

where 〈k〉 = (1 + k2)1/2.Now that we have introduced the proper Sobolev spaces, let us just show

a corollary of Theorem 6.6.

Corollary 6.10 (Corollary of Theorem 6.6). — Assume a1 ∈ Ω1 andη−a1 = 0. Then there exists w =

∑k>1 wk ∈ L2

xH−1θ (SM) such that

X · w = 0, w1 = a1, each wk is in C∞(SM) and ||w||L2xH−1θ6 C||a1||L2 .

It will be important insofar as, on a non-hyperelliptic surface, the productof elements a1 ∈ Ω1 such that η−a1 = 0 generates ker η− ∩ Ω2.Démonstration. — Consider w the distribution given by Theorem 6.6

with a−1 = 0. We denote by w its holomorphic projection. Since w−1 =w0 = 0, we have w =

∑∞k=1 wk and X ·w = 0. Now, the smoothness of the

wk comes from a bootstrap argument, using the ellipticity of the operatorsη±, just like in the proof of Theorem 6.3.

We can now prove the

Theorem 6.11. — Let u, v ∈ D′(SM) such that u =∑k>0 uk, v =∑

k>0 vk, where u ∈ L2xH−sθ , v ∈ L2

xH−tθ , for s, t > 0. We define for k > 0

the Cauchy product

(6.2) wk =k∑j=0

ujvk−j ∈ Hk

If N is an integer such that N > s+ t+1/2, then the sum∑k wk converges

in H−N−2(SM) to some w with ||w||H−N−2 6 C||u||L2xH−sθ||v||L2

xH−tθ.

Moreover :

(6.3) ||wk||L1(SM) 6 〈k〉s+t||u||L2xH−sθ||v||L2

xH−tθ


If X · u = X · v = 0, then X · w = 0.

Démonstration. — For k > 0, we have by Cauchy-Schwarz

||wk||L1 6k∑j=0||ujvk−j ||L1

6k∑j=0||uj ||L2 ||vk−j ||L2

6

k∑j=0||uj ||2L2

1/2 k∑j=0||vj ||2L2

1/2

6

k∑j=0〈j〉2s〈j〉−2s||uj ||2L2

1/2 k∑j=0〈j〉2t〈j〉−2t||vj ||2L2

1/2

6 〈k〉s+t||u||L2xH−sθ||v||L2

xH−tθ

Let us define W l =∑lj=0 wj and consider N > s+ t+ 1/2. Since we are in

dimension three on a compact manifold, by the Kato-Rellich theorem, thecontinuous embedding H2(SM) → L∞(SM) holds, that is, there existsa constant C > 0, such that for any ϕ ∈ H2(SM), ϕ ∈ L∞(SM) and||ϕ||L∞(SM) 6 C||ϕ||H2(SM). For ϕ ∈ HN+2(SM), we have :

|〈W l, ϕ〉| =

∣∣∣∣∣∣l∑

j=0〈wj , ϕj〉

∣∣∣∣∣∣6

l∑j=0||wj ||L1(SM)||ϕj ||L∞(SM)

6 ||u||L2xH−sθ||v||L2

xH−tθ

l∑j=0〈j〉s+t||ϕj ||L∞(SM)

6 C||u||L2xH−sθ||v||L2

xH−tθ

l∑j=0

j−δjs+t+δ||ϕj ||H2(SM)

6 C||u||L2xH−sθ||v||L2

xH−tθ

l∑j=0

j2(s+t+δ)||ϕj ||2H2(SM)

,

using the Cauchy-Schwarz inequality in the last line, for any δ > 1/2. Wemay take δ = N−s−t > 1/2. Let us define Y1 = η+, Y2 = η−, Y3 = V . Thisgenerates an equivalent norm on the Sobolev spaces Hk to that generated


by the moving frame X,H, V . We have :

j2(s+t+δ)||ϕj ||2H2 6 jN (||ϕj ||2L2 +3∑q=1||Yqϕj ||2L2 +

3∑q,r=1

||YqYrϕj ||2L2)

6 ||V Nϕj ||2L2 +3∑q=1||V NYqϕj ||2L2 +

3∑q,r=1

||V NYqYrϕj ||2L2

So :

|〈W l, ϕ〉| 6 C||u||L2xH−sθ||v||L2

xH−tθ

l+2∑j=−2

(||(V Nϕ)j ||2L2 +

3∑q=1||(V NYqϕ)j ||2L2

+3∑

q,r=1||(V NYqYrϕ)j ||2L2

)6 C||u||L2

xH−sθ||v||L2

xH−tθ||ϕ||2H2+N

Therefore, W l ∈ H−N−2 and ||W l||H−N−2 6 C||u||L2xH−sθ||v||L2

xH−tθ. Using

the same inequalities with Cauchy sequences, we can define in a similarfashion, for any ϕ ∈ HN+2(SM) :

〈w,ϕ〉 = liml→∞〈W l, ϕ〉

Thus w ∈ H−N−2 and ||w||H−N−2 6 C||u||L2xH−sθ||v||L2

xH−tθ. Now, using

the definition with the Cauchy product, the conditions X · u = X · v = 0immediately imply X · w = 0.

Now, the problem with this proof is that it is rather hard to generalizeto a product of more than three distributions because there is no naturaldecompositions of functions using Fourier analysis on a space like L3(SM)(and it is no longer a Hilbert space).

6.4.2. Attempt for a general proof

The general strategy would be to prove the injectivity of the ray trans-form Im by induction on m. We are going to try such a proof and showwhere it breaks down. We assume in this paragraph that the surface isnon-hyperelliptic.Let us fixm > 0 and assume that Il is s-injective for any l < m. Consider

a smooth real-valued symmetric m-tensor f (which we still confuse withits function Φm(f) ∈ Rm ⊂ C∞(SM)) such that Im(f) = 0. By Livcic’stheorem, we know that there exists u ∈ C∞(SM) such that X · u = f . Wecan decompose f = fm + fm−2 + ... + f−(m−2) + f−m ∈ ⊕mj=0Hm−2j . By


Corollary 3.24, we can write fm = η+(hm−1) + qm, where hm−1 ∈ Ωm−1,qm ∈ Hm ∩ ker (η−). Now, we use on qm Max Noether’s Theorem 3.26which allows us to write qm as a finite sum

(6.4) qm =∑k∈Nm

ak1 ...akm ,

where each aki is in Ω1 ∩ ker (η−). Now, by Theorem 6.6, we know thatthere exists wki ∈ H−1

inv(SM) such that wki,1 = aki , wki,p = 0 for p 6 0. Wehave seen in the previous paragraph that it is possible to give a sense to themultiplication of two distributions, as long as they lie in some "good" mixedSobolev spaces. The problem is that it is rather unclear that this is stillwell-defined when there are more than two distributions in the product (andhere there are m), and the estimates produced in the previous paragraphsfail to conclude the argument. Actually, this will only allow us to proof theinjectivity of Im for the case m = 2.

For the reader’s convenience, we simply explain how the proof wouldend if we were able to give a sense to the product (6.4). Assume w =∑k∈Nm wk1 ...wkm =

∑k>m wk makes sense as a distribution in someH−N (SM).

Then, X · w = 0 and wm = qm. Also note that since f is real-valued,fk = f−k. And qm = X · (u − hm−1) + η−(hm−1) − (fm−2 + ...+ f−m).Thus :

||qm||2L2 = 〈wm, qm〉 = 〈w,X(u− qm)〉 = 〈X · w, u− qm〉 = 0,

where the penultimate equality holds because w =∑k>m qk and the last

equality holds because u− qm is smooth. As a consequence :

X ·(u− hm−1 − hm−1

)= −η+(hm−1)− η−(hm−1) + fm−2 + ...+ f−(m−2)

∈m−2⊕j=0

Hm−2−2j

The hypothesis tells us that u − hm−1 − hm−1 ∈⊕m−3

j=0 Hm−3−2j , so u ∈⊕m−1j=0 Hm−1−2j , which "proves" the induction.As explained earlier, we will only be able to carry out this argument with

m = 2.

6.4.3. Injectivity of I2

Recall that η− : Ω1 → Ω0 has a kernel of dimension g, where g is thegenus of the surface M , according to Proposition 3.22. We define Λ =Span ab, a, b ∈ Ω1 ∩ ker (η−).


Theorem 6.12. — Assume q ∈ Ω2 is in the linear span Λ. Then thereexists w =

∑k>2 wk ∈ H−5(SM) such that X · w = 0, w2 = q, ||w||H−5 6

C||q||L2 and each wk is in C∞(SM).

Démonstration. — Assume q ∈ Λ, that is q =∑Nj=1 a

jbj (with N 6dim(Λ)), where aj , bj ∈ Ω1 and η−a

j = η−bj = 0. By Corollary 6.10, we

know that there exists distributions uj , vj ∈ L2xH−1θ , such that X · uj =

X · vj = 0 and uj1 = aj , vj1 = bj and ||uj ||L2xH−1θ6 C||aj ||L2 , ||vj ||L2

xH−1θ6

C||bj ||L2 . By Theorem 6.11, we know that the product wj = ujvj makessense a distribution, namely wj ∈ H−5(SM) (actually wj ∈ H−9/2−ε, forany ε > 0) satisfies X · wj = 0, wj =

∑k>2 w

jk, w

j2 = ajbj and ||wj ||H−5 6

C||aj ||L2 ||bj ||L2 . Note that the Fourier coefficients of wj are in C∞(SM)by (6.2), since the Fourier coefficients of uj and vj are in C∞(SM) (byCorollary 6.10). Then, w =

∑Nj=1 w

j satisfies the properties stated in thetheorem.

Thanks to this theorem, the argument detailed in the previous paragraphapply here with m = 2 (and using the injectivity of I0 in the end), thusshowing the

Theorem 6.13. — If (M, g) is an Anosov non-hyperelliptic surface,then I2 is s-injective.

Let us now explain how we can recover the general case of an Anosovsurface and remove the assumption of non-hyperellipticity :

Proof of Theorem 6.1. — Recall that (M, g) has genus g > 2. We admitthat for any integer n > 1, the hyperelliptic surface M admits a Galoiscover p : N → M of degree n, where N has genus n(g − 1) + 1. By takingn > 5, it is possible to ensure that N will not be hyperelliptic.Now, the metric g on M can be lifted on N so that p becomes a local

isometry and the geodesic flow will still be Anosov. The transport equationX ·u = f also lifts to X · u = f on N and by applying the previous theorem,we can conclude that u has degree one, and so does u.

Remark 6.14. — This problem has actually been solved very recently byC. Guillarmou in [11], using a rather different approach. Indeed, his proofrelies on semi-classical analysis and on the use of "good" Sobolev spaces,that is anistotropic Sobolev spaces (see [7] for a reference) which take intoaccount the splitting into stable and unstable subbundles. Such a techniqueallows to define properly the product of m distributions.


7. Injectivity of the X-ray transform on simple surfacesWe present in this section the proof of Paternain-Salo-Uhlmann [23] of

the injectivity of the X-ray transform on simple surfaces.

7.1. IntroductionWe are going to prove the

Theorem 7.1 (Paternain, Salo, Uhlmann, 2012). — Let (M, g) be asimple Riemannian surface. Then Im is s-injective for any m > 0.

As mentioned in a previous remark, when no assumption is made onthe curvature, the Pestov identity — in the form given in (7.1) — canno longer be used to conclude to the injectivity of the ray transform. Theproblem comes from the fact that we have no control on the sign of theterm −(KV u, V u) in the identity. Let us explain the fundamental idea ofPaternain-Salo-Uhlmann [23] at the root of their proof. In order to controlthe term −(KV u, V u) in Pestov identity, they introduce an attenuation A(which is a 1-form on SM) and compute a new Pestov identity, taking intoaccount this attenuation. This gives birth to a new term in the identity,whose sign can be easily controlled when taking the "good" attenuation A.We will explain this more formally but let us first begin with the Pestovidentity in presence of an attenuation.

7.2. Pestov identity in presence of an attenuationLet A be complex-valued 1-form on M . In the following, we will still

denote by A the smooth function Φ1(A) associated to A on SM . Therefore,given u ∈ C∞(SM), what we write Au is just the multiplication of u byΦ1(A). On M , d+A is a connection, where d denotes the flat connection.We denote by FA its curvature.

Proposition 7.2. — Let u ∈ C∞(SM). Then :(7.1)||(X+A)V u||2−(KV u, V u)+||(X+A)u||2−||V (X+A)u||2+(?FAV u, u) = 0

Remark 7.3. — The last term can be written :

(?FAV u, u) =+∞∑

k=−∞ik(?FAuk, uk)

Therefore, if A is chosen well enough such that i?FA > 0 (and big enough),we may obtain a strong positive contribution in the Pestov identity, thuscontrolling the term −(KV u, V u).


Démonstration. — Just like for the Pestov identity, the proof relies onthe commutant formula

||Pu||2 = (Pu, Pu) = (u, P ∗Pu) = (u, [P ∗, P ]u) + ||P ∗u||2,

where P = V (X +A) (and P ∗ = (X +A)V since A = −A), u ∈ C∞(SM).All we have to show is that

([P ∗, P ]u, u) = (KV u, V u)− ||(X +A)u||2 − (?FAV u, u)

The proof is rather long and tedious so we will only give the main steps.Note that we still confuse the 1-forms and their associated function via theapplication Φ1. It relies on the following lemmas, which are easy to obtainby some elementary computations :

Lemma 7.4. —

?FA = H ·A−X · (?A) + [?A,A]

Lemma 7.5. —[V,A] = − ? A

[V, ?A] = A

[X +A,H + ?A] = −KV − ?FA

Now, we have :

[P ∗, P ] = [XV, V X] + [AV, V X] + [XV, V A] + [AV, V A]

We already know by a previous computation that [XV, V X] = −X2 +V KV . Using a factorisation trick and the previous formulas, one can provethat :

[P ∗, P ] = −X2 + V KV + [V A,H] + [?A, V X] + [?A,AV ]

The three last brackets can be computed :

[V A,H] = −XA+ V [A,H]

[?A, V X] = −AX + V [?A,X]

[?A,AV ] = [?A,A]V −A2

Note in particular that [A,H], [?A,X] are the respective multiplication byH ·A,X · (?A). This yields to :

[P ∗, P ] = V KV − (X2 +AX +XA+A2) + [?A,A]V + V [A,H] + V [?A,X]

= V KV − (X +A)2 + (?FA − [A,H] + [?A,X])V + V [A,H]− V [?A,X]

= V KV − (X +A)2 + ?FAV + V [A,H]− [A,H]V + [?A,X]V − V [?A,X]


The last four terms can actually be written V · (X · ?A−H ·A) but accor-ding to the previous lemma H ·A−X · (?A) is in Ω0 so its V -derivative iszero. Taking the inner product, we obtain the sought identity.

Let us also mention a lemma which will be useful in the sequel. It isa generalization to the case with attenuation of the inequality for u ∈C∞(SM), u|∂(SM) = 0 :

||XV u||2 − (KV u, V u) > 0

Lemma 7.6. — If (M, g) is simple, u ∈ C∞(SM), u|∂(SM) = 0, then :

||(X +A)(V u)||2 − (KV u, V u) > 0

The proof of this Lemma is actually completely similar to that of theinequality without attenuation (using the solutions of the Riccati equation).

7.3. End of the proofIn order to prove the injectivity of the ray transform, we already ex-

plained that one has to prove that given f ∈ Rm, the transport equationX · u = −f in SM and u|∂(SM) = 0 admits a solution u ∈ Rm−1. Thisproblem is solved by the two following propositions :

Proposition 7.7. — Let (M, g) be a simple surface, and assume thatu ∈ C∞(SM) satisfies X ·u = −f in SM with u|∂(SM) = 0. If m > 0 and iff ∈ C∞(SM) is such that fk = 0 for k 6 −m−1, then uk = 0 for k 6 −m.

Proposition 7.8. — Let (M, g) be a simple surface, and assume thatu ∈ C∞(SM) satisfies X · u = −f in SM with u|∂(SM) = 0. If m > 0 andif f ∈ C∞(SM) is such that fk = 0 for k > m+ 1, then uk = 0 for k > m.

Now, let us explain the heuristic behind these results. f and u are smoothfunctions SM → C and thus can be seen as sections of the trivial bundleE = SM×C. Therefore, on can write the transport equation as∇0

Xu = −f ,where∇0 = d is the flat connection on the trivial bundle E. Any connectionon E is of the form ∇Γ = d + Γ, where Γ is a complex-valued 1-form onSM . Consider a complex-valued 1-form A on M . It can be pulled back viathe projection π : SM → M to A = π∗A. Thus ∇A is a connection on E.For c ∈ C∞(SM), one has :

∇AX(cu) = c(X · u) + u∇AX(c) = −cf + u(X · c+ cA(X))

If c is a non-vanishing function on SM and if A = −c−1dc, then the previousequality reduces to :

∇AX(cu) = −cf


For x ∈ M , dπ(x,v)(X) = v so A(X) = A(v). Therefore, by identifyingA with the function Φ1(A) defined on SM , the previous equality can bewritten :

(7.2) (X +A)(cu) = −cf

In our reasoning, the 1-form A will be prescribed since we want to obtaini ? FA > 0. We therefore need to find a function c such that A = −c−1dc.Note that we require c to be holomorphic (or either antiholomorphic). In-deed, when using Pestov identity with attenuation, we will obtain somehowconditions on the Fourier coefficients of cu. In order to recover conditionson the Fourier coefficients of u, we need at least a certain control on c, andthe least we can ask is that c be holomorphic.If we set c = ew for some w ∈ C∞(SM), then equality (7.2) becomes

(7.3) X · w = −A

Now, the following theorem is the key to the proof of the injectivity of theray transform. We postpone its proof to the next section :

Theorem 7.9. — Let (M, g) be a simple surface. If A is a smooth 1-formon M , then there exists a holomorphic w ∈ C∞(SM) and an antiholomor-phic w ∈ C∞(SM), such that X · w = X · w = A.

Remark 7.10. — Actually, one can lower the hypothesis on this theoremto obtain the following statement — which we will not need in our case (itcan be found in [23]) :Theorem 7.11. — Let (M, g) be a compact nontrapping surface with

strictly convex boundary and assume that I∗0 is surjective. If A is a smooth1-form on M , then there exists a holomorphic w ∈ C∞(SM) and an anti-holomorphic w ∈ C∞(SM), such that X · w = X · w = A.Our statement of the Theorem 7.9 comes from the fact that a simple

manifold is nontrapping and that I∗0 is surjective on such a simple surface(see the next paragraph).

We will only prove Proposition 7.8 since the proof of Proposition 7.7 istotally equivalent.

Proof of Proposition 7.8. — Note that we can always reduce the proofto the case when f is even or odd. Indeed, we can decompose the transportequation with respect to the odd and the even parts, which gives thatX · u∓ = −f± in SM , u∓|∂(SM) = 0. In the sequel, we assume that f iseven (so u is odd), and the proof of the other case is similar.


We fix s > 0. Let us consider ϕ, the real-valued 1-form defined on M

such that dϕ = dvol (3) . We define A = −isϕ. Therefore i ? FA = s > 0since dA = −is · dvol. By Theorem 7.9, we know that we can find anantiholomorphic w ∈ C∞(SM) such that X ·w = iϕ and we can assume wis even. By the previous remarks, we obtain that the functions u := eswu

and f := eswf satisfy (X + A)u = −f in SM , with u|∂(SM) = 0. Since wis antiholomorphic and even, esw is antiholomorphic and even too, fk = 0for k > m + 1 and f is even. Since u is odd and m is even, we know thatum = 0. We define

v :=∞∑

k=m+1uk

and our aim is to show that v is zero on SM , which will imply the soughtresult. We already know that v ∈ C∞(SM), v|∂(SM) = 0 and v is odd. Mo-reover, for k > m+ 2, we have ((X +A)v)k = ((X +A)u)k = fk = 0. Fork 6 m− 1, we have ((X +A)v)k = 0 since vk = 0 for k 6 m. By assump-tion, m is even and v is odd so (X +A)v is even and ((X +A)v)m+1 = 0.Therefore, the only non-vanishing Fourier coefficient is ((X +A)v)m and :

(X +A)v = µ−vm+1, v|∂(SM) = 0

We now apply the Pestov identity with attenuation (7.1) A to v :

||(X+A)V u||2−(KV u, V u)+||(X+A)u||2−||V (X+A)u||2+(?FAV u, u) = 0

By Lemma 7.6, since (M, g) is simple and v|∂(SM), we know that :

||(X +A)V u||2 − (KV u, V u) > 0

Moreover :

(?FAV v, v) =∞∑

k=m+1ik(?FAuk, uk) = s

∞∑k=m+1

k||uk||2 > 0

Now, the remaining two terms are easy to compute

||(X +A)u||2 − ||V (X +A)u||2 = (1−m2)||µ−vm+1||2,

and using Pestov identity, it is clear that it is non-positive. Moreover, ifm = 0, 1 the conclusion is immediate, so we may assume from now on thatm > 2. In order to conclude, we use the same iterating trick in the spirit of

3. This is always possible since M is simple and therefore diffeomorphic to R2 (andthus H1(M,R) = 0)


Guillemin and Kazhdan’s proof. First, note that for all k ∈ Z (this is theequivalent of equation (3.15) :

||µ+vk||2 = ||µ−vk||2+ i

2(KV vk+(?FA)vk, vk) = ||µ−vk||2+s

2 ||vk||2−k2 (Kvk, vk)

For k > m+ 1, we have

µ+vk−1 + µ−vk+1 = ((X +A)v)k = fk = 0,

thus giving ||µ+vk−1||2 = ||µ−vk+1||2 (this is the equivalent of equation(5.2)). As a consequence :

||(X +A)u||2 − ||V (X +A)u||2 = (1−m2)||µ−vm+1||2

= (1−m2)(||µ+vm+1 ||2 −

s

2 ||vm+1||2 + m+ 12 (Kvm+1, vm+1)

)= .(1−m2)

(||µ−vm+3 ||2 −

s

2 ||vm+1||2 + m+ 12 (Kvm+1, vm+1)

)= (1−m2)

(||µ+vm+3 ||2 −

s

2(||vm+1||2 + ||vm+3||2

)+(m+ 1

2 (Kvm+1, vm+1) + m+ 32 (Kvm+3, vm+3)

))We iterate this process. Since µ−v ∈ L2(SM), we have ||µ−vk ||2 → 0.

Moreover, we know that this term is non-positive and m2 − 1 > 0. Thus :

−s2 ||v||2 +

∑ k

2 (Kvk, vk) > 0

Since K is bounded, by letting s→∞, we obtain that v = 0. Thus, uk = 0for k > m and uk = 0 too, since esw is antiholomorphic.

7.4. Surjectivity of I∗0

We recall that I∗0 is defined by

I∗0 :

∣∣∣∣∣ L2(∂+(SM), µ)→ L2(M)

h 7→(x 7→ I∗0h(x) =

∫Sxhψ(x, v)dSx

),,

and we are here concerned with its restriction I∗0 : C∞α (∂+(SM))→ C∞(M),where

C∞α (∂+(SM)) = h ∈ C∞(∂+(SM)), hψ ∈ C∞(SM)More precisely, we want to prove that

Theorem 7.12. — I∗0 : C∞α (∂+(SM))→ C∞(M) is surjective.


Recall that I∗0 I0 is an elliptic pseudodifferential operator of order −1that is formally self-adjoint. We also know, by injectivity of I0 (see Section4.4.2) that it is injective. But since M is not a closed manifold, we cannotconclude immediately, thanks to the Fredholm operator theory, to the sur-jectivity of I∗0 I0 : C∞(M) → C∞(M). Moreover, had we known that I∗0 I0was surjective, this could not even allow us to draw any conclusion becauseit is rather unclear that, given u ∈ C∞(M), I0u ∈ C∞α (∂+(SM)). Indeed,according to Lemma 4.18, this is true if u has compact support in

M , but

not in the general case. We thus need to use an embedding trick.

NMe

M

(x; v)

γ(x; v)

'τ(x;v)(x; v)

'−τ(x;−v)(x; v)

Figure 7.1. The embedding of (M, g) into (N,h).

Démonstration. — Consider (N,h) a smooth Riemannian surface wi-thout boundary such that (M, g) embeds into (N,h). Consider a smoothtubular neighborhood of ∂M in N (see below), which we denote byMe. Wedenote by J0 the ray transform on Me. We take χ, a cut-off function thatis constant equal to 1 on M (and even on a compact K which is strictlylarger than M), and with support in

Me. The operator χJ∗0J0χ is elliptic

on supp(χ). We will use the following

Lemma 7.13. — On a closed manifold N , for any m ∈ R, there exists aself-adjoint and invertible pseudo-differential operator P such that σ(P )(x, ξ) =|ξ|mg−1 .


Proof of the lemma. — Consider B = Op(|ξ|m/2). Then B∗B is self-adjoint with principal symbol |ξ|m. It is elliptic, thus Fredholm on N , withindex 0. Assume that it has a non-trivial kernel (finitely dimensional sinceit is Fredholm) and let us call ϕ1, ..., ϕN an orthonormal basis of thekernel. We consider the operator P = B∗B + Πker B∗B , where

Πker B∗B =N∑i=1〈ϕi, u〉ϕi,

is the orthogonal projection on the kernel (it is self-adjoint on L2(N)).By construction, P is invertible. Moreover, the perturbation of Πker B∗B issmooth since the ϕi are smooth and it is finite dimensional (thus compact).

We take P = B2, defined on N , with B invertible, self-adjoint andσ(B) = |ξ|−1/2, so that σ(P ) = |ξ|−1. We now define :

Q = χJ∗0J0χ+ (1− χ)B2(1− χ)

Q is clearly self-adjoint with principal symbol

σ(Q)(x, ξ) =(χ2(x) + (1− χ(x))2) |ξ|−1

and is therefore elliptic of order −1. Let us prove that it is injective. Thisis rather formal since Qu = 0 implies, taking the L2-scalar product withu :

||J0χu|| = 0, ||B(1− χ)u|| = 0

By injectivity of J0 and B, we conclude that u = 0 on both supp(χ) andsupp(1− χ), that is on N . Since Q — defined on the closed manifold N—is Fredholm of index 0, we deduce that Q is surjective.

Consider h ∈ C∞(M). We can extend it in order to obtain a smoothfunction h ∈ C∞(N) with support in χ−1(1) ⊂

Me. Now, we know that

there exists a unique smooth function f ∈ C∞(N) such that

(7.4) Qf = χJ∗0J0χf + (1− χ)B∗B(1− χ)f = h

Let us call w = J0χf . Since χf is smooth and supported inMe, we know

that w is smooth on ∂+(SMe) and wψ ∈ C∞(SMe) (which means, in otherwords, that w ∈ C∞α (∂+(SMe)). The function wψ is constant along thegeodesics (see the figure). We now define on ∂+(SM), w = wψ|∂+(SM).Then, by construction, w ∈ C∞(∂+(SM)) and wψ = wψ|SM ∈ C∞(SM),that is w ∈ C∞α (∂+(SM)).


Now, we go back to (7.4). On M , by construction, the second term va-nishes since χ ≡ 1 on supp(h) and we are left with

J∗0J0χf(x) = J∗0 w = I∗0w = h(x),

which concludes the proof.

As a corollary, we obtain Theorem 7.9, which we state once again herefor the reader’s convenience.

Corollary 7.14. — Let (M, g) be a simple surface. If A is a smooth1-form on M , then there exists a holomorphic w ∈ C∞(SM) and an anti-holomorphic w ∈ C∞(SM), such that X · w = X · w = A.

Démonstration. — Both cases are analogous, so we only deal with theholomorphic case. M being simply connected, the 1-form A can be decom-posed into A = da+ ?db, where a, b ∈ C∞(M). By Lemma 3.12, we obtain,going back to functions on SM : A = X · a+H · b. We can always replacew by w − a so we can now assume that A = H · b.The trick is to try a holomorphic solution of the form w = (Id + iH)w,

for some even w. Then, by Lemma 3.7 :

X · w = (Id+ i H)X · w − [H, X]w = (Id+ iH)X · w − iH · w0

It would be sufficient to find a w invariant by the geodesic flow, i.e. suchthat X ·w = 0 and such that w0 = −ib. Now, remember that I∗0 is surjectiveby the previous theorem and if h ∈ C∞(∂+(SM)), I∗0h(x) = 1

2π (hψ)0(x).Therefore, we take h ∈ C∞(∂+(SM)) such that I∗0 = −2πib and considerthe function hψ on SM . It satisfies X · hψ = 0, (hψ)0 = −ib. In order toconclude, we take w = (hψ)+.

7.5. Application to the deformation boundary rigidity problem

In this section, we deduce the deformation boundary rigidity of a simplemanifold from Theorem 7.1. This can be seen as a linearized version of theboundary rigidity property (see [26] or [20] for further details). We recallthe

Definition 7.15. — We say that a Riemannian manifold with boun-dary (M, g) is deformation boundary rigid if any smooth (or at least C1)family of metrics (gs)s∈(−ε,+ε), such that dgs = dg on ∂M × ∂M andg0 = g, is trivial, that it there exists a smooth isotopy (ψs)s∈(−ε,ε) suchthat ψ0 = Id, ψs|∂M = Id and ψ∗sg0 = gs.


Theorem 7.16. — A simple Riemannian surface is deformation boun-dary rigid.

This can be seen as a linearization of the boundary rigidity problem, thatis, proving that if dg and d′g agree on ∂M , then there exists a diffeomor-phism ψ such that ψ|∂M = Id and ψ∗g′ = g. The following result is provedin the next paragraph.

Theorem 7.17 (Pestov, Uhlmann, 2003). — A simple Riemannian sur-face is boundary rigid.

Note that the property of being simple for a Riemannian manifold (M, g)is open, that is a small C2 deformation of the metric will still be simple.We call simple a deformation (gs) such that the (M, gs) are simple. Theproof of Theorem 7.16 only relies on the following fact :

Proposition 7.18. — Let (M, g) be a simple Riemannian manifold.If I2 is s-injective for any simple deformation, then (M, g) is deforma-tion boundary rigid. If (M, g) is deformation boundary rigid, then I2 iss-injective.

Démonstration. — We assume that I2 is s-injective. Consider a smoothdeformation (gs)s∈(−ε,+ε) such that ds = d0 on ∂M × ∂M . We define βs =∂gs∂s

. Now, consider x, y ∈ ∂M and γs the unique (since (M, gs) is simple)geodesic in M joining x to y under gs, parametrized by arc-length. Wedefine T := d0(x, y). Since ds = d0 on ∂M × ∂M by assumption, it isimmediate that all the γs are defined on [0, T ]. For γ : [a, b] → M , anypiecewise C1 curve, we define the usual energy functional

Es(γ) :=∫ b

a

|γ(t)|2sdt

It is clear that Es(γs) = T so by differentiating with respect to s, weobtain :

0 = ∂T

∂s

∣∣∣∣s=0

= dEs(γs)ds

∣∣∣∣s=0

=∫ T

0

dgsds

∣∣∣∣s=0

(γ0(t), γ0(t))dt+ dE0(γs)ds

∣∣∣∣s=0

Since (γs)s∈(−ε,+ε) is a variation of γ0 which is a geodesic for g0, the lastterm vanishes. Therefore, we obtain that I2(β)(γ0) = 0. Since x and y werearbitrary, we can conclude that I2(β0) = 0. Of course, this also holds forany s, namely I2(βs) = 0. By assumption, I2 is s-injective, so there existsa smooth family 1-form (αs)s∈(−ε,+ε) such that βs = dαs. Setting Ys = α[sand integrating the family of vector fields (Ys), we obtain a smooth family


of diffeomorphisms (ψ)s such that ψ∗sg0 = gs, just like in the proof of Theo-rem 5.4.

Assume (M, g) is deformation boundary rigid. Consider a smooth sym-metric 2-tensor β such that I2(β) = 0. We are going to prove that β is apotential tensor. Let us consider a smooth deformation (gs)s∈(−ε,+ε) such

that g0 = g and β = ∂gs∂s

∣∣∣∣s=0

. By repeating the previous argument back-

wards, it is clear that ds = d0 on ∂M × ∂M . Thus, by assumption, thereexists a smooth family of diffeomorphisms (ψs)s∈(−ε,ε) such that ψ∗sg0 = gs.

We define Ys := ∂ψs∂s

. Then, using the same computation as in the proofof Theorem 5.4 :

βx(v, w) = ∂

∂s

∣∣∣∣s=0

g0(dψs(v), dψs(w)) = g0(∇vY,w) + g0(∇wY, v)

Setting α = Y [, we obtain that β = dα. This also proves Theorem 7.16.


8. The boundary rigidity property

This paragraph relates a theorem of L. Pestov and G. Uhlmann provedin [26]. For the reader’s conveniance, let us state once again this result :

Theorem 8.1 (Pestov, Uhlmann, 2003). — A simple Riemannian sur-face is boundary rigid.

We thus consider two simple Riemannian surfaces (M, g1) and (M, g2)such that dg1 and dg2 agree on ∂M × ∂M . Let us first introduce somenotations that have not been used so far.

8.1. Notations

Let us define on the boundary ∂(SM) the odd part of τ , given by

τ−(x, ξ) = 12 (τ(x, ξ)− τ(x,−ξ)) ,

for (x, ξ) ∈ ∂(SM). It is zero on ∂−(SM) and actually smooth on ∂(SM)(the potential singularity in ∂0(SM) is killed by the antisymmetrization).

Definition 8.2. — The scattering relation α : ∂(SM) → ∂(SM) isgiven by :

α(x, ξ) =(ϕ2τ−(x,ξ)(x, ξ), ϕ2τ−(x,ξ)

)It actually defines a diffeomorphism α : ∂(SM) → ∂(SM) such that

α : ∂−(SM) → ∂+(SM), α : ∂+(SM) → ∂−(SM), α|∂0(SM) = Id∂0(SM)and α is an involution. In other words, it maps an inward pointing vectorξ on ∂M to the outward pointing vector η on ∂M obtained by looking atthe exit point of the geodesic passing through (x, ξ).Given f ∈ C∞(M), its gradient ∇f (which is given by df ], where ] is the

musical isomorphism) can also be obtained by XXADD DEFINITION OPERATORS A

8.2. A first reduction

A reference for this paragraph is [10], Chapter 2.

Proposition 8.3. — Assume (M, g1) and (M, g2) are two simple Rie-mannian surfaces such that dg1 and dg2 agree on ∂M × ∂M . Then, thereexists a diffeomorphism ψ : M → M , such that ψ|∂M = Id and ψ∗g2 = g′2and αg1 = αg′2 .


Note that this requires to be accurate because even though dg1 = dg2 , itmay not be obvious that the two unit tangent bundles agree (on ∂M), andthere is no reson for it to be the case. We thus need the

Lemma 8.4. — Assume (M, g1) and (M, g2) are two simple Riemanniansurfaces such that dg1 and dg2 agree on ∂M × ∂M . Then, there exists adiffeomorphism ψ : M → M , such that ψ|∂M = Id and ψ∗g2 = g′2 andg1 = g′2 on T∂MM × T∂MM (that is on TxM × TxM , for any x ∈ ∂M).

Démonstration. — First, consider x ∈ ∂M and ξ ∈ Tx(∂M), and γ acurve in ∂M adapted to (x, ξ). There exists a unique g1-geodesic (resp.g2-geodesic) running from x to γ(s) and :

|ξ|g1 = lims→0

dg1(x, γ(s))s

= lims→0

dg2(x, γ(s))s

= |ξ|g2

By polarization, this implies that g1 and g2 agree on Tx∂M × Tx∂M . Butthere is still one direction missing. Let us denote by ν(x) the inward nor-mal unit vector field defined for x ∈ ∂M . As seen before, the applicationΨi : ∂M → [0, εi) → M,Ψi(x, t) = expix(tνi(x)) for i ∈ 1, 2 is a diffeo-morphism on some collar neighborhood of ∂M and thus ψ = Ψ2 Ψ−1

1 is adiffeomorphism from an annulus to another annulus (in a vicinity of ∂M)such that ψ|∂M = Id. Now, it is possible to extend ψ to a whole diffeomor-phism of the manifold M (this is a rather classical statement is the theoryof dynamical systems, see [10], Chapter 3 for instance). We claim that ψsatisfies the required properties. Indeed, consider a geodesic γ1(t) adaptedto the point (x, ν1(x)) and a geodesic γ2 adapted to (x, ν2(x)). Then, wehave γ2(t) = ψ(γ1(t)) and differentiating at t = 0, we obtain :

dxψ(ν1(x)) = ν2(x)

As a consequence, setting g′2 = ψ∗g2, we have for v ∈ Tx∂M :

g′2(v, ν1(x)) = g2(dxψ(v), dxψ(ν1(x))) = g2(v, ν2(x)) = 0,

where the second equality comes from the fact that dxψ = Id on Tx∂M . Asa consequence, since TxM = Tx∂M ⊕⊥Rν(x), we obtain that g′2 = ψ∗g2 =g1 on T∂M M × T∂MM .

We can now go on to proving the previous Proposition.Démonstration. — From the previous lemma, we can assume that g1 =

g2 on T∂M M × T∂MM . Consider (x, v1) ∈ ∂+(S1M) (= ∂+(S2M) by theprevious lemma) and denote by (y, w1) = α1(x, v1). We denote by γ1

x,y

the unique g1-geodesic joining x to y and we necessarily have γ1x,y(0) =


v1, γ1x,y(τ1(x, v1)) = w1. Let us denote by γ2 the unique g2-geodesic joining

x to y. We want to prove that :

v2 = γ2x,y(0) = v1, w2 = γ2

x,y(τ2(x, v2)) = w1

Consider the two smooth functions ri : M → R defined by ri(p) = di(x, p),for i ∈ 1, 2. Now, we have ∇r1(y) = w1. Indeed, these two vectors arecolinear because they are both orthogonal to the spheres r1 = cst : thisstems from Gauss lemma for w1 and by definition of the gradient for ∇r1.But these vectors are both unitary and pointing outward the manifold, sothey are equal. This also holds for ∇r2(y) = w2.Consider hi = ri|∂M . Then, ∇hi(y) is the projection of ∇ri(y) (which

belongs to the hemisphere ∂−(SM)) onto Ty(∂M) (see the figure below).Noe in particular that ∇hi determines ∇ri. But we have h1 = h2 by as-sumption, so ∇h1(y) = ∇h2(y) and thus ∇r1(y) = w1 = ∇r2(y) = w2.Considering the geodesics backwards (with initial vectors −w1,−w2), weobtain in the same fashion that v1 = v2.

In other words, we have proved that if d1 = d2 on ∂M ×∂M and g1 = g2on T∂MM × T∂MM , then α1 = α2.

8.3. The scattering relation determines the Dirichlet-to-Neumannmap

In this paragraph, we are going to prove that the scattering relationdetermines the Dirichlet-to-Neumann (DN) map and even, more precisely,the traces of conjugate harmonic functions on the boundary. The DN mapis defined by the following procedure. Consider f ∈ C∞(∂M) and h thesolution to the Dirichlet problem :

h|∂M = f

∆gh = 0, in M

Then, the DN map Λg is defined by

Λg(f) = 〈∇h, ν〉,

where ν is the normal unit outward vector field on ∂M . We are going toprove the

Theorem 8.5. — Assume (M, g1) and (M, g2) are simple manifolds.Then α1 = α2 implies that Λ1 = Λ2.


Now, consider h∗ the conjugate harmonic function of h that is the func-tion satisfying dh∗ = ?dh, where ? is the Hodge operator and denote byf∗ its trace on ∂M . It does exist since H1(M,R) = 0 (because M isdiffeomorphic to a closed disk) and ?dh is closed (because d ? dh = 0 isequivalent to ?d ? dh = ∆gh = 0). But we have :

Λg(f) = 〈∇h, ν〉 = dh(ν) = − ? dh∗(ν) = −dh∗(ν⊥) = −df∗(ν⊥),

where we assume that ∂M has been oriented andν, ν⊥

form a positively

oriented basis of T∂MM . Thus, if we can recover f∗ from f and the know-ledge of the scattering relation α, then it becomes clear that α determinesΛg.Also note, that according to Section 3.2.4, we can see conjugate harmonic

functions as smooth functions on SM (constant in the fibers, consideringtheir pullback) and the equality dh∗ = ?dh thus becomes X · h∗ = H · h.We first state the

Lemma 8.6. — Take w ∈ C∞α (∂+(SM)), then on ∂+(SM) :

2πA∗−H+A+w = IHI∗w

Démonstration. — For the different notations, we refer to the previoussections. We have I∗w = 2π(wψ)0 and thus

HI∗w = 2πH · (wψ)0 = 2π(H−X · wψ −X · H+wψ) = −2πX · H+wψ,

using the equality (3.1) and the fact that wψ is constant along the geodesics.And eventually, using w|∂(SM) = A+w, we obtain :

IHI∗w = −2πIXH+wψ = −2πA∗−H+A+w

Note that the operators A are known (they depend on the scatteringrelation α), as well as the operator H because we are on the boundary ∂Mand we assume that we know g on ∂M (and the Hilbert transform onlydepends on the metric). Thus, the lemma proves that the quantity IHI∗wonly depends on things we know (it does not depend on g in

M for instance,

which is unknown).Démonstration. — We know that g1 = g2 on T∂MM × T∂MM and α1 =

α2. Thus, we only have to show that we can determine the traces of conju-gate harmonic functions from the knowledge of g on ∂M and α. Both thesurjectivity of I∗ and the injectivity of I will appear to be crucial in theproof.Consider a smooth function h0

∗ on ∂M and denote by h∗ its harmoniccontinuation onM and h its conjugate harmonic function (such thatH ·h =


−X · h∗). Consider on ∂+(SM) a function w ∈ C∞α (∂+(SM)) such that2πA∗−H+A+w = −A∗−h0

∗. First, we have to prove that such a w exists.Since I∗ : C∞(∂+(SM)) → C∞(M) is surjective (see Theorem XX), thereexists at least one w such that I∗w = h. But then :

IHI∗w = −2πA∗−H+A+w = IH · h = −IX · h∗ = A∗−h0∗

As a consequence, we can find such a w and it only depends on the data gon ∂M and α (since the equation which defines it only depends on knowndata).

The converse is also true. Namely, if w satisfies the previous equation,then h = I∗w is a conjugate harmonic function to h∗, the harmonic conti-nuation of h0

∗. Indeed, we only need to prove that H · h = −X · h∗. But wehave :

IHh = IHI∗w = −2πA∗−H+A+w = A∗−h0∗ = −IXh∗

By injectivity of the X-ray transform, we obtain thatH ·h+X ·h∗ = X ·p forsome smooth p ∈ C∞(M) such that p|∂M = 0 and p is obviously harmonicso p ≡ 0. Now, h0 = I∗w|∂M = 2π(A+w)0 is the trace of the conjugateharmonic function h. So h0 is obtained from h0

∗ and the knowledge of g on∂M and α/

8.4. The DN map determines the conformal class

The reference for this paragraph is [?], where the following result is pro-ved.

Theorem 8.7 (Lassas-Uhlmann). — The DN map Λg determines theconformal class of the metric g.

In other words, this theorem states that given any f ∈ C∞(∂M) and uthe harmonic solution of the Dirichlet problem u|∂M = f , if one knows theimage Λg(f) = 〈∇u, ν〉 by the DN map, then it is possible to reconstruct theconformal class of the metric. This is actually the best result one can expectbecause if g and g′ are two conformal and simple metrics on the manifoldM , that is g′ = σg for some smooth σ > 0, then ∆g′u = ∆σgu = σ−1∆gu,which means that Λg = Λσg.

8.5. Conclusion of the proof

From the previous sections, we thus know that given (M, g1) and (M, g2)two simple Riemannian surfaces such that d1 = d2 on ∂M × ∂M , we can


find a diffeomorphism ψ such that ψ|∂M = Id and g1 = fψ∗g2, for somesmooth f > 0 such that f |∂M = 1.

TO BE CONTINUED


9. Conclusion

As a conclusion, let us sum up what is known so far about the injectivityof the X-ray transform. This résumé is partly taken from [24] :

(1) On simple manifolds of dimension n > 2 :— I0 is injective and I1 is s-injective : the proof relies on the exten-

sion of the Pestov identity to greater dimension and the use of a"0-control", like in Section 4.4.2,

— Im is s-injective for all m if n = 2 as we have proved it,— Im is s-injective for all m on manifolds of negative sectional cur-

vature (see [3] for instance), or under certain other curvature as-sumptions : the proof also relies on the use of Pestov identity likewe did in Section 5.

(2) On Anosov manifolds of dimension n > 2 :— I0 is injective and I1 is s-injective,— Im is s-injective for all m if n = 2 (see [11]) (we have proved it in

the cases m = 0, 1, 2) and the proof for m > 3 heavily relies onmicrolocal analysis,

— Im is s-injective for all m on non-positively curved manifolds asmentioned before,

— But it is not known whether Im is s-injective on Anosov manifoldsof dimension n > 3, and not even I2.


Annexe A. (Pseudo)differential operators

For this section, we refer to [2] or [29].

A.1. Differential operators

We consider (M, g) a n-dimensional smooth Riemannian manifold. LetE,F be two vector bundles over M . A linear operator P : Γ(M,E) →Γ(M,F ) is a differential operator of order d if in any local coordinates (xi),one can write :

Pu(x) =∑|α| 6d

aα(x)∂αu(x),

where α = (α1, ..., αk) ∈ 1, ..., nk, |α| = k, ∂α = ∂α1 ...∂αn and aα(x) is amatrix standing for an linear application Ex → Fx.

Its principal symbol is defined for x ∈M, ξ ∈ T ∗xM by :

σP (x, ξ) =∑|α|=d

aα(x)ξα,

with ξα = ξα1 ...ξαd , where ξ = ξidxi. It is a homogeneous polynomial in

the variable ξ (of degree d) with values in the linear application Ex → Fx.Actually, one can prove that this principal symbol is well defined, regardlessof any coordinate system. For instance, given f ∈ C∞(M), t ∈ R and u ∈Γ(M,E), one can check that

e−tf(x)P (etf(x)u(x))

is a polynomial of degree d in the variable t, whose monomial of degree dis given by :

tdσP (x, df(x))u(x)We have the two

Lemma A.1. —σPQ = σP σQ

Lemma A.2. — Given P : Γ(M,E) → Γ(M,F ) of order d, it has aformal adjoint P ∗, whose principal symbol is given by :

σP∗(x, ξ) = (−1)dσP (x, ξ)∗

We now introduce the main class of operators, for which the main theo-rem of this paragraph will be stated.


Definition A.3. — We say that P : Γ(M,E) → Γ(M,F ) is elliptic iffor any x ∈M, ξ ∈ T ∗xM with ξ 6= 0, the principal symbol σP (x, ξ) : Ex →Fx is injective.

Theorem A.4. — Assume rg(E) = rg(F ) and P : Γ(M,E)→ Γ(M,F )is an elliptic operator. Then ker (P ) is finite dimensional and there is anL2 orthogonal splitting :

C∞(M,F ) = ker (P ∗)⊕ P (C∞(M,E))

Note that ker (P ∗) is also finite dimensional since P ∗ is elliptic. Thus Pis a Fredholm operator of index :

ind(P ) = dim ker (P )− dim ker (P ∗)

In particular, a formally self-adjoint elliptic operator has index zero, andthe invariance of the Fredholm index under continuous deformation impliesthat any elliptic operator with the same symbol has also index zero.One can also prove that a differential operator P : Γ(M,E) → Γ(M,F )

induces continuous operators on Sobolev spaces as P : Hs+d(M,E) →Hs(M,F ), for any s ∈ R. Given U and V two open sets ofM , we will denoteU ⊂⊂ V the fact that there exists a compact K such that U ⊂ K ⊂ V .We have the following local elliptic estimate :

Lemma A.5. — Assume P : Γ(M,E)→ Γ(M,F ) is elliptic and considerU, V two open sets of M such that U ⊂⊂ V ⊂⊂ M . If u ∈ D′(M) is suchthat Pu ∈ Hs(M), then u ∈ Hs+m

loc (M) and for each σ < s + m, thereexists a constant C(U, V, s, σ) such that :

||u||Hs+m(U) 6 C(||Pu||Hs(V ) + ||u||Hσ(V )

)This is a rather difficult lemma, which can be proved by freezing the

coefficients of P , but we will omit its proof. In the case M is closed andcompact, this generalizes to the whole manifold :

Lemma A.6. — Assume M is closed compact, P : Γ(M,E)→ Γ(M,F )is elliptic. If u ∈ D′(M) is such that Pu ∈ Hs(M), then u ∈ Hs+m(M)and for each σ < s+m, there exists a constant C(s, σ) such that :

||u||Hs+m(M) 6 C(||Pu||Hs(M) + ||u||Hσ(M)

)A.2. Pseudodifferential operatorThe notion of pseudodifferential operators generalizes that of differential

operators. We refer to [29] for further details. On a compact Riemanniann-dimensional manifold (M, g), let us introduce the symbol class Sm(M) :


Definition A.7. — The symbol class Sm(M) of order m ∈ R consistsof C∞(T ∗M) complex-valued functions such that :

∀α, β, |∂αξ ∂βxp(x, ξ)| 6 Cα,β〈ξ〉m−|α|,

where 〈ξ〉 =√

1 + |ξ|2.

There are more general classes of symbols (with three indices Smρ,δ forinstance), but they are not used in this memoire, so we will not introducethem. Choosing a chart system and a related partition of unity on M , onecan define the pseudodifferential operator P = Op(p) related to the symbolp, acting on C∞(M) and defined locally by :

P : u 7→ (P u)(x) =∫ ∫

eiξ·(x−y)p(x, ξ)u(y)dydξ

The symbol p does not carry a geometrical meaning insofar as a change ofcoordinate systems will change the symbol. In other words, it is not definedintrinsically. However, this will only change p up to a symbol of subleadingorderm−1, which means that the principal symbol of p, denoted by σ(P ) =p mod Sm−1 is a well defined function of C∞(T ∗M), independent of thecharts.For example, if X is a vector field on M , the operator H = −iX admits

the symbol H(x, ξ) = 〈ξ,X〉. It does not depend on the charts, which issomething particular to operators of order 1. Let us state a few importantoperational properties on the symbols :

Proposition A.8. — (1) If p ∈ Sm, then P : Hs(M)→ Hs−m(M).(2) If p ∈ Sm1 , q ∈ Sm2 , then pq ∈ Sm1+m2 and the principal symbol of

P Q is the product σ(P )σ(Q) (it is an operator of order m1 + m2in particular).

(3) If p ∈ Sm, then P ∗ is of order m and has principal symbol σm(P ) ∈Sm.

(4) If p ∈ Sm1 , q ∈ Sm2 , then [P , Q] is of order m1 + m2 − 1 and hasprincipal symbol p, q mod Sm1+m2−2.

To an operator P of symbol p ∈ Sm, one can associate its kernel K,given in local coordinates by

K(x, y) = (2π)−n∫p(x, ξ)ei(x−y)·ξdξ

It is possible to prove rather sharp estimates on the kernel K, but let ussimply state the

Proposition A.9. — K is C∞ off the diagonal in M ×M .


The singularities of K along the diagonal actually determine the beha-viour of the operator P .Whereas the differential operators are local, the pseudodifferential are

not, which makes their definition more delicate on manifolds with boundary.This the reason why we use an embedding trick in the study of the operatorI∗0 I0.

Annexe B. Existence of isothermal coordinates on a surface

The reference for this part is [29].

We say that a smooth map ϕ between two Riemannian manifolds (M, g)and (N,h) is conformal if there exists λ ∈ C∞(M) such that ϕ∗h = e2λg.This is strictly equivalent to the fact that the application ϕ preserves theangle. We want to show the following theorem

Theorem B.1. — Let (M, g) be a Riemannian surface and p ∈ M .There exists a local coordinate system around p which is conformal, thatis there exists a neighborhood U ⊂ M of p, a chart ϕ : U → ϕ(U), and afunction λ ∈ C∞(ϕ(U)) such that ϕ∗(e2λ(dx2 + dy2)) = g.

We call this coordinate system an isothermal coordinate system. It israther useful since it simplifies a lot of computations when carried out inlocal coordinates. The previous definition clearly shows that the composi-tion of two conformal maps is still conformal. In particular, given two localconformal charts (U,ϕ), (V, ψ) on M around p, we see that Φ := ψ ϕ−1

is conformal where it is defined. Since an orientation-preserving conformalmap between two oriented open sets of the plane is holomorphic when seenas a function of the complex variable (and note that the converse is alsotrue), we conclude that Φ = f+ ig is holomorphic. Now, this also immedia-tely implies that f and g (which are respectively the real and the imaginarypart of Φ) are harmonic functions. As a consequence, the conformal mapϕ : U → ϕ(U) is harmonic (that is each of its coordinate is harmonic).Now, let us see that there exists somehow a converse to this fact. Assume

that we can find a map f : U → R which is harmonic on U , and such thatdfp 6= 0. We would like to find a function g : U → R such that ϕ = f + ig

is holomorphic (given an orthonormal basis in U , it satisfies the Cauchy-Riemann equations) and dfp and dgp are independent (and therefore df anddg will be independent in a vicinity of p). It is easy to see that ϕ is conformal(i.e. holomorphic) if and only if ?df = dg, where ? is the Hodge operator.


This is just a rewriting of the Cauchy-Riemann equations. Since we canalways assume U to be simply connected (if not, we can always shrink U sothat it becomes true), the existence of g is equivalent by Poincaré’s lemmato the fact that d ? df = 0 But this is also equivalent to ?d ? df = −∆f = 0and since f is assumed to be harmonic, the existence of g is guaranteed.Note that it is clear that dfp and dgp are independent since df = ?dg andwe assumed that dfp 6= 0. Thus, for a neighborhood V ⊂ U small enougharound p, dfx and dgx will still be independent for x ∈ V . In other words,we have obtained that (f, g) is a conformal coordinate system defined onV .Thus, the proof of Theorem B.1 reduces to proving the following

Proposition B.2. — Given p ∈ M , there exists a neighborhood U ⊂M around p and a real valued function f such that ∆f = 0 and dfp 6= 0.

Démonstration. — The proof mainly relies on a standard Dirichlet pro-blem. Indeed, choose a centered coordinate system ϕ : x 7→ (x1, x2) ina neighborhood U around p. In these coordinates, we have, according toequation (2.3) :

∆f(x) = gij(x)∂j∂kf + bk(x)∂kf,

for some smooth functions bk. Now, let us choose a disk D small enough inϕ(U) (of radius ε) centered at 0. We could define for instance f to be thesolution of the Dirichlet problem ∆f = 0 in D and such that f = x1 on∂D, but this will not guarantee that df0 6= 0. In order to ensure this, weconsider for ε > 0 the dilated coordinate system (xε1, xε2) = (x1/ε, x2/ε),sending the disk Dε of the original coordinate system to the unit disk in thedilated coordinate system. Note that in this coordinate system, we have :

∆εf(x) = gij(εx)∂j∂kf + εbk(εx)∂kf,

Consider the function fε which is harmonic in the unit disk D1 and equalto xε1 on the boundary (in the dilated coordinate system). We define thefunction vε such that ∆εvε = b1(εx) = b1ε(x) on the unit disk and vε = 0 onits boundary. Then we remark that fε = x1−εvε by unicity of the Dirichletproblem. Now let us see that for ε small enough, ∂1fε(0) = 1− ε∂1vε(0) isnot zero. Since vε = 0 on ∂D1, the Poincaré inequality provides :

||vε||2L2(D1) 6 C||∇vε||2L2 6 C|〈∆εvε, vε〉| = C|〈b1ε, vε〉| 6 C||vε||L2 ||b1ε||L2

Since b1ε is bounded in L2(D1) (and in each Hk(D1)) independently of ε,we obtain : ||vε||L2 6 C. Applying the global elliptic estimate (A.6), we


thus obtain :

||v||2Hk+1 6 C(||∆εvε||2Hk−1 + ||vε||2L2) = C(||b1ε||2Hk−1 + ||vε||2L2) 6 C

Since Hk(D1) → C1(D1) for k large enough, we obtain a uniform boundon the C1 norm of vε, independant of ε, which concludes the proof.

Annexe C. On Anosov flows

C.1. Definition

The reference for this section is mainly [20]. Let (M, g) be a Riemannianmanifold and φ : R×M →M a flow.

Definition C.1. — We say that a closed set Λ ⊂ M is φ-invariant(or simply invariant) is φt(Λ) ⊂ Λ for all t ∈ R. A closed invariant setΛ ⊂ N is called hyperbolic if there exists subbundles Es, Eu ⊂ TΛM =(x, v) ∈ TM, x ∈ Λ such that for all x ∈ Λ,

TxM = RX(x)⊕ Es(x)⊕ Eu(x)

, and for all t ∈ R,dφt(Es(x)) ⊂ Es(φt(x)),

dφt(Eu(x)) ⊂ Eu(φt(x)),and for all t > 0,

||dφt|Es || 6 Ce−µt,

||dφ−t|Eu || 6 Ce−µt,

where C, µ > 0 are constants.

Remark C.2. — Note that it is always possible to find a new metric g′on M such that the constant C is equal to 1.

Definition C.3. — If M is itself a hyperbolic set, then we say thatthe flow φ is Anosov. The subbundles Es and Eu are called the stable andunstable bundles respectively.

Not every manifold can carry an Anosov flow. Actually, this is such aspecific property that there is only a limited number of known examplesof manifolds which can carry an Anosov flow. For instance, the followingresult was proved by Ghys (see [9]) in 1984 :


Theorem C.4 (Ghys, 1984). — Let M be a closed three-dimensionalmanifold that is a circle bundle and φ : R×M →M an Anosov flow. Thenthere exists a closed surface N of genus g > 2 such that π : M → SN is afinite cover of SN and φ is continuously orbit equivalent to the lift to M ofthe geodesic flow on SN corresponding to a metric g0 of constant negativecurvature −1.

In our case, we are particularly interested in the situations when thegeodesic flow is Anosov on the unit tangent bundle. Ghys’ theorem provesin particular that the sphere and the 2-torus cannot carry any metric forwhich the geodesic flow is Anosov.

Definition C.5. — We define the stable and unstable manifold of φ atx as the sets :

W s(x) = y ∈ N, d(φt(x), φt(y))→t→+∞ 0

Wu(x) = y ∈ N, d(φt(x), φt(y))→t→−∞ 0

Remark C.6. — Actually, the term manifold is too ambitious in so faras these sets are not manifolds but embedded manifolds inM . For instance,on the two torus, given an initial direction with irrational angle, the traceof the geodesic flow starting from (0, 0) is a dense set of parallel straightlines, and therefore not a submanifold of the torus.It is also possible to require the convergence to be exponentially fast in

the definition of the previous sets.

We can also define the previous sets locally. For U ⊂M , we set :

W s(x, U) = y ∈ U, (φt(x), φt(y))→t→+∞ 0

Wu(x, U) = y ∈ U, d(φt(x), φt(y))→t→−∞ 0

Theorem C.7 (Local stable manifold theorem). — There exists a ε > 0such that for each x ∈M , the local stable and unstable manifolds

W sloc(x) = W s

loc(x,B(x, ε)), Wuloc(x) = Wu

loc(x,B(x, ε))

are embedded discs such that TxW sloc(x) = Es(x), TxWu

loc = Eu(x) for allx ∈M and

φt(W sloc(x)) ⊂W s

loc(φt(x)), φt(Wuloc(x)) ⊂Wu

loc(φt(x)),

for all t > 0.

In particular, one can check that for any neighborhood U(t) of φt(x), thefollowing equalities hold

W s(x) = ∪t>0φ−t(W sloc(φt(x), U(t)))


Wu(x) = ∪t>0φt(Wuloc(φ−t(x), U(t)))

and TxW s(x) = Es(x), TxWu(x) = Eu(x).We will not give the proof of this subtle theorem. It can be found in [15]

(Section 17.43).

C.2. The Anosov theorem

We now end this section with a few properties satisfied by Anosov flows.We first state Anosov’s celebrated result on the behaviour of the geodesicflow on a negatively curved manifold.

Theorem C.8 (Anosov, 1967). — If (M, g) is a closed Riemannian ma-nifold with negative curvature (K < 0), then the geodesic flow on SM isAnosov.

Even though we will not detail any proof here, we explicit the mainideas leading to this result. There are mainly two ways to prove Anosov’stheorem :

— The first one (see [15], Section 17.6) relies on the study of the geo-desic variations of a given geodesic γ, that is on the study on theJacobi fields along this geodesic. One can actually prove, thanks tothe hypothesis on the curvature and the use of Jacobi equation, thatthe vector space of normal (to γ) Jacobi fields along γ which vanishin γ(0) (this is a (n − 1)-dimensional vector space according to thetheory of Jacobi vector fields) splits in two vector spaces : one onwhich the Jacobi fields expand exponentially fast and one on whichthey contract exponentially fast. Then, from the Jacobi fields, it iseasy to recover the derivative of the exponential map on the manifold.

— The second proof is more subtle and relies on a theorem which westate here for cultural purposes. Since d(φt)x(X(x)) = X(φt(x)), thegeodesic flow on TM descends to the quotient TM , defined as vectorbundle over M such that TxM = TxM/RX(x), and defines a flow onTM .

Theorem C.9 (Wojtkowksi, 2000). — LetM be a closed manifoldand φ : R×M →M a non-singular flow with infinitesimal generatorX. We assume that there exists a quadratic form Q : TM → R onTM satisfying the following properties :— For each x ∈ M , the form Qx : Q|TxM : TxM → R depends

continuously on x.


— For all x ∈ M, v ∈ TxM,a ∈ R : Qx(v + aX(x)) = Qx(v). Thismeans that Q descends to the quotient bundle Q : TM → R.

— Q : TM → R is non-degenerate.— The Lie derivative LXQ must be continuous and if L denotes the

projection of the Lie derivative LXQ to TM (it is well definedaccording to the previous point), then L must be positive definiteon TM .

Then, φ is Anosov.

Thus, one only has to find a suitable quadratic form Q in order toprove the Anosov property of the geodesic flow. Somehow, this alsorelies on a proper use of Jacobi vector fields.

C.3. Livcic’s periodic theorem

We now assume that M is a closed compact manifold. We can now statethe

Theorem C.10. — Let f : M → R be a Ck function such that itsintegral over every periodic integral curve of X is zero. Then, there existsa Ck function u such that X · u = f .

Remark C.11. — This result still holds for the α-Hölder regularity. Na-mely, if f : M → R is α-Hölder such that its integral over every periodicintegral curve of X is zero, then there exists an α-Hölder function u whichis differentiable in the flow direction and such that X · u = f .

We will partially prove this result, namely we will only consider thecase k = 1. For higher regularity, this a very subtle question which wastackled in the 1980’s by De la Llave-Marco-Moriyón (see [4]). Using theanisotropic Sobolev spaces introduced in [7], C. Guillarmou (see [11]) wasable to extend Livcic’s theorem to the Sobolev regularity Hk(M), thus alsoproviding the Ck(M) regularity.The main idea in the proof presented below relies on Anosov’s closing

lemma. We refer to [15] (Corollary 18.1.8) for a proof :

Theorem C.12. — Let φ : R×M →M be a transitive Anosov flow ona closed manifold M . Then there exists ε > 0,K > 0, T0 > 0 such that iffor some T > T0, d(φT (x), x) < ε, then there exists a unique periodic pointp ∈ M with period T + τ such that max d(x, p), d(φT (x), p), |τ | 6 Kε

andW sloc(p)∩Wu

loc(x) 6= ∅. In fact, this unique point p satisfies in addition :

max d(x, p), d(φT (x), p), |τ | 6 Kd(φT (x), x),


and there exists a unique point z ∈ N such that :

W sloc(p) ∩Wu

loc(x) = z

From this, we obtain the following proposition :

Proposition C.13. — Let φ : R×M →M be a transitive Anosov flowon the closed compact manifold M and f : M → R an α-Hölder function.Then there exists ε > 0,K0 > 0, T0 > 0 such that if d(φT (x), x) < ε forsome T > T0, then there exists a closed orbit Γ with period T + τ for someτ > 0 such that :

(C.1)

∣∣∣∣∣∫ T+τ

0f(φt(p)) dt−

∫ T

0f(φt(x))dt

∣∣∣∣∣ 6 K0d(φT (x), x)α,

where p is some point in Γ.

Démonstration. — Note that we can always assume ε < 1 (the theoremis actually used with ε 1). By the previous theorem, there exists aperiodic point p of period T + τ such that max d(x, p), d(φT (x), p), |τ | 6K min(ε, d(φT (x), x)) andW s

loc(p)∩Wuloc(x) = z. Moreover, we also have

d(p, z) 6 d(x, p), d(φT (x), φT (z)) 6 d(φT (x), p). Decomposing the integral(C.1), we obtain :∣∣∣∣∣∫ T+τ

0f(φt(p)) dt−

∫ T

0f(φt(x))dt

∣∣∣∣∣6∫ T+τ

T

|f(φt(p))| dt+∫ T

0|f(φt(p))− f(φt(z))|dt+

∫ T

0|f(φt(z))− f(φt(x))|dt

The first term is bounded by

τ ||f ||∞ 6 K||f ||∞d(x, φT (x)) 6 K||f ||∞d(x, φT (x))α,

while the second is bounded by∫ T

0d(φt(p), φt(z))αdt 6

∫ T

0e−µαtd(p, z)αdt 6 1

µαd(p, x)α 6 K

µαd(x, φT (x))α

and the third by∫ T

0d(φt(x), φt(z))αdt 6

∫ T

0e−µαtd(φT (z), φT (x))αdt

61µα

d(p, φT (x))α

6K

µαd(x, φT (x))α


Démonstration. — We first prove Theorem C.10 in the α-Hölder case.Since u is differentiable in the flow direction and X · u = f , a necessarycondition on u is that

u(φt(x)) = u(x) +∫ t

0f(φs(x)) ds,

by the fundamental theorem of calculus.Since φ is Anosov, it admits at least a dense orbit Γ0. We take x0 ∈ Γ0

and define u : Γ0 → R by :

(C.2) u(φt(x0)) =∫ t

0f(φs(x0)) ds

We are going to prove that u is α-Hölder continuous on Γ0. Let us takex, y ∈ Γ0 such that d(x, y) < ε, where ε is the one provided by Proposition(C.13). We write x = φr(x0), y = φs(x0) and we assume that s > r. Wecan also assume that s − r > T0 since Γ0 is dense. Therefore, we haved(φs−r(x), x) < ε. By Proposition (C.13), we know that there exists aclosed orbit Γ such that :∣∣∣∣∫ s−r+τ

0f(φt(z0))dt−

∫ s−r

0f(φt(x))dt

∣∣∣∣ 6 K0d(φs−rx, x)α = K0d(x, y)α,

for some τ > 0, z0 ∈ Γ. By assumption, the first term is zero. Therefore,we obtain :

|u(x)− u(y)| =∣∣∣∣∫ s−r

0f(φt(x))dt

∣∣∣∣ 6 K0d(x, y)α

As a consequence, u is α-Hölder on Γ0 and, in particular, uniformly conti-nuous. Thus, there exists a unique extension u : Γ0 = M → M satisfying(C.2), α-Hölder on M , differentiable along the flow direction and such thatX · u = f .

We now assume that f is C1. We may take α = 1 in the previous para-graph, so u is Lipschitz on M . If x ∈M,y ∈W s(x), we have :

u(y)− u(x) = u(φt(x))− u(φt(y)) +∫ t

0(f(φs(x))− f(φs(y))) ds

By assumption, since the flow is Anosov and f is C1, we have

|f(φs(x))− f(φs(y))| 6 Cd(φs(x), φs(y)) 6 Ce−λsd(x, y),

which allows us to conclude that the integral in the previous line convergesas t→∞. Thus :

u(x)− u(y) =∫ ∞

0(f(φs(x))− f(φs(y))) ds


Now, consider a curve γ : (−ε,+ε) → W s(x) with γ(0) = x, γ(0) = v ∈Es(x) (= TxW

s(x)). Then :u(γ(r))− u(γ(0))

r= −1

r

∫ ∞0

(f(φs(γ(r)))− f(φs(γ(0)))) ds

The term in the integral converges to df(dφs(v)) as r → 0. Since dφs(v)→ 0converges exponentially fast to 0, we can pass to the limit in the integraland obtain :

d

dr

∣∣∣∣r=0

u(γ(r)) = −∫ ∞

0df dφs(v))ds

u is differentiable in the direction of the stable manifoldW s(x) (and a simi-lar argument applies toWu(x)) and the derivative is continuous. Therefore,u is C1.

Annexe D. Decomposition of symmetric tensors

D.1. Decomposition in potential and solenoidal parts

The reference for this paragraph is mostly [28]. We consider a smoothcompact Riemannian manifold (M, g) with boundary.

Theorem D.1. — Let k > 1 and m > 0 be integers. For every tensorfield f ∈ Hk(M,⊗mS T ∗M), there exists a unique fs ∈ Hk(M,⊗mS T ∗M)and a unique v ∈ Hk+1(M,⊗m−1

S T ∗M) such that :

f = fs + dv, δfs = 0, v|∂M = 0

Moreover, there exists a constant C such that :

||fs||Hk 6 C||f ||Hk , ||v||Hk+1 6 C||δf ||Hk−1

In particular, this theorem immediately implies Theorem 3.8 : if f issmooth, then so are fs and v.Démonstration. — Assume f ∈ Hk(M,⊗mS T ∗M) satisfies the previous

decomposition. Then δf = δfs + δdv = δdv. Setting

u = δf ∈ Hk−1(M,⊗mS T ∗M),

we are reduced to solving the problem :

(D.1) δdv = u, v|∂M = 0

If we prove, that there exists a unique solution to this equation satisfyingthe previous estimates, then setting fs = f − dv ∈ Hk(M,⊗mS T ∗M), wewill obtain the sought result.


Thus, we are going to study the differential operator

P = δd : Hk+1(M,⊗mS T ∗M)→ Hk−1(M,⊗mS T ∗M)

In particular, we are going to prove that it is elliptic of order 2 with zero ker-nel. Using Theorem A.4, this will imply unicity and existence of a solutionv to the problem (D.1). The estimate will be provided by the global ellipticestimate of Lemma A.6. Note that this estimate is still valid even thoughthe manifold has boundary because we consider a differential operator withDirichlet conditions on the boundary. The proof relies on a succession oflemmas.

Lemma D.2. — The principal symbol of the inner derivative

d : Hk(M,⊗mS T ∗M)→ Hk−1(M,⊗m+1S T ∗M)

is σd(x, ξ)u 7→ σ (ξ ⊗ u) (where the second σ denotes the operator of sym-metrization).

Proof of the lemma. — Let us do the computation in local coordinates.It is sufficient to do the computation for the operator ∇ since the resultis recovered by linearity of the operator of symmetrization σ. Moreover,∇, is a differential operator of order 1 and since we want to compute itsprincipal symbol, we can forget the part of the operator which is of order0. Let us denote by ∇ the homogeneous part of order 1 of ∇. If T =Ti1...imdx

i1 ⊗ ...⊗ dxim , then :

∇T = ∂Ti1...im∂xk

dxk ⊗ dxi1 ⊗ ...⊗ dxim =n∑k=1

ak(x)∂kT,

where ak(x) : Γ(M,⊗mS T ∗M) → Γ(M,⊗m+1S T ∗M) is the operator defined

by :ak(x)(dxi1 ⊗ ...⊗ dxim) = dxk ⊗ dxi1 ⊗ ...⊗ dxim

Therefore, we have :

σ∇(x, ξ) =n∑k=1

ak(x)ξk = ξ ⊗ ·,

which provides the sought result.

Lemma D.3. — The principal symbol of the divergence (the adjoint ofd)

δ : Hk(M,⊗m+1S T ∗M)→ Hk−1(M,⊗mS T ∗M)

is given by :σδ(x, ξ) = −(σd(x, ξ))∗ = −Cm(·)(ξ])


Here, Cm denotes the contraction according to the last factor, namely incoordinates :

Cm(dxi1 ⊗ ...⊗ dxim)(X) = dxim(X) · dxi1 ⊗ ...⊗ dxim−1

Proof of the lemma. — Recall that in coordinates, one has :

(δT )i1...im−1 = −∂Ti1...im−1j

∂xkgjk

Thus :δT =

∑k

bk(x)∂kT,

where bk(x) : Γ(M,⊗m+1S T ∗M)→ Γ(M,⊗mS T ∗M) is defined by :

bk(x)(dxi1 ⊗ ...⊗ dxim−1 ⊗ dxim) = −gimk(x)dxi1 ⊗ ...⊗ dxim−1

Thus its symbol is given by :

σδ(x, ξ)(T ) =∑k

ξkbk(x)T = −Cm(T )(ξ])

Lemma D.4. — The principal symbol of σδd is given by :

σδd(x, ξ) = σδ(x, ξ) σd(x, ξ) = − 1m+ 1 |ξ|

2Id+ m

m+ 1σd(x, ξ) σδ(x, ξ)

Démonstration. — This lemma is proved using the two previous identi-ties. Consider a symmetric m-tensor T = Ti1...imdx

i1 ⊗ ...⊗ dxim . Then :

σd(x, ξ) = σ(ξ ⊗ T ) = 1m+ 1

m∑k=0

Ti1...im dxi1 ⊗ ...⊗ ξ︸︷︷︸

k−fold product

⊗...⊗ dxim

So :

σδ(x, ξ) (σd(x, ξ)T ) = − 1m+ 1

m−1∑k=0

Ti1...imξlgimldxi1 ⊗ ...⊗ ξ ⊗ ...⊗ dxim−1

− 1m+ 1 |ξ|

2g−1T

On the other hand, one can compute :

σd(x, ξ) (σδ(x, ξ)T ) = − 1m

m−1∑k=0

Ti1...imξlgimldxi1 ⊗ ...⊗ ξ ⊗ ...⊗ dxim−1

Thus :σδ σd = − 1

m+ 1 |ξ|2g−1Id+ m

m+ 1σd σδ


Since d and δ are formally dual and of order 1, the symbol σδd operator isdefinite negative as long as ξ 6= 0. Thus it is elliptic and Fredholm accordingto Theorem A.4. But δd is formally self-adjoint, so its index is zero. Let usprove that it admits 0 as kernel. Consider v ∈ ker (δd) (it is smooth byellipticity of the operator). Then, since v|∂M = 0, one has :

〈δdv, v〉 = −〈dv, dv〉 = 0

So dv = 0. Thus, identifying the tensor v with its associated function inSM , one has v ≡ 0 by integrating dv = X · v along geodesics (and usingthe fact that v|∂M = 0).

Moreover, the estimates are provided by Lemma A.6. We have :

||v||2Hk+1 6 C(||δdv||2Hk−1 + ||v||2L2) = C(||δf ||2Hk−1 + ||v||2L2)

The Poincaré inequality (written for the function canonically associated tothe tensor v on SM and still denoted v) holds since v|∂M = 0 :

||v||2L2 6 C||dv||2 = C|〈δdv, v〉| = C|〈δf, v〉| 6 C||δf ||L2 ||v||L2

Thus : ||v||L2 6 C||δf ||L2 6 C||δf ||Hk−1 , which gives the second estimate.As to the first estimate, we have : fs = f − dv. Thus :||fs||Hk 6 ||f ||Hk + ||dv||Hk 6 ||f ||Hk + C||v||Hk+1 6 ||f ||Hk + C||δf ||Hk−1

6 C||f ||Hk

D.2. Decomposition in a vicinity of the boundary

Assume (M, g) is an n-dimensional compact manifold with boundary.Then, there exists an annulus ψ : [0, ε)×∂M →M such that ψ∗g = dr2+hr,where r ∈ [0, ε) and hr is a family of metrics on ∂M . Indeed, one canconsider

ψ(r, u) = expu(rν(u)),for ε strictly less than the radius of injectivity of exp on (M, g) and whereν is the inward unitary normal vector to ∂M . This gives the sought resultusing Gauss’ lemma.

Theorem D.5. — Given f ∈ C∞(M,⊗mS T ∗M), there exists, in a vici-nity of ∂M , h ∈ C∞(M,⊗m−1

S T ∗M) and p ∈ C∞(M,⊗mS T ∗M) such thath|∂M = 0 and i ∂

∂rp = 0 (where i denotes the interior product) and :

f = dh+ p,

where d = σ∇ is the symmetric covariant derivative.


For the sake of simplicity, and because it is the only case studied in thismemoire, we will only prove this result in the two-dimensional case. Thearguments given can be extend to any dimension, but their writing maybecome more tedious.

Démonstration. — Let us assume that such a decomposition exists. Inthe sequel, our arguments will show that it is indeed uniquely determined.We identify ∂M ' S1 and use the coordinates (r, θ) ∈ [0, ε)× S1. We writein short

f = f0(r, θ)drm + f1drm−1 ⊗ dθ + ...+ fmdθ

m,

h = h1drm−1 + h2dr

m−2 ⊗ dθ + ...+ hmdθm−1,

where by drk⊗dθm−k we actually mean its symmetrized σ(drk ⊗ dθm−k

).

Note that, since the tensors are symmetric, we know that the coefficientin front of each term containing the same number of dr (or dθ, whichis equivalent) will be equal. The fact that i ∂

∂rp = 0 simply means that

p = a(r, θ)dθm for some smooth function a. The fact that ∇ ∂∂r

∂

∂r= 0

implies that∇(dr) = α(dr ⊗ dθ + dθ ⊗ dr) + βdθ2

∇(dθ) = λ(dr ⊗ dθ + dθ ⊗ dr) + µdθ2

One can actually show, using the Koszul formula (2.5) that α = 0. Inparticular, ∇ applied on dr or dθ does not raise the number of dr in thetensor product.Now, we prove by induction that we recover h in a unique way. The

equation f = dh+ p projected on the coordinate drm provides :

f0(r, θ) = ∂h1

∂θ(r, θ)

Moreover, since h|∂M = 0, we have h1(0, θ) = 0 and thus, h1 is given bythe formula :

h1(r, θ) =∫ r

0f0(s, θ)ds

By the previous remark made on ∇(dr) and ∇(dθ), using the Leibniz for-mula for tensors, we see that the coefficient in front of the terms drm−1⊗dθin the equality f = dh+ p will be obtained by :

h1λ2 + ∂h1

∂rβ2 + h2γ2 + ω2

∂h2

∂r= f1,

where λ2, β2 and γ2 are smooth coefficients which can be expressed in termsof the α, β, λ, µ and some constants of symmetrization (independent of thetensors) and ω2 6= 0 is a constant of symmetrization. We already know h1


and since h2(0, θ) = 0, by intgegrating this ordinary differential equation,we obtain the formula :

h2(r, θ) = 1ω2

∫ r

0exp

(−∫ s

0

γ(u, θ)ω2

du

)(f1 − h1λ−

∂h1

∂rβ

)(s, θ)ds

Now, iterating this process, we see that we can always write the coefficientin front of the term drk ⊗ dθm−k in the equality f = dh+ p :

hkλk+1 + ∂hk∂r

βk+1 + hk+1γk+1 + ωk+1∂hk+1

∂r= fk,

with hk+1(0, θ) = 0. Thus, integrating this differential equation, we gethk+1. Iterating this process, we see that we can recover h and that it isunique.Setting p = f − dh, we see that the only term left will be of the form

a(r, θ)dθm, and p is unique.

Annexe E. The Riemann-Roch theorem

E.1. Holomorphic line bundles

Let L be a complex line bundle over an oriented Riemannian surface(M, g). We assume L is endowed with a hermitian metric and a metricconnection. A holomorphic structure on L is the choice of a covering of Mby a collection of charts ϕj : Uj → ϕj(Uj) ⊂ C and local sections sj overϕj(Uj) such that sk = ψjksj , where the ψjk are holomorphic.

We can look at the complexified tangent and cotangent bundles. Wedenote by T ∗CM = C⊗ TM∗. By Appendix B, we know that the orientedmanifold M admits local isothermal coordinates which, as we have seen,is equivalent to the existence of holomorphic charts. Thus, we can definein holomorphic coordinates dz = dx + idy and dz = dx − idy and this isindependent of the choice of holomorphic coordinates. We define κ to be theholomorphic line bundle generated by dz and κ the line bundle generatedby dz. In particular, we have :

T ∗CM = κ⊕ κ

If L is a holomorphic line bundle over M , then we have a naturallydefined operator

∂ : C∞(M,L)→ C∞(M,L⊗ κ),which we can define as follows. Given a local (nowhere vanishing) holomor-phic section S of L on an open set U ⊂M , then any arbitrary section u in


U is of the form u = vS for some complex-valued smooth function v. Weset :

∂u = ∂v

∂zS ⊗ dz,

and one can check that this is both independent of the choice of holomorphiccoordinates and of the local section S.If L is a holomorphic line bundle, then we can define its inverse L−1

(up to an isomorphism) such that L⊗ L−1 'M × C is the trivial bundle.Actually, since for any vector bundle E over M E ⊗E∗ ' End(E), we cantake L−1 = L∗, the dual vector bundle of L. Indeed, End(L) is trivial (it isisomorphic to the trivial bundleM×C) because there exists a non-vanishingsection (one can take id ∈ End(L) such that id(x) = idLx ∈ End(Lx)).

Note that in this setting, TCM = C ⊗ TM splits in two holomorphicline bundles, and we have, according to the paragraph above : κ−1 is theholomorphic line bundle generated by ∂/∂z = 1/2(∂/∂x− i∂/∂y) and κ−1

is the holomorphic line bundle generated by ∂/∂z = 1/2(∂/∂x + i∂/∂y)and

TCM = κ−1 ⊕ κ−1

On κ−1 the multiplication by i acts as the endomorphism J on TM (thecomplex structure) induced by the conformal class of the metric and theorientation.

E.2. The first Chern class

Let M be a Riemannian manifold and L be a holomorphic line bundleover M . Consider two connections ∇ and ∇′ on L, then since the space ofconnections is affine, ∇−∇′ ∈ Λ1(M,End(L)). But since End(L) is trivial(that is isomorphic to M × C), we can consider ∇ − ∇′ as an element ofΛ1(M,C), that is as a complex-valued 1-form on M . We write ∇−∇′ = α.

In the same fashion, one can prove that F∇ is a tensor (it is linear ineach argument), which means in other words that F∇ ∈ Λ2(M,End(L)) 'Λ2(M,C). Here, using the crucial fact that L is a complex line bundle,a computation yields to F∇ − F∇′ = dα. Now, since M is a surface, weautomatically have that F∇ is closed, that is dF∇ = 0 (actually, this keepsbeing true even in dimension greater than 2).As a consequence, [F∇], the class in de Rham cohomology of F∇, does

not depend on ∇ but only on L. We now endow L with a hermitian metricand consider the Levi-Civita connection ∇. Since ∇ is metric, it satisfiesthe relation

〈F∇(X,Y )s, t〉 = −〈F∇(X,Y )t, s〉,


and consider a non-vanishing local section s and t = s, one can show thatF∇ is imaginary or iF∇ is real. Thus, i

2πF∇ is a real-valued 2-form and

we define the first Chern class of L as the class in cohomology of [ i2πF∇],

which is independent of ∇. We define the first Chern number as

c1(L) =∫M

i

2πF∇

Now if L1 and L2 are two holomorphic vector bundles over M endowedwith a hermitian metric, the tensorial product L1⊗L2 is still a holomorphicline bundle. If we chose two unitary connections∇1,∇2, then they naturallygive rise to a connection ∇ = ∇1 ⊗∇2 defined on L1 ⊗ L2 by :

∇(s1 ⊗ s2) = ∇1s1 ⊗ s2 + s1 ⊗∇2s2

From this, and using the definition of the curvature (2.6), a computationleads to :

F∇(X,Y )s1 ⊗ s2 =(F∇1(X,Y ) + F∇2(X,Y )

)s1 ⊗ s2

Thus :c1(L1 ⊗ L2) = c1(L1) + c1(L2)

In particular, if 1 = M × C denotes the trivial bundle, then, taking theflat connection d, one has c1(1) = 0. Considering κ ⊗ κ−1 = 1 and theprevious equality, we obtain c1(κ) + c1(κ−1) = c1(1) = 0, that is :

c1(κ−1) = −c1(κ)

But c1(κ−1) can be computed thanks to the Gauss-Bonnet formula. In-deed, κ−1 is TM considered as a holomorphic line bundle. Thus, c1(κ−1) isnothing but the integral of the curvature, which is the Euler characteristicby the Gauss-Bonnet formula :

c1(κ−1) =∫M

F∇ = χ(M) = 2(1− g),

where ∇ is the metric connection associated to any metric g on M . Andwe obtain :

c1(κ) = −c1(κ−1) = 2(g − 1)

E.3. The Riemann-Roch formula

Let (M, g) be a compact connected Riemannian surface. We considerthe holomorphic line bundle Lk = κ⊗k over M . We recall that a functionu ∈ Ωk = C∞(SM) ∩ Hk (where Hk is the eigenspace associated to the


eigenvalue k of the operator −iV ) can be identified with a smooth sectionof Lk. As mentioned in a previous paragraph, we can define the operator

∂k : C∞(M,Lk)→ C∞(M,Lk ⊗ κ)

and the kernel O(k) = ker (Lk) of this operator consists of holomorphicsections of Lk. As explained in Section 3.4.1, the dimension of the kernelof the operator η− : Ωk → Ωk−1 is that of the kernel of ∂k.

Theorem E.1 (The Riemann-Roch formula). — If L is a holomorphicline bundle over the compact Riemannian surface M , then :

dim O(L)− dim O(L−1 ⊗ κ) = c1(L)− 12c1(κ)

The Riemann-Roch formula can be interpreted as a particular case of thegeneral Atiyah-Singer formula, linking the index of an elliptic pseudodiffe-rential operator to the topology of the manifold. In our case, the operatoris ∂L (which is elliptic). Let us explain how we can recover the dimensions

If L0 = 1 = M ×C denotes the trivial bundle then we have dim O(0) =1, since M is connected and the holomorphic functions on M are theconstants. It is a standard fact (but not obvious, stemming from Hodgetheory) that dim O(1) = g, where g denotes the genus of M . It can also beseen from the Riemann-Roch formula, using the fact that c1(κ) = 2(g− 1).Indeed, from the Riemann-Roch formula, taking L = κ−1, we obtain :

dim O(1) = dim O(0) + 12c1(κ) = 1 + (g − 1) = g

It is possible to prove, using arguments which involve meromorphic sec-tions on M for instance, that for k < 0, dim O(k) = 0. Thus, takingL2 = κ⊗ κ, the previous paragraph gives us c1(κ⊗ κ) = 2c1(κ) = 4(g − 1)and, using the fact that L−1

2 ⊗κ ' κ−1, the Riemann-Roch formula writes :

dim O(2) = dim O(−1)+c1(κ⊗κ)− 12c1(κ) = 0+4(g−1)−(g−1) = 3(g−1)

Iterating this computation :

dim O(k) = (2k − 1)(g − 1)


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Thibault LEFEUVREÉcole Polytechnique91128 PalaiseauFranceCurrent address:Université de Paris VIFaculté des sciences de Jussieu75005 [email protected]

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