A Geometric Transformation Theory for PDE

Pseudospherical Surfaces Proving Lie’s Theorem Geometric Exterior Differential Systems Future Directions Appendix

A Geometric Transformation Theory for PDE

M. NoonanCornell University


Table of Contents

1 Pseudospherical Surfaces

2 Proving Lie’s Theorem

3 Geometric Exterior Differential Systems

4 Future Directions

5 Appendix


Classical Motivation: Pseudospherical Surfaces

A surface in E3 is pseudospherical if it has constant Gaussiancurvature K = −1.

The graph of a function f : R2 −→ R defines a pseudosphericalsurface if and only if f satisfies the PDE

(∂2f∂x∂y

)2

−∂2f∂x2

∂2f∂y2 +

(1 +

(∂f∂x

)2

+

(∂f∂y

)2)2

= 0

Note that this equation fails to be even quasilinear — it can onlybe classified as “rather unpleasant”.





(∂2f∂x∂y

)2

−∂2f∂x2

∂2f∂y2 +

(1 +

(∂f∂x

)2

+

(∂f∂y

)2)2

= 0






(∂2f∂x∂y

)2

−∂2f∂x2

∂2f∂y2 +

(1 +

(∂f∂x

)2

+

(∂f∂y

)2)2

= 0



The Pseudosphere

By looking for solutions with a rotational symmetry, we canderive a new equation for the profile curve of a rotationallysymmetric pseudospherical surface.This leads to the classical pseudosphere:


Bianchi’s Theorem

Theorem (Bianchi)

Let f , f parametrize two surfaces in R3, and let N, N be thecorresponding normal maps. Suppose further that the fourrelations

|f − f | = 1N ⊥ NN ⊥ f − f

N ⊥ f − f

hold at each point. Then f and f are both parameterizations ofpseudospherical surfaces.


Bianchi’s Relations

The Bianchi relations are geometric in the sense that if f ∼ fand g is any Euclidean motion, g · f ∼ g · f .

Two surfaces are Bianchi-related at p, p exactly when thetangent planes are in the geometric configuration depictedbelow:

Figure: Two planes related by a 90 unit-distance screw motion.


Lie’s Backlund Transformation

Bianchi’s theorem describes a relation which can only existbetween pseudospherical surfaces. Can we use Bianchi’stheorem as a bridge to build new K = −1 surfaces from oldones?

Theorem (Lie)Let f parameterize a pseudospherical surface. Then thereexists a pseudospherical surface f which is Bianchi-related to f .The transformed surface f may be computed from f by solvinga series of ordinary differential equations.

Translation: Once you have one solution to the K = −1equations, it is “easy” to compute a new one!












Table of Contents




4 Future Directions

5 Appendix


Proving Lie’s Theorem with Lie GroupsEuclidean Frames

Let f : M −→ E3 be a parameterized surface.

DefinitionA Euclidean frame over f is a map F : M −→ ASO(3) such that

F ·O = f

where O is the origin of E3.

Using the standard representation of ASO(3), F must take theblock form

F =

[1 0f R

], R ∈ SO(3)


Proving Lie’s Theorem with Lie GroupsEuclidean Frames

Let f : M −→ E3 be a parameterized surface.

DefinitionA Euclidean frame over f is a map F : M −→ ASO(3) such that

F ·O = f

where O is the origin of E3.

Using the standard representation of ASO(3), F must take theblock form

F =

[1 0f R

], R ∈ SO(3)


Proving Lie’s Theorem with Lie GroupsAdapted Euclidean Frames

Definition

A Euclidean frame F =

[1 0f R

]is adapted if

Re3 = N

where N is the normal map of f .

Lemma

A Euclidean frame F is adapted if and only if e3(F−1dF) = 0.


Proving Lie’s Theorem with Lie GroupsAdapted Euclidean Frames

Definition

A Euclidean frame F =

[1 0f R

]is adapted if

Re3 = N

where N is the normal map of f .

Lemma

A Euclidean frame F is adapted if and only if e3(F−1dF) = 0.


Proving Lie’s Theorem with Lie GroupsTransformed Frames

Let βθ ∈ ASO(3) be a 90 unit-displacement screw motion inthe cos(θ)e1 + sin(θ)e2 direction.

Lemma

If f , f : M −→ E3 are Bianchi-related and F is an adapted frameover f , then there is a unique function θ : M −→ S1 such that

F = F · βθ

is an adapted frame over f .


Proving Lie’s Theorem with Lie GroupsWhen is F · βθ Adapted?

To find the function θ, we need to know when F · βθ is adapted.Let

F−1dF =

0 0 0 0τ1 0 λ ν1

τ2 −λ 0 ν2

0 −ν1 −ν2 0

Then F · βθ is adapted if and only if

0 = e3(Ad(β−1θ )(F−1dF) + β−1

θ dβθ)

= λ+ sin(θ)τ1 − cos(θ)τ2 − dθ

...but why should this equation have any solutions?


Proving Lie’s Theorem with Lie GroupsWhen is F · βθ Adapted?

To find the function θ, we need to know when F · βθ is adapted.Let

F−1dF =

0 0 0 0τ1 0 λ ν1

τ2 −λ 0 ν2

0 −ν1 −ν2 0

Then F · βθ is adapted if and only if

0 = e3(Ad(β−1θ )(F−1dF) + β−1

θ dβθ)

= λ+ sin(θ)τ1 − cos(θ)τ2 − dθ

...but why should this equation have any solutions?


Proving Lie’s Theorem with Lie GroupsA Consistency Condition

We have no reason to expect that a function θ exists which alsosatisfies

0 = λ+ sin(θ)τ1 − cos(θ)τ2 − dθ

But if there were such a θ, differentiating the above equationand using the relation d(F−1dF) = −F−1dF ∧ F−1dF gives

0 = dλ+ cos(θ)dθ∧ τ1 + sin(θ)dτ1

+ sin(θ)dθ∧ τ2 − cos(θ)dτ2

= ν1 ∧ ν2︸︷︷︸K |df |2

+ τ1 ∧ τ2︸︷︷︸|df |2

Note the miracle — all dependence on θ has vanished!








= ν1 ∧ ν2︸︷︷︸K |df |2

+ τ1 ∧ τ2︸︷︷︸|df |2









= ν1 ∧ ν2︸︷︷︸K |df |2

+ τ1 ∧ τ2︸︷︷︸|df |2



Proving Lie’s Theorem with Lie GroupsThe Underlying ODE

We can interpret the previous calculation as saying that theoverdetermined system of PDEs

dθ = λ+ sin(θ)τ1 − cos(θ)τ2

is consistent if and only if K = −1 on the surface f .

LemmaA consistent overdetermined system of first-order PDEs canalways be solved by a sequence of ordinary integrations.

This proves Lie’s theorem: if f is pseudospherical, then bysolving a sequence of ODEs we may find a secondBianchi-related pseudospherical surface f .


Proving Lie’s Theorem with Lie GroupsThe Underlying ODE

We can interpret the previous calculation as saying that theoverdetermined system of PDEs

dθ = λ+ sin(θ)τ1 − cos(θ)τ2

is consistent if and only if K = −1 on the surface f .

LemmaA consistent overdetermined system of first-order PDEs canalways be solved by a sequence of ordinary integrations.

This proves Lie’s theorem: if f is pseudospherical, then bysolving a sequence of ODEs we may find a secondBianchi-related pseudospherical surface f .


Application: Transforming the PseudosphereUnderlying Equations

Using the curvature-line adapted frame F on the pseudosphere:

∂θ

∂x= sin θ tanh x,

∂θ

∂y= (1 − cos θ) sech x

Starting with θ(0, 0) = π, the second equation may beintegrated along the curve x = 0 to obtain

θ(0, y) = π+ 2 tan−1(y)

Then integrate along each curve y ′ = 0 to get

θ(x, y) = π+ 2 tan−1 (y sech x)




∂θ


∂θ


Starting with θ(0, 0) = π,

the second equation may beintegrated along the curve x = 0 to obtain

θ(0, y) = π+ 2 tan−1(y)






∂θ


∂θ



θ(0, y) = π+ 2 tan−1(y)






∂θ


∂θ



θ(0, y) = π+ 2 tan−1(y)




Application: Transforming the PseudosphereResults

Using the θ constructed above, F = F · βθ is an adapted framefor a new pseudospherical surface f .

In this case we obtainKuen’s Surface, a very nontrivial surface with K = −1.


Application: Transforming the PseudosphereResults

Using the θ constructed above, F = F · βθ is an adapted framefor a new pseudospherical surface f . In this case we obtainKuen’s Surface, a very nontrivial surface with K = −1.


Table of Contents




4 Future Directions

5 Appendix


Homogeneous Spaces

Definition (Homogeneous Space)A homogeneous space is a smooth manifold M equipped with asmooth transitive left action of a Lie group G. We call G thestructure group of M.


Examples of Homogeneous Spaces

1 The round n-sphere Sn is a homogeneous space for therotation group SO(n + 1).

2 Euclidean n-space En is a homogeneous space for theEuclidean group ASO(n) = Rn n SO(n).

3 The conformal sphere CP1 is a homogeneous space forthe Mobius group PSL(2, C). Note that this is a distinctspace from S2, even though they are diffeomorphic!

4 If H is any Lie subgroup of G, then the coset space G/H isa homogeneous space with structure group G.




















Invariant Relations

An invariant relation on a homogeneous space M is a subset ofM×M which is fixed under the diagonal action of G.

TheoremThe double coset space RM = H\G/H is isomorphic to thespace of atomic invariant relations on M = G/H.

CorollaryGiven any point x ∈M and atomic invariant relation r ∈ RM,there is an H-family of related points y ∼r x.

For UTE3, SO(2)\ASO(3)/SO(2) is coordinatized by the fourrelations which appear in Lie’s theorem.


Invariant Relations

An invariant relation on a homogeneous space M is a subset ofM×M which is fixed under the diagonal action of G.

TheoremThe double coset space RM = H\G/H is isomorphic to thespace of atomic invariant relations on M = G/H.

CorollaryGiven any point x ∈M and atomic invariant relation r ∈ RM,there is an H-family of related points y ∼r x.

For UTE3, SO(2)\ASO(3)/SO(2) is coordinatized by the fourrelations which appear in Lie’s theorem.


Generalized Bianchi and Lie Theorems

Now we can seek a generalization of Bianchi’s relations andLie’s theorem to other homogeneous spaces, along the lines of

Theorem (Generalized Bianchi)

Let f , f : X −→M be two surfaces in the homogeneous spaceM and r ∈ RM a relation such that [??]. If fp ∼r fp for all p ∈ X,then f and f satisfy the differential equations ∆, ∆ respectively.

Theorem (Generalized Lie)

If f satisfies ∆ and r ∈ RM is as above, then there is a surface fsuch that f ∼r f and f satisfies ∆. f may be constructed from fby solving a sequence of ODEs.


Exterior Differential SystemsContact Geometry

Every smooth first-order ODE ∆ for one function of one variableis of the form

F∆(x, y, y ′) = 0

for some smooth function F∆ : R3 −→ R. We may think offormal solutions to ∆ as being curves γ : (−ε, ε) −→ R3 suchthat F∆ γ = 0.

Giving R3 the coordinates x, y, p, a formal solution is an actualsolution to ∆ exactly when

γ∗(dy − p dx) = 0


Exterior Differential SystemsContact Geometry

Every smooth first-order ODE ∆ for one function of one variableis of the form

F∆(x, y, y ′) = 0

for some smooth function F∆ : R3 −→ R. We may think offormal solutions to ∆ as being curves γ : (−ε, ε) −→ R3 suchthat F∆ γ = 0.

Giving R3 the coordinates x, y, p, a formal solution is an actualsolution to ∆ exactly when

γ∗(dy − p dx) = 0


Exterior Differential SystemsEncoding PDEs

The same idea works for general differential equations, of anyorder and in any number of dependent and independentvariables.

LemmaTo any PDE ∆ there is a manifold M∆ and a differential idealΘ∆ 6 Ω•M∆

(R) such that solutions to ∆ are in a naturalcorrespondence with maps f : U −→M∆ satisfying f ∗Θ∆ = 0.

The pair (M∆,Θ∆) is called an exterior differential system, orEDS.


Exterior Differential SystemsEncoding PDEs

The same idea works for general differential equations, of anyorder and in any number of dependent and independentvariables.

LemmaTo any PDE ∆ there is a manifold M∆ and a differential idealΘ∆ 6 Ω•M∆

(R) such that solutions to ∆ are in a naturalcorrespondence with maps f : U −→M∆ satisfying f ∗Θ∆ = 0.

The pair (M∆,Θ∆) is called an exterior differential system, orEDS.


Exterior Differential SystemsExample of an EDS for a PDE

Let ∆ be the Euler-Tricomi equation ∂2u∂x2 = x∂

2u∂y2 . Then ∆ is

equivalent to the EDS generated by the 1-forms

α = du − p dx − q dy, β = p dy + xq dx

on R5 with coordinates x, y, u, p, q.

If f : R2 −→ R5 satisfies f ∗Θ = 0 then from f ∗α = 0 we get

p(x, y) =∂u∂x

, q(x, y) =∂u∂y

Combined with f ∗dβ = 0,

0 = dp ∧ dy + x dq ∧ dx =

(∂2u∂x2 − x

∂2u∂y2

)dx ∧ dy









p(x, y) =∂u∂x

, q(x, y) =∂u∂y


0 = dp ∧ dy + x dq ∧ dx =

(∂2u∂x2 − x

∂2u∂y2

)dx ∧ dy









p(x, y) =∂u∂x

, q(x, y) =∂u∂y


0 = dp ∧ dy + x dq ∧ dx =

(∂2u∂x2 − x

∂2u∂y2

)dx ∧ dy


Integrable Extensions of EDSsIntegrability

Let Θ be an EDS which is differentially generated by a set I of1-forms. We call Θ integrable (or Frobenius) if

dI = 0 mod I

so that Θ is algebraically generated by I as well. Θ is integrableif and only if I⊥ is a Frobenius distribution.

Let π : Y −→ X be a submersion, Θ an EDS on X, and Θ anEDS on Y such that π∗Θ ⊆ Θ.

Definition

If Θ is generated by π∗Θ and J ⊆ Ω1Y(R) with J a basis of

(ker dπ)∗, then Θ is called an extension of Θ. If additionallydJ = 0 mod J,π∗Θ then the extension is called integrable.


Integrable Extensions of EDSsIntegrability

Let Θ be an EDS which is differentially generated by a set I of1-forms. We call Θ integrable (or Frobenius) if

dI = 0 mod I

so that Θ is algebraically generated by I as well. Θ is integrableif and only if I⊥ is a Frobenius distribution.

Let π : Y −→ X be a submersion, Θ an EDS on X, and Θ anEDS on Y such that π∗Θ ⊆ Θ.

Definition

If Θ is generated by π∗Θ and J ⊆ Ω1Y(R) with J a basis of

(ker dπ)∗, then Θ is called an extension of Θ. If additionallydJ = 0 mod J,π∗Θ then the extension is called integrable.


Integrable Extensions of EDSsThe Estabrook-Wahlquist Theorem

If Θ is an extension of Θ then there are solutions to Θ lifting anysolution to Θ. If the extension is integrable, then finding asolution to Θ over a solution of Θ only involves solving aFrobenius system.

Theorem (Estabrook-Wahlquist)

Let Θ be an integrable extension of Θ, f : U −→ X a solution toΘ, and q ∈ π−1f (p) for some p ∈ U. Then there is a unique liftf : U −→ Y of f through q, and f may be constructed by solvinga sequence of ordinary differential equations.


Geometric Exterior Differential SystemsDefinition

Not every PDE on a manifold M is interesting. In order tonarrow our focus, we will define a class of PDEs which onlycontains equations involving geometric quantities.

DefinitionLet M be a homogeneous space with structure group G. Ageometric exterior differential system (gEDS) on M is adifferential ideal Θ 6 Ω•M(R) such that Θ is invariant under theaction of G.

A gEDS is only general enough to encode “geometricallymeaningful” differential equations.


Geometric Exterior Differential SystemsDefinition

Not every PDE on a manifold M is interesting. In order tonarrow our focus, we will define a class of PDEs which onlycontains equations involving geometric quantities.

DefinitionLet M be a homogeneous space with structure group G. Ageometric exterior differential system (gEDS) on M is adifferential ideal Θ 6 Ω•M(R) such that Θ is invariant under theaction of G.

A gEDS is only general enough to encode “geometricallymeaningful” differential equations.


Geometric Exterior Differential SystemsExample: UTE3

Let UTE3 be the unit tangent bundle of Euclidean 3-space, soUTE3 ∼= E3 × S2. An element (T, R) of the Euclidean groupASO(3) acts on (p, n) ∈ E3 × S2 by

(T, R) · (p, n) = (T + R · p, R · n)

This action is clearly transitive, so UTE3 is a homogeneousspace for the Euclidean group.

Claim: The 1-form 〈n, dp〉 is invariant.

L∗(T,R)〈n, dp〉 = 〈R · n, d(T + R · p)〉

= 〈R · n, R · dp〉= 〈n, dp〉




(T, R) · (p, n) = (T + R · p, R · n)



L∗(T,R)〈n, dp〉 = 〈R · n, d(T + R · p)〉

= 〈R · n, R · dp〉= 〈n, dp〉




(T, R) · (p, n) = (T + R · p, R · n)



L∗(T,R)〈n, dp〉 = 〈R · n, d(T + R · p)〉

= 〈R · n, R · dp〉= 〈n, dp〉


Geometric Exterior Differential SystemsExample: UTE3 continued

Let Θ be the differential ideal on UTE3 generated by 〈n, dp〉.Since 〈n, dp〉 is G-invariant, so is Θ. Therefore, Θ is a gEDS.

Integral manifolds of Θ are maps f : U −→ E3, n : U −→ S2 suchthat n is the normal map of f . In this sense, Θ is a geometricversion of a contact ideal.


Geometric Exterior Differential SystemsExample: UTE3 continued

Let Θ be the differential ideal on UTE3 generated by 〈n, dp〉.Since 〈n, dp〉 is G-invariant, so is Θ. Therefore, Θ is a gEDS.

Integral manifolds of Θ are maps f : U −→ E3, n : U −→ S2 suchthat n is the normal map of f . In this sense, Θ is a geometricversion of a contact ideal.


Characterization of gEDSs on M

Let M be a homogeneous space with structure group G, and letH 6 G be the stabilizer of some point p ∈M.

Theorem (N–, 2008)There is a one-to-one correspondence between gEDSs on Mand ad∗(h)-submodules of h⊥ ⊆ g∗.

Usage: This theorem allows us to replace gEDS calculationswith much simpler computations on g∗.


The Exterior Algebra on a Homogeneous SpaceThe Exterior Derivative

A local frame on U ⊆M relative to q is a map σ : U −→ G suchthat σ(p) · q = p.

Lemma

Let δ : g∗ −→∧2 g∗ be the negative dual of the Lie bracket,

(δφ)(x, y) = −φ([x, y])

and σ a local frame. Then if Θ is a gEDS we have

µ∗σ−1(δΘ) = d(µ∗σ−1Θ)

where µg(p) = g · p is the action of G on M.

Moral: δ replaces d as the exterior derivative for a gEDS.


The Exterior Algebra on a Homogeneous SpaceThe Exterior Derivative

A local frame on U ⊆M relative to q is a map σ : U −→ G suchthat σ(p) · q = p.

Lemma

Let δ : g∗ −→∧2 g∗ be the negative dual of the Lie bracket,

(δφ)(x, y) = −φ([x, y])

and σ a local frame. Then if Θ is a gEDS we have

µ∗σ−1(δΘ) = d(µ∗σ−1Θ)

where µg(p) = g · p is the action of G on M.

Moral: δ replaces d as the exterior derivative for a gEDS.


The Exterior Algebra on a Homogeneous SpaceThe Lie Derivative

Since we have a differential on g∗ and a pairing y of g∗ with g,we can define for any ξ ∈ g the Lie derivativeLξ :

∧k g∗ −→∧k g∗ by

Lξω = ξ y (δω) + δ(ξ y ω)

Lemma

Let h be a subalgebra of g, and Θ 6∧

h⊥ an ideal closed underδ. Then Θ is ad∗(h)-invariant (and therefore a gEDS on G/H) ifand only if

LξΘ = 0 mod Θ

for all ξ ∈ h.


The Exterior Algebra on a Homogeneous SpaceThe Lie Derivative

Since we have a differential on g∗ and a pairing y of g∗ with g,we can define for any ξ ∈ g the Lie derivativeLξ :

∧k g∗ −→∧k g∗ by

Lξω = ξ y (δω) + δ(ξ y ω)

Lemma

Let h be a subalgebra of g, and Θ 6∧

h⊥ an ideal closed underδ. Then Θ is ad∗(h)-invariant (and therefore a gEDS on G/H) ifand only if

LξΘ = 0 mod Θ

for all ξ ∈ h.


The Exterior Algebra on a Homogeneous SpaceA Special Operator

For each ξ ∈ h there is a special operator ∇ξ : g∗ −→∧2 g∗,

useful in determining which relations lead to Backlundtransformations. This operator is defined by the equation

∇ξω = ξ y (ω∧ δω)

∇ is used to single out a lift of f which realizes the relation to f .


The Main Theorem

Theorem (N–, 2009)

Let Θ ⊆ h⊥ generate a gEDS on the homogeneous spaceM = G/H, and let [β] ∈ H\G/H be a geometric relation. Define∆, ∆ ∈ Hom(h×Θ,

∧2 g∗) by ∆(ξ, ϑ) = ∇ξAd∗(β−1)ϑ,∆(ξ, ϑ) = ∇ξAd∗(β)ϑ.

If β is such that for all ξ ∈ h we have Lξ∆ = Lξ∆ = 0 mod Θthen:

1 (Generalized Bianchi) If f and f are Θ-adapted and f ∼[β] f ,then f , f satisfy the differential equations ∆, ∆ resp.

2 (Generalized Lie) If f is Θ-adapted and satisfies ∆, then wecan construct a f which is [β]-related to f , is Θ-adapted,and which satisfies ∆. Furthermore, f may be constructedby integrating a series of ODEs.


The Main TheoremProof Outline

Define Θ1 = Θ ∪ ∆, Θ2 = Θ ∪ ∆, where Θ is a gEDS generatedby 1-forms. The goal is to construct a gEDS which is anintegrable extension of both Θ1 and Θ2.

Equip g× g with the projections

g× gπ1

||zzzz

zzzz

z Ad(β−1)π2

""DDDD

DDDD

D

g g

Denote these projections by π, π for short.



Define Θ1 = Θ ∪ ∆, Θ2 = Θ ∪ ∆, where Θ is a gEDS generatedby 1-forms. The goal is to construct a gEDS which is anintegrable extension of both Θ1 and Θ2.

Equip g× g with the projections

g× gπ1

||zzzz

zzzz

z Ad(β−1)π2

""DDDD

DDDD

D

g g

Denote these projections by π, π for short.



1 Define a gEDS Ω on g× g differentially generated by the2-forms π∗∆, π∗∆, the 1-forms π∗Θ, and

π∗ϕ− π∗ϕ∣∣∣ ϕ ∈ g∗

A map (F, F) : X −→ G× G is an integral manifold of Ω iffthere is an element g ∈ G such that g · F ∼[β] F.

2 The 1-forms in Ω fail to give an integrable extension of Θ.The additional curvature is measured by∆ = ∇Ad∗(β−1)Θ.

3 Ω is a gEDS for the diagonal action of h only whenLhΩ = 0. This is automatic, except for Lh∆ = 0.




π∗ϕ− π∗ϕ∣∣∣ ϕ ∈ g∗







π∗ϕ− π∗ϕ∣∣∣ ϕ ∈ g∗





Applying the Main TheoremCanonical Form of Euclidean Relations

As an extended example, let us use the main theorem to findthe geometric Backlund transformations for surfaces inEuclidean 3-space. A generic relation [β] on UTE3 has arepresentation of the form

1 0 0 0X 1 0 0Y 0 cosϕ − sinϕZ 0 sinϕ cosϕ

=⇒

|f − f |2 = X2 + Y2 + Z2

〈n, n〉 = cosϕ〈n, f − f 〉 = Z〈n, f − f 〉 = Z cosϕ− Y sinϕ

[β] is a symmetric relation ([β] = [β−1]) when

Y sinϕ = Z(1 + cosϕ)


Applying the Main TheoremCanonical Form of Euclidean Relations

As an extended example, let us use the main theorem to findthe geometric Backlund transformations for surfaces inEuclidean 3-space. A generic relation [β] on UTE3 has arepresentation of the form

1 0 0 0X 1 0 0Y 0 cosϕ − sinϕZ 0 sinϕ cosϕ

=⇒

|f − f |2 = X2 + Y2 + Z2

〈n, n〉 = cosϕ〈n, f − f 〉 = Z〈n, f − f 〉 = Z cosϕ− Y sinϕ

[β] is a symmetric relation ([β] = [β−1]) when

Y sinϕ = Z(1 + cosϕ)


Applying the Main TheoremUsing the ∇ Operator

The contact system Θ on UTE3 is generated by the singleelement e3 ∈ aso(3)∗. Since h is generated by e2

1, we get thelone constraint

Le21∇e2

1Ad∗(β−1)e3 = 0 mod Θ

The Differential EquationThe corresponding differential operator ∆ is

(sin2ϕ)e1 ∧ e2 + (Y sinϕ)(e31 ∧ e2 − e3

2 ∧ e1) + (X2 + Y2)e31 ∧ e3

2


Applying the Main TheoremUsing the ∇ Operator

The contact system Θ on UTE3 is generated by the singleelement e3 ∈ aso(3)∗. Since h is generated by e2

1, we get thelone constraint

Le21∇e2

1Ad∗(β−1)e3 = 0 mod Θ

The Differential EquationThe corresponding differential operator ∆ is

(sin2ϕ)e1 ∧ e2 + (Y sinϕ)(e31 ∧ e2 − e3

2 ∧ e1) + (X2 + Y2)e31 ∧ e3

2


Applying the Main TheoremGeometric Backlund Transformations in UTE3

Theorem (Backlund Transformations in UTE3)

Let [β] ∈ RUTE3 be a geometric relation of the form

|f − f |2 = X2 + Z2 csc2(ϕ/2)

〈n, n〉 = cosϕ〈n, f − f 〉 = Z〈n, f − f 〉 = −Z

Then [β] induces a geometric Backlund transformation betweensurfaces satisfying the affine Weingarten equation

sin2ϕ+ 2H((1 + cosϕ)Z) + K(X2 + Z2 cot2 ϕ

2) = 0


Affine Weingarten Surfaces

Specialize to surfaces satisfying the affine Weingarten equation2K + 2H + 1 = 0.

The unit-radius cylinder is a simple solution to the Weingartenequation 2K + 2H + 1 = 0. Let us parameterize the cylinder byf (u, v) = (cos u, sin u, v)T.


Transform of the Cylinder

The simplest transform of the cylinder is a surface of revolutionsatisfying 2K + 2H + 1 = 0 with profile curve

γ(t) =1

4π2e2t + 1

(4π2e2t − 4πet − 1

4π2(t − 1)e2t − 4πet + t + 1

)



There is actually a 1-parameter family of transformations actingon solutions to 2K + 2H + 1 = 0. The case just analyzedcorresponds to the parameter −π/2. Here is a solution wherethe parameter is nearly −π:



More generally, the cylinder transforms to a solution of2K + 2H + 1 = 0 parameterized by

f(u, v) =1

e2u cotϕ + e2v cscϕ ·

(−2eu cotϕ+v cscϕ + e2u cotϕ) sinϕ sin u + e2u cotϕ cosϕ cos u + cos(u +ϕ)e2v cscϕ(2eu cotϕ−v cscϕ + e2u cotϕ) sinϕ cos u + e2u cotϕ cosϕ sin u + sin(u +ϕ)e2v cscϕ

(v − sinϕ)e2v cscϕ + ve2u cotϕ +(2eu cotϕ−v cscϕ + e2u cotϕ) sinϕ

where ϕ is the parameter mentioned previously.


Table of Contents




4 Future Directions

5 Appendix


Future DirectionsInfinitesimal Relations I

There are many interesting Backlund transformations which are“infinitesimally geometric”, in the sense that the relevant relationis a relation on g× g rather than G× G. Let f be a surface and

F−1dF = ω =

0 0 0 0τ1 0 λ ν1

τ2 −λ 0 ν2

0 −ν1 −ν2 0

A surface f is minimal if and only if

ω =

0 0 0 0

−τ2 0 λ ν1

τ1 −λ 0 ν2

0 −ν1 −ν2 0

is integrable.


Future DirectionsInfinitesimal Relations I

There are many interesting Backlund transformations which are“infinitesimally geometric”, in the sense that the relevant relationis a relation on g× g rather than G× G. Let f be a surface and

F−1dF = ω =

0 0 0 0τ1 0 λ ν1

τ2 −λ 0 ν2

0 −ν1 −ν2 0

A surface f is minimal if and only if

ω =

0 0 0 0

−τ2 0 λ ν1

τ1 −λ 0 ν2

0 −ν1 −ν2 0

is integrable.


Future DirectionsInfinitesimal Relations II

A surface f has constant mean curvature 1 (CMC) if andonly if

ω =

0 τ2 −τ1 0

−τ2 0 λ ν1

τ1 −λ 0 ν2

0 −ν1 −ν2 0

is integrable in so(4).

The surface corresponding to ω is minimal in S3.Combined with the Dorfmeister-Pedit-Wu method, we get anew Weierstrass representation for CMC surfaces (N–,2006).

(N–, 2008) There are similar transformations takingNambu-Goto strings in L2,1 to themselves, and takingNambu-Goto strings in (2, 1)-dimensional deSitter space tostrings subject to an additional field in L2,1.


Future DirectionsInfinitesimal Relations II

A surface f has constant mean curvature 1 (CMC) if andonly if

ω =

0 τ2 −τ1 0

−τ2 0 λ ν1

τ1 −λ 0 ν2

0 −ν1 −ν2 0

is integrable in so(4).

The surface corresponding to ω is minimal in S3.Combined with the Dorfmeister-Pedit-Wu method, we get anew Weierstrass representation for CMC surfaces (N–,2006).(N–, 2008) There are similar transformations takingNambu-Goto strings in L2,1 to themselves, and takingNambu-Goto strings in (2, 1)-dimensional deSitter space tostrings subject to an additional field in L2,1.


Future DirectionsAssociated PDEs

In the classical case, it happens that θ is a solution to thesine-Gordon equation

∂2θ

∂u2 −∂2θ

∂v2 =12

sin(2θ)

The geometric Backlund transformations also induce Backlundtransformations of the sine-Gordon equation.

ConjectureBy comparing the special frames appearing in thetransformations to canonical frames, we should be able tosystematically associate a totally integrable PDE for H-valuedfunctions to each geometric transformation on G/H.


Future DirectionsAssociated PDEs

In the classical case, it happens that θ is a solution to thesine-Gordon equation

∂2θ

∂u2 −∂2θ

∂v2 =12

sin(2θ)

The geometric Backlund transformations also induce Backlundtransformations of the sine-Gordon equation.

ConjectureBy comparing the special frames appearing in thetransformations to canonical frames, we should be able tosystematically associate a totally integrable PDE for H-valuedfunctions to each geometric transformation on G/H.


Future DirectionsBianchi Permutability

In the classical Backlund transform, the relations |f − f | = 1 and〈n, n〉 = 0 can be replaced with |f − f | = sinα, 〈n, n〉 = cosα toget a Backlund transform βα.

Theorem (Bianchi’s Permutability Theorem)Let f be a pseudospherical surface, and βαf its transform. Then

βαβα ′ f = βα ′βαf

Furthermore, βαβα ′ f is an algebraic function of f , βαf , andβα ′ f .

Leads to a nonlinear superposition principle for these surfaces.Can this be generalized?


Future DirectionsMore General Differential Equations

The techniques described here only pick out “elementary”equations, so we get things like aK + bH + c = 0 but notH2 − K = 0, despite the fact that H2 − K = (κ1 − κ2)

2 is aEuclidean invariant.

Is there a natural analog of prolongation which gives access tothese invariants?


Thank you for your time!


Table of Contents




4 Future Directions

5 Appendix


Integrating an Overdetermined System

To integrate an overdetermined PDE df = Φ(~x, f ), pick acomplete flag X1 ⊂ X2 ⊂ · · · ⊂ X on the independent variablesand solve for f inductively. Moving up each step in the flag onlyinvolves integrating an ODE.














Kuen’s Surface


Kuen’s Surface

f (u, v) =

2

1+v2 sech2 u(sech u cos v + v sech u sin v)

21+v2 sech2 u

(sech u sin v − v sech u cos v)

u − 21+v2 sech2 u

tanh u

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A Geometric Transformation Theory for PDE

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