We show that the results of Foulon (1997 an 2002) and Kim (2007)
(independently, Deng and Hou (2007)) about the nonexistence of locally
symmetric Finsler metrics of positive or negative flag curvature are in fact
local

Matveev, Vladimir S.

Comptes Rendus Mathematique

English

arXiv

Vladimir S. Matveev

Comptes Rendus Mathématique

C. R. Acad. Sci. Paris, Ser. I 353 (2015) 81–83Contents lists available at ScienceDirectC. R. Acad. Sci. Paris, Ser. Iwww.sciencedirect.comDifferential geometryThere exist no locally symmetric Finsler spaces of positive or negative flag curvatureIl n’existe pas d’espace de Finsler localement symétrique de courbure de drapeau positive ou négativeVladimir S. MatveevMathematisches Institut, Friedrich-Schiller Universität Jena, 07737 Jena, Germanya r t i c l e i n f o a b s t r a c tArticle history:Received 12 June 2014Accepted 23 October 2014Available online 11 November 2014Presented by Étienne GhysWe show that the results of Foulon [5,6] and Kim [7] (independently, of Deng and Hou [4]) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.r é s u m éNous montrons que les résultats de Foulon [5,6] et de Kim [7] (et indépendamment, de Deng et Hou [4]) sur l’inexistence de métriques de Finsler localement symétriques, de courbure de drapeau positive ou négative, sont en fait locaux.© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.Finsler metric on a smooth manifold M is a continuous function F : T M → [0, ∞) such that for every point p ∈ M the restriction F p = F |T p M on the tangent space at p is a Minkowski norm, that is F p is positively homogeneous and convex and it vanishes only at y = 0:(a) F p(λ · y) = λ · F p(y) for any λ ≥ 0.(b) F p(y + ỹ) ≤ F p(y) + F p( ỹ).(c) F p(y) = 0 ⇒ y = 0.We will also assume that our Finsler metric is of class C2, that is the restriction of F to the slit tangent bundle T M0 =T M \ (the zero section) is a function of class C2, and that it is strongly convex, that is the Hessian of the restriction of 12 F 2to T p M \ {0} (which will be denoted by g = gij later) is positively definite for any p ∈ M and any nonzero vector y ∈ T p M .The Finsler manifold (M, F ) is called locally symmetric if, for every point p ∈ M , there exists r = r(p) > 0 and an isometry I p : Br(p) → Br(p) (called the reflection at p) such that I p(p) = p and dp(I p) = −id : T p M → T p M . Here Br(p) denotes the ball of radius r around p.E-mail address: vladimir.matveev@uni-jena.de.http://dx.doi.org/10.1016/j.crma.2014.10.0221631-073X/© 2014 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.82 V.S. Matveev / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 81–83Our main result is:Theorem 1. Suppose the flag curvature of a locally symmetric Finsler metric is negative or positive. Then, the metric is actually a Riemannian metric, that is, there exists a Riemannian metric h such that for all p ∈ M, y ∈ T p M we have F p(y) =√hp(y, y).Note that there exist examples of locally (and even globally) symmetric Finsler metrics such that the flag curvature changes the sign, or is nonpositive, or is nonnegative. Actually, the (reversible) Minkowski space is already an example of globally symmetric Finsler space of nonpositive and of nonnegative flag curvature, since its flag curvature is zero. One can also take the direct product of a Minkowski space with the round sphere and/or with the hyperbolic space constructing ex-amples such that the flag curvature is nonpositive and is negative somewhere, or such that the flag curvature is nonnegative and is positive somewhere, or such that it changes the sign. The holonomy group of all these examples is reducible, but one can also construct irreducible examples by perturbing a locally symmetric Riemannian metric of rank ≥2 by an arbitrary (smooth, small and homogeneous of degree 1) function of the Chevalley’s polynomials. Since the Chevalley’s polynomials are preserved by any isometry of the initial Riemannian metric (see, e.g., [9,3]), the obtained Finsler metric is still locally symmetric.Special cases of Theorem 1 are the following two statements, which were known before in the “global” setting; the initial proof of these statements is also “global”, i.e., it requires the assumption that the manifold M is compact, and is very different from our proof, which is local.Special Case. (Corollary 1 of Foulon [6]; also follows from theorem (A) of Foulon [5].) Let (M, F ) be a Finsler locally symmetric space of negative flag curvature. If M is compact, then F is actually a Riemannian metric.Special Case. (Independently Deng and Hou [4] and Kim [7].) Let (M, F ) be a Finsler locally symmetric space of positive flag curvature. If M is compact, then F is actually a Riemannian metric.Remark. Actually, Foulon in [5] has a less restrictive definition of locally symmetric spaces (locally symmetric spaces in the our definition are also locally symmetric in the definition of [5] but not vice versa), so his result is in fact stronger and we cannot repeat it by our methods or prove its local version.Proof of Theorem 1. Our proof of Theorem 1 is based on the following recent result:Fact. ([8], Remark (A) in Section 8 + Theorem 9.2.) Let (M, F ) be a C2-smooth Finsler manifold. If (M, F ) is locally symmetric, then F is Berwald. Moreover, the associated connection is the Levi–Civita connection of a locally symmetric Riemannian metric. (We denote this Riemannian metric by h, in paper [8] it is called the Binet–Legendre metric associated with the Finsler metric.)Recall that a Finsler manifold is Berwald, if there exists a torsion free linear connection ∇ called associated connectionsuch that the parallel transport preserves the Finsler metric.It is well known (see for example [1,2]) that the flag curvature K p(y, V ) for the Berwald metrics can be calculated by the following procedure. Compute the curvature tensor R jik of the associated connection; for every p we view the curvature tensor as the mappingR : T p M × T p M × T p M → T p M, R(a,b)c = ci R jikakb. (1)Then, for every two linearly independent vectors yi, V i ∈ T p M we have:K p(y, V ) = g y(V , R(V , y)y)g y(y, y)g y(V , V ) − g y(V , y)2 . (2)Here g y is the second differential of the restriction of the function 12 F2 to T p M; g y = gij is a (0, 2)-tensor whose compo-nents depend on the point p in M and on the tangent vector y ∈ T p M .Let us now consider the Riemannian metric h whose existence we recalled in fact above: it is locally symmetric and its Levi–Civita connection is the associated connection of F . Let us now show that if the flag curvature of F is positive (for all linearly independent y and V ∈ T p M) then the sectional curvature of h is also positive, and if the flag curvature of F is negative, then the sectional curvature of h is also negative.In order to do this, for each vector y ∈ TxM , let us consider the endomorphism:A y : T p M → T p M, V 	→ R(V , y)y.In the tensor notation, (A y)jk = R jik yi y . Since the bilinear form (ξ, ν) 	→ h(ξ, R(ν, y)y) is symmetric with respect to ξand ν because of the symmetries of the curvature tensor, for each y the endomorphism A y is diagonalizable. Clearly, y is an eigenvector of A y with eigenvalue 0. Comparing the formula for A y with (2), we see that the flag curvature is given byV.S. Matveev / C. R. Acad. Sci. Paris, Ser. I 353 (2015) 81–83 83K p(y, V ) = g y(V , A y(V ))g y(y, y)g y(V , V ) − g y(V , y)2 .If K p(y, V ) is positive for all linearly independent y and V , then for each y = 0, all eigenvalues of A y except of the eigenvalue 0 corresponding to the eigenvector y are positive, otherwise the pair (y, V ), where V is an eigenvector with nonpositive eigenvalue has K p(y, V ) ≤ 0. Then, the sectional curvature of h (which is given by h(V ,A y(V ))h(y,y)h(V ,V )−h(V ,y)2 ) is also positive for all linearly independent y and V as we claim.The case when K p(y, V ) is negative for all linearly independent y and V is virtually the same—one needs to replace “positive” by “negative” in the previous arguments.Finally, if the flag curvature is positive, or if it is negative, the sectional curvature of h is positive resp. negative, i.e., is never equal to zero. But then the holonomy group of h is transitive by [10, Theorem 9]. Since both the Finsler function Fand the metric h are preserved by the holonomy group, the function F p(y)2hp(y,y)does not depend on y, so the metric F is a Riemannian metric as we claimed. Theorem 1 is proved.AcknowledgementsThe paper was started during the conference XVII Escola de Geometria Diferencial at Manaus and was initiated by discussions with J.-H. Eschenburg. The last part of the proof of Theorem 1 was explained to me by S. Deng during my visit to Nankai University. I am grateful to the organizational committee of the above-mentioned conference and to Nankai University for partial financial support and their hospitality. I also thank the Friedrich-Schiller University of Jena for partial financial support.References[1] D. Bao, C. Robles, Ricci and Flag Curvatures in Finsler Geometry, Riemann–Finsler Geometry MSRI Publications, vol. 50, 2004.[2] D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann–Finsler Geometry, Graduate Texts in Mathematics, vol. 200, Springer-Verlag Inc., New York, 2000.[3] C. Chevalley, Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955) 778–782.[4] S. Deng, Z. Hou, On symmetric Finsler spaces, Isr. J. Math. 162 (2007) 197–219.[5] P. Foulon, Locally symmetric Finsler spaces in negative curvature, C. R. Acad. Sci. Paris, Ser. I 324 (10) (1997) 1127–1132.[6] P. Foulon, Curvature and global rigidity in Finsler manifolds, Houst. J. Math. 28 (2) (2002) 263–292.[7] C.-W. Kim, Locally symmetric positively curved Finsler spaces, Arch. Math. 880 (2007) 378–384.[8] V.S. Matveev, M. Troyanov, The Binet–Legendre metric in Finsler geometry, Geom. Topol. 16 (2012) 2135–2170.[9] R.S. Palais, C.-L. Terng, Critical Point Theory and Submanifold Geometry, Lecture Notes in Mathematics, vol. 1353, Springer-Verlag, New York, 1988.[10] J. Simons, On transitivity of holonomy systems, Ann. of Math. (2) 76 (1962) 213–234.

There exist no locally symmetric Finsler spaces of positive or negative flag curvature

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There exist no locally symmetric Finsler spaces of positive or negative flag curvature

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