ANALELE S ¸TIINT ¸ IFICE ALE UNIVERSIT ˘ AT ¸ II “AL.I. CUZA” DIN IAS ¸I (S.N.) MATEMATIC ˘ A, Tomul LIII, 2007, Supliment ON THE SECTIONAL CURVATURE OF DESZCZ BY STEFAN HAESEN and LEOPOLD VERSTRAELEN Dedicated to Academician Radu Miron at his 80th anniversary Abstract. A scalar valued curvature invariant is constructed which in general de- pends on two tangent planes at a point. This invariant, which is called the sectional curvature of Deszcz, can be geometrically interpreted in terms of the parallelogramoids of Levi-Civita and isotropy of this invariant with respect to both planes characterises the pseudo-symmetric spaces. Mathematics Subject Classification 2000: 53B20. Key words: sectional curvature of Deszcz, pseudo-symmetry. 1. Introduction. One of the most natural kinds of transformations to perform on a Riemannian manifold (M,g) which takes into account some- thing essential with respect to the underlying differential structure, i.e. the co-ordinate patches, and, of course, also something essential with respect to the geometrical structure, i.e. the Riemannian connection, is the parallel transport around co-ordinate parallelograms, and around infinitesimal co- ordinate parallelograms for that matter, g after all being conceived as an infinitesimal measure of lengths on M . No doubt, the simplest objects to move around ”this way” are the vectors ; the symmetry of this operation was studied by Schouten who doing so obtained the geometrical interpretation of the Riemann-Christoffel curvature tensor R of (M,g), which nowadays mostly even serves as its definition. Parallely, Levy-Civita introduced the parallel transport of vectors on Riemannian manifolds to define his paral- lelogramoids in terms of which he succeeded to give a beautiful geometrical interpretation of the Riemann or sectional curvature K of (M,g). The next
Transcript
ANALELE STIINTIFICE ALE UNIVERSITATII “AL.I. CUZA” DIN IASI (S.N.)MATEMATICA, Tomul LIII, 2007, Supliment
ON THE SECTIONAL CURVATURE OF DESZCZ
BY
STEFAN HAESEN and LEOPOLD VERSTRAELEN
Dedicated to Academician Radu Miron at his 80th anniversary
Abstract. A scalar valued curvature invariant is constructed which in general de-pends on two tangent planes at a point. This invariant, which is called the sectionalcurvature of Deszcz, can be geometrically interpreted in terms of the parallelogramoidsof Levi-Civita and isotropy of this invariant with respect to both planes characterises thepseudo-symmetric spaces.
Mathematics Subject Classification 2000: 53B20.
Key words: sectional curvature of Deszcz, pseudo-symmetry.
1. Introduction. One of the most natural kinds of transformations toperform on a Riemannian manifold (M, g) which takes into account some-thing essential with respect to the underlying differential structure, i.e. theco-ordinate patches, and, of course, also something essential with respectto the geometrical structure, i.e. the Riemannian connection, is the paralleltransport around co-ordinate parallelograms, and around infinitesimal co-ordinate parallelograms for that matter, g after all being conceived as aninfinitesimal measure of lengths on M . No doubt, the simplest objects tomove around ”this way” are the vectors; the symmetry of this operation wasstudied by Schouten who doing so obtained the geometrical interpretationof the Riemann-Christoffel curvature tensor R of (M, g), which nowadaysmostly even serves as its definition. Parallely, Levy-Civita introduced theparallel transport of vectors on Riemannian manifolds to define his paral-lelogramoids in terms of which he succeeded to give a beautiful geometricalinterpretation of the Riemann or sectional curvature K of (M, g). The next
182 STEFAN HAESEN and LEOPOLD VERSTRAELEN 2
simplest objects to move around ”this way” are likely to be precisely thensectional curvatures. That is what the authors have been doing for sometime now and on some of the results obtained along this way this paper willreport.
2. The sectional curvature of a plane. Let (Mn, g) be an n-dimensional manifold with Riemannian metric g. Denote the Levi-Civitaconnection by ∇ and its related Riemann-Christoffel (1, 3)-curvature tensorby R. The (0, 4)-curvature tensor R is related to the (1, 3)-curvature tensorby R(X,Y, U, V ) = g(R(X, Y )U, V ). The endomorphisms V ∧g W andR(V, W ) of the Lie algebra of vector fields X(M) of M are defined by
(V ∧g W )Z = g(W,Z)V − g(V,Z)W,
andR(V,W )Z = ∇V∇W Z −∇W∇V Z −∇[V,W ]Z.
Here and in the following, vectors will be denoted by lower case letters, whilevector fields will be denoted by capital letters. As is well-known, and whichgoes back to Schouten [11], R(~x, ~y)~z measures the second order change ofa vector ~z ∈ TpM at p ∈ M after parallel transport around an infinitesimalco-ordinate parallelogram P cornered at p with sides of parameter changes∆x and ∆y and with tangent vectors ~x and ~y at p to the x- and y-sides ofP, namely,
~z? = ~z + [R(~x, ~y)~z]∆x∆y +O>2(∆x,∆y).
A vector (~x ∧g ~y)~z can be geometrically interpreted as follows. Assumethat ~x, ~y ∈ TpM are orthonormal and choose vectors ~e3, . . . , ~en so that{~x, ~y,~e3, . . . , ~en} is an orthonormal basis of TpM . Then, ~z ∈ TpM can bedecomposed as
~z = g(~z, ~x)~x + g(~z, ~y)~y +n∑
i=3
g(~z,~ei)~ei.
By rotating the projection of ~z onto the plane ~x ∧ ~y, spanned by ~x and~y, over an infinitesimal angle ∆ϕ, while keeping the projection of ~z ontothe (n − 2)-plane spanned by ~e3, . . . , ~en fixed, a new vector ~z is obtained,namely,
3 ON THE SECTIONAL CURVATURE OF DESZCZ 183
x
y
x x+∆x
y
y+∆y
P
p
Figure 1: A co-ordinate parallelogram.
P
z
z
Figure 2: Parallel transport of a vector around a co-ordinate parallelogram.
~z = ~z + [g(~z, ~y)~x− g(~z, ~x)~y]∆ϕ +O≥2(∆ϕ).
Thus, the vector (~x ∧g ~y)~z measures the first order change of the vector ~zafter such an infinitesimal rotation of ~z in the plane ~x ∧ ~y at the point p.Therefore, it seems natural to consider (~x ∧g ~y)~z as some kind of normali-sation for R(~x, ~y)~z, which leads to the following definition.
184 STEFAN HAESEN and LEOPOLD VERSTRAELEN 4
Definition 1. At any point p ∈ M , let π = ~v ∧ ~w be any plane tangentto M at p, spanned by two of its vectors ~v and ~w. Then, the real number
K(p, π) =g(R(~v, ~w)~w,~v)g((~v ∧g ~w)~w,~v)
only depends on the point p and on the plane π and is called the sectionalcurvature of M at p for the plane section π.
As shown by Cartan, the knowledge of the full curvature tensor R isequivalent to the knowledge of the sectional curvatures K. A Riemannianmanifold is said to be a space of constant curvature c when all its sectionalcurvatures K(p, π) are equal to c, i.e., when these curvatures are indepen-dent of both the points p and the planes π. By Schur’s theorem, for n > 2, itsuffices for this to hold that at all points p the sectional curvatures K(p, π)are independent of the planes π at p. The spaces of constant curvature care characterised by their (0, 4)-curvature tensor R being given by R = c G,where the (0, 4)-tensor G is defined as G(X, Y, U, V ) = g((X ∧g Y )U, V ).
A geometrical interpretation of the sectional curvature of a plane π,spanned by the vectors ~v and ~w, at a point p in terms of lengths of geodesicswas given by Levi-Civita using his so-called parallelogramoids as follows (seee.g. [4, 10]). Consider through p the geodesic α with tangent ~v and let qbe the point on this geodesic at an infinitesimal distance A from p. Denoteby ~w? the vector obtained after parallel transport of ~w from p to q along α.Then, through p and q consider the geodesics βp and βq with tangents ~w and~w?, respectively. Fix on them the points p and q, respectively, at the sameinfinitesimal distance B from p and q, respectively. The parallelogramoidcornered at p with sides tangent to ~v and ~w is then completed by thegeodesic α through p and q. Let A′ be the geodesic distance between pand q. Levi-Civita showed that, in first order approximation, the sectionalcurvature of the plane π = ~v ∧ ~w can be expressed as
K(p, π) ≈ A2 −A′2
(AB sinφ)2,
whereby φ is the angle between the vectors ~v and ~w. In particular, let ~vand ~w be orthonormal vectors at p ∈ M . Consider the Levi-Civita squaroidbased on ~v and ~w with side ε, i.e., the parallelogramoid for which A = B = ε.Then, when ε′ is the length of the closing geodesic, the sectional curvature
5 ON THE SECTIONAL CURVATURE OF DESZCZ 185
K(p, π) is given by
K(p, π) ≈ ε2 − ε′2
ε4.
v
w w
p ε
ε
ε
ε'
Figure 3: A squaroid of Levi-Civita.
3. The sectional curvature of Deszcz of two planes. Considerthe (0, 6)-tensor R · R, obtained by the action of the curvature operatorR(X, Y ) on the (0, 4)-curvature tensor R,
whereby X1, X2, X3, X4, X, Y ∈ X(M). A Riemannian manifold M is saidto be semi-symmetric when the tensor R · R vanishes, i.e., R · R = 0.Consider any two linearly independent vectors ~v and ~w at any point p ofM and any co-ordinate parallelogram P cornered at p with sides of lengths∆x and ∆y tangent to the linearly independent vectors ~x and ~y at p. Then,by parallel transport of ~v and ~w around P we obtain the vectors ~v? =
186 STEFAN HAESEN and LEOPOLD VERSTRAELEN 6
~v + [R(~x, ~y)~v]∆x∆y + O>2(∆x,∆y) and ~w? = ~w + [R(~x, ~y)~w]∆x∆y +O>2(∆x,∆y), so that
Thus, the (0, 6)-tensor R ·R of M measures the change in sectional curva-ture at any point p for any plane π under parallel transport of π aroundany infinitesimal co-ordinate parallelogram P cornered at p [5]. As a conse-quence it follows that a Riemannian manifold M is semi-symmetric if andonly if its sectional curvature function K(p, π) is invariant, up to secondorder, under parallel transport of any plane π at any point p of M aroundany infinitesimal co-ordinate parallelogram cornered at p.
Probably the simplest (0, 6)-tensor on an n(≥ 3)-dimensional Rieman-nian manifold having the same symmetry properties as R·R is the Tachibanatensor Q(g, R), defined by
Q(g,R)(X1, X2, X3, X4; X, Y ) := −((X ∧g Y ) ·R)(X1, X2, X3, X4)= R((X ∧g Y )X1, X2, X3, X4)
+R(X1, (X ∧g Y )X2, X3, X4)+R(X1, X2, (X ∧g Y )X3, X4)+R(X1, X2, X3, (X ∧g Y )X4).
A classical result states that the vanishing of this tensor, i.e., Q(g,R) = 0, isa necessary and sufficient condition for M to be of constant curvature. Usingthe above geometrical interpretation of (~x ∧g ~y)~z, a geometrical meaningof the components Q(g,R)(~v, ~w, ~w,~v; ~x, ~y) of the Tachibana tensor can beobtained as follows. Let {~x, ~y,~e3, . . . , ~en} be an orthonormal basis of TpM
and consider orthonormal vectors ~v, ~w ∈ TpM . The vectors ~v and ~w arethe vectors obtained after an infinitesimal rotation of the projection of ~vand ~w in the plane ~x ∧ ~y, namely ~v = ~v + [(~x ∧g ~y)~v]∆ϕ + O≥2(∆ϕ), and~w = ~w +[(~x ∧g ~y)~w] ∆ϕ+O≥2(∆ϕ). Comparing the sectional curvatures ofthe planes π = ~v ∧ ~w and π = ~v ∧ ~w, we find
Thus, the components Q(g, R)(~v, ~w, ~w,~v; ~x, ~y) measure the change of sec-tional curvature K(p, π) under an operation involving infinitesimal rotationsperformed at the point p, without leaving this point, in contrast to thecomponents (R · R)(~v, ~w, ~w,~v; ~x, ~y), which measure the change of sectionalcurvature K(p, π) after the movement of the plane π in an infinitesimalneighbourhood of the point p. It seems therefore natural to consider thecomponents Q(g, R)(~v, ~w, ~w,~v; ~x, ~y) as some kind of normalisation for thecomponents (R ·R)(~v, ~w, ~w,~v; ~x, ~y).
Definition 2. Let (Mn, g) be an n(≥ 3)-dimensional Riemannian man-ifold which is not of constant curvature and denote by U the set of pointswhere the Tachibana tensor Q(g,R) is not identically zero, i.e.,
U = {x ∈ M | Q(g, R)x 6= 0}.
Then, at a point p ∈ U , a plane π = ~v ∧ ~w ⊂ TpM is said to be curvature-dependent with respect to a plane π = ~x∧~y ⊂ TpM if Q(g, R)(~v, ~w, ~w,~v; ~x, ~y)6= 0.
This definition is independent of the choice of bases for π and π. In analogywith Definition 1 we propose the following.
Definition 3. At a point p ∈ U ⊂ M , let the tangent plane π = ~v∧ ~w becurvature-dependent with respect to π = ~x∧~y. Then, the sectional curvatureof Deszcz L(p, π, π) of the plane π with respect to π at p is the scalar
This definition is again independent of the choice of bases for the tangentplanes π and π. Analogously to the result of Cartan concerning the deter-mination of the Riemann curvature tensor through the sectional curvatures,one can show that at any point p ∈ U , the tensor R ·R is completely deter-mined by the knowledge of the sectional curvatures of Deszcz L(p, π, π) ofcurvature-dependent planes π, π ⊂ TpM .
The analogy between the sectional curvature of a plane and the sec-tional curvature of Deszcz of two curvature-dependent planes goes further,in the sense that a geometrical interpretation of the sectional curvature ofDeszcz can be given in terms of the squaroids of Levi-Civita as follows. Ata point p ∈ M , consider two planes π = ~v ∧ ~w and π = ~x ∧ ~y and parallel
188 STEFAN HAESEN and LEOPOLD VERSTRAELEN 8
transport the vectors ~v and ~w around the infinitesimal co-ordinate parallel-ogram formed by the tangents ~x and ~y at p. We construct the two squaroidsstarting from the vectors ~v, ~w and ~v?, ~w?, respectively, with equal sides ε.In general, the lengths of the closing geodesics, ε′ and ε?′, will be different.We find, up to second order with respect to the sides ∆x and ∆y of theco-ordinate parallelogram that
(R ·R)(~v, ~w, ~w,~v; ~x, ~y) ≈ (ε?′)2 − (ε′)2
ε4
1∆x∆y
.
Let ~v, ~w be the vectors which are obtained after an infinitesimal rotationas before from the vectors ~v, ~w with respect to the plane π = ~x ∧ ~y, andconstruct for the plane ~v ∧ ~w the squaroid of Levi-Civita, with the side ε.Denote the lengths of the completing geodesics by ε′. We find, with respectto the angle ∆ϕ of infinitesimal rotation, that
Q(g,R)(~v, ~w, ~w,~v; ~x, ~y) ≈ (ε′)2 − (ε′)2
ε4
1∆ϕ
.
Thus, calibrating the changes of the Riemann sectional curvatures underparallel translation (?) around a parallelogram P with infinitesimal pa-rameter growths ∆x and ∆y by the changes of the same curvatures underrotation (∼) over an infinitesimal angle ∆ϕ = ∆x∆y with respect to π,we find the following approximate geometrical expression in terms of thesquaroids of Levi-Civita of sides ε, for the sectional curvature of Deszcz L,
L(p, π, π) ≈ (ε?′)2 − (ε′)2
(ε′)2 − (ε′)2.
In the particular case that, at a point p ∈ U ⊂ M , the sectional curva-ture of Deszcz L(p, π, π) is independent of the planes π and π, the manifoldM is said to be pseudo-symmetric in the sense of Deszcz at p. If the mani-fold M is pseudo-symmetric at all points of U ⊂ M , the manifold M is saidto be pseudo-symmetric in the sense of Deszcz. In this case, there holdsthat R ·R = L Q(g, R). We observe that there does not hold a strict analogof the theorem of Schur in the case of pseudo-symmetric manifolds, i.e.,there are many examples of pseudo-symmetric manifolds with non-constantsectional curvature of Deszcz. However, a partial analog does hold in thesense that if the sectional curvatures of Deszcz L(p, π, π) at p ∈ U are in-dependent of π, i.e., L(p, π, π) = L(p, π) for every tangent plane π which is
9 ON THE SECTIONAL CURVATURE OF DESZCZ 189
curvature-dependent with respect to π, then the Riemannian manifold Mis pseudo-symmetric at p.
Following Kowalski and Sekizawa, a pseudo-symmetric space for whichthe sectional curvature of Deszcz is constant is said to be pseudo-symmetricof constant type. The three-dimensional Riemannian pseudo-symmetricspaces of constant type are obtained in [8, 9]. For example, the eight three-dimensional Thurston metrics have either constant sectional curvature Kequal to 0, 1 or −1, or have constant sectional curvature of Deszcz L equalto 0, 1 or −1 [1].
That the pseudo-symmetric spaces are natural generalisations of thespaces of constant curvature can be seen from both intrinsic and extrinsicpoints of view. Extrinsically, it was shown by Deszcz [3] that the extrinsicspheres Mn, i.e., totally umbilical submanifolds with parallel mean curva-ture vector, of semi-symmetric spaces Mn+m are pseudo-symmetric, whichextends the result that ordinary spheres in Euclidean space are of constantcurvature. And similar to the fact that the extrinsic spheres Mn in spheresSn+m are themselves of constant curvature, the extrinsic spheres Mn inpseudo-symmetric spaces Mn+m are also pseudo-symmetric. From an in-trinsic point of view, pseudo-symmetry appears in the study of geodesicmappings. If a Riemannian manifold Mn admits a geodesic mapping ontoa locally flat Riemannian space Mn, then Mn itself must be a space of con-stant curvature. Further, if a space Mn admits a geodesic mapping ontoa space Mn of constant curvature, then Mn must itself have constant cur-vature. Accordingly, results of Mikesh and Venzi [12] and Defever andDeszcz [2] learn that when a Riemannian manifold Mn admits a geodesicmapping onto a semi-symmetric manifold Mn, then Mn must be pseudo-symmetric, and when a manifold Mn admits a geodesic mapping onto apseudo-symmetric manifold, then Mn itself must also be pseudo-symmetric.So, in some sense, the extension of space-symmetry along these lines termi-nates with the pseudo-symmetry of Deszcz.
Acknowledgments. S. Haesen was partially supported by the Span-ish MEC Grant MTM2007-60731 with FEDER funds and the Junta deAndalucıa Regional Grant P06-FQM-01951. S. Haesen and L. Verstraelenwere partially supported by the Research Foundation - Flanders projectG.0432.07.
190 STEFAN HAESEN and LEOPOLD VERSTRAELEN 10
REFERENCES
1. Belkhelfa, M.; Deszcz, R.; Verstraelen, L. – Symmetry properties of 3-
dimensional d’Atri spaces, Kyungpook Math. J. 46 (2006), 367–376.
2. Defever, F.; Deszcz, R. – A note on geodesic mappings of pseudosymmetric Rie-
