On two point homogeneous Finsler spaces - CiteSeerX

On two point homogeneous Finsler spaces ∗ Shaoqiang Deng and Zixin Hou School of Mathematics and LPMC, Nankai University, Tianjin 300071, P. R. China. E-mail: [email protected], [email protected].

Abstract In this note, we use the result that the group of isometries of a Finsler space is a Lie transformation group of the underlying manifold to prove that a two point homogeneous Finsler space is a Riemannian space. Key words: Two point homogeneous Finsler spaces, group of isometries, Minkowski spaces. Mathematical Subject Classification (2000): 53C60, 58B20, 22E46.

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Introduction

The study of Finsler geometry recently becomes active due to the excellent works of many geometers. In particular, the publication of several substantial books has attracted more and more people to this field (cf [3], [4], [9], etc.) One of the important motivation of the study of Finsler geometry is that it has important applications in Physics and Biology ([2]). While many works have been done on the general geometric properties of Finsler geometry, such as connections, geodesics and curvatures, only very little attention has been paid to the group aspects of this interesting field. This may be mainly due to the reason that the Myers-Steenrod theorem in Riemannian geometry was not successfully generalized to the Finslerian case for a rather long period. This goal was achieved in our previous paper [5]. Namely we proved that the group of isometries of a Finsler space (not necessary reversible) is a Lie transformation group of the underlying manifold. This result opens a door to using Lie group theory to study Finsler geometry ([7]). ∗

Supported by NSFC (No.10371057), EYTP and NCET of China.

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The purpose of the present note is to study two point homogeneous Finsler spaces. These are the Finsler spaces which occupy most symmetry. The definition of two point homogeneous Finsler spaces is similar to the Riemannian case. Let (M, F ) be a connected Finsler manifold, where F is positively homogeneous (but perhaps not absolutely homogeneous) of degree one. Then (M, F ) is called two point homogeneous if for any two pairs of points (p1 , q1 ) and (p2 , q2 ) satisfying d(p1 , q1 ) = d(p2 , q2 ) there exists an isometry σ of (M, F ) such that σ(p1 ) = p2 and σ(q1 ) = q2 . The main result of the paper can be stated as the following: Main Theorem Let (M, F ) be a Finsler space, where F is positively homogeneous (but not necessary absolutely homogeneous) of degree one. If (M, F ) is two point homogeneous, then F is Riemannian. Intuitively, the theorem says that a non-Riemannian Finsler manifold cannot occupy too much symmetry. It should be noted that Szab´o proved in [10] that on each irreducible Riemannian globally symmetric space of rank ≥ 2 there exists infinitely many invariant non-Riemannian Berwald metrics whose connection coincides with the Levi-Civita connection of the underlying Riemannian metric.

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The case of Minkowski spaces

We first consider the special case of Minkowski spaces. In this case our result is closely related to a classical problem in functional analysis, namely the BanachMazur rotation problem. This problem can be stated as the following. Let X be a separate Banach space. Suppose that the group of linearly isometries of X acts transitively on the unit sphere of X, is X necessary a Hilbert space? The finite-dimensional case was solved affirmatively by Mazur in [8]. Other proofs can be found in [1] (page 52), [11] (page 83). However, all these proofs depend on the assumption that the norm under the consideration is reversible, i.e., ||x|| = || − x||. For example, in [1] (page 52), this result is stated as a characterization labelled (6.12) (which is just Mazur’s theorem). The author proved that (6.12) is equivalent to another characterization (6.11) and any norm satisfying (6.11) is an inner product. But in the proof of (6.11), he used the property that if a cone C of a linear space E is convex (i.e., u, v ∈ C implies u + v ∈ C) and satisfies spanC = E, then C = E. This evidently is true only when C is reversible (i.e., x ∈ C implies −x ∈ C). However, if the norm is not reversible, then the cone C defined there is necessary not reversible. Therefore this proof can not be adapted 2

to the non-reversible case. Some similar analysis of other existing proofs shows that they are all only valid for the reversible case. In this section we will generalize Mazur’s result to the non-reversible case, but under the assumption that the norm is Minkwoski. Namely, the norm F in the real vector space V is smooth away from the origin and for any point y 6= 0, the quadratic form gy (u, v) =

1 ∂ 2 [F 2 (y + su + tv)] |s=t=0 , 2 ∂s∂t

u, v ∈ V,

is positive definite. In the process of the proof of the following proposition we use the following result of [5]: Let (M, F ) be a Finsler space and σ be a distance-preserving mapping of M onto itself. Then σ is a differmorphism of M and ∀p ∈ M , dσp is a linear isometry from the Minkowski space (Tp (M ), F ) onto (Tσ(p) (M ), F ). Proposition Let (Rn , F ) a Minkowski space. If (V, F ) is two point homogeneous when viewed as a Finsler space in the canonical way (see [3], page 14), then F is the Euclidean norm of an inner product. Proof Let I denote the group of isometries of (Rn , F ) and Io be the subgroup of I which leaves the origin o fixed. Then [5] proved that I is a Lie group with respect to the compact-open topology and Io is a compact subgroup of I. Furthermore, it is also true that Io consists of linear transformations of Rn (see the proof of Proposition 2.1 of [5]). Thus Io is a compact subgroup of GL(n, R). Let Ioe be the unit component of Io . Then we assert that Ioe is a subgroup of SL(n, R) (i.e., each element of calI e0 has determinant equal to 1) . In fact, the determinant function is continuous on GL(n, Rn ). Therefore it is bounded on the compact subgroup Ioe . Then each element of Ioe must have determinant ≤ 1. On the other hand, if φ ∈ I0e has determinant < 1 (and certainly > 0), then φ−1 has determinant > 1. This is a contradiction. This proves our assertion. Now SL(n, R) is a connected semisimple Lie group and SO(n) is a maximal compact subgroup of SL(n, R). By conjugacy of maximal compact subgroup of connected semisimple Lie groups ([6]), there exists g ∈ SL(n, R) such that g −1 Ioe g ⊂ SO(n). Now consider the indicatrix Io = {y ∈ Rn |F (y) = 1}. Since (Rn , F ) is two point homogeneous, for any two points y1 , y2 ∈ I0 , there exists φ ∈ Io such that φ(y1 ) = y2 . That is, Io is transitive on Io . We assert that Ioe is also transitive on Io . In fact, for any y ∈ Io , the orbit Ioe · y is an open and 3

closed subset of Io · y = I0 . Since Io is connected, we conclude that Ioe · y = Io . Now for any x1 , x2 ∈ S n−1 (S n−1 is defined with respect to the standard inner g(x1 ) g(x2 ) product of Rn ), F (g(x ∈ Io , F (g(x ∈ Io . Thus there exists g1 ∈ Ioe such that 1 )) 2 )) g1 (

g(x1 ) g(x2 ) )= . F (g(x1 )) F (g(x2 ))

Then we have g1 g(x1 )) =

F (g(x1 )) g(x2 ). F (g(x2 ))

Thus g −1 g1 g(x1 ) = Since g −1 g1 g ∈ SO(n), x1 , x2 ∈ S n−1 and

F (g(x1 )) x2 . F (g(x2 )) F (g(x1 )) F (g(x2 ))

> 0, we have

F (g(x1 )) F (g(x2 ))

= 1. Thus

g −1 g1 g(x1 ) = x2 . This means that g −1 Ioe g is transitive on S n−1 , or in other words, Ioe is transitive on the hypersurface g · S n−1 . Hence F is constant on g · S n−1 . Suppose F (g · S n−1 ) = λ > 0. Then it is easy to see that F coincides with the Euclidean norm of the inner product defined by hy1 , y2 i1 = λhg −1 · y1 , g −1 · y2 i, where h , i is the standard inner product of Rn . 2

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Proof of the main theorem

Now we can give the proof of the main result of the paper. Proof of the main theorem Let (M, F ) be a connected two point homogeneous Finsler manifold and p ∈ M . We proceed to prove that (Tp (M ), F ) is a two point homogeneous Minkowski space. Since for any y ∈ Tp (M ), the parallel translate πy : πy (x) = x − y is an isometry with respect to F and F is positively homogeneous of degree one, we only need to show that the isotropy subgroup Io (Tp (M )) of I(Tp (M )) at the origin acts transitively on the set Io (λ) = {x ∈ Tp (M )|F (x) = λ}, for some λ > 0. Now we select r > 0,  > 0 such that the exponential mapping expp is a C 1 -differmorphism from the tangent ball Bp (r + ) = {x ∈ Tp (M )|F (x) < r + } 4

onto its image. Let Sp (r) = {x ∈ Tp (M )|F (x) = r}, Sp+ (r) = {q ∈ M |d(p, q) = r}. Then we have ([3], page 155, Theorem 6.3.1) expp [Sp (r)] = Sp+ (r). Therefore for any y1 , y2 ∈ Io (r), expp (y1 ), expp (y2 ) ∈ Sp+ (r). Since (M, F ) is two point homogeneous, there exist an isometry σ of (M, F ) such that σ(p) = p and σ(expp (y1 )) = expp (y2 ). This means that the geodesics σ(expp (ty1 )), expp (ty2 ), 0 ≤ t ≤ 1 coincide. Therefore we have dσp (y1 ) = y2 . Since dσp is a linear isometry of (Tp (M ), F ), we conclude that Io (Tp (M )) acts transitively on Io (r). Thus (Ip (M ), F ) is a two point homogeneous Minkowski space. By the Proposition, F |Tp (M ) is the Euclidean norm of an inner product. Since p is arbitrary, F is Riemannian. 2 The compact two point homogeneous Riemannian spaces were classified by H. C. Wang [13] and the noncompact ones were classified by J. Tits [12]. Therefore their lists are also the classification of two point homogeneous Finsler spaces. Acknowledgements The author is grateful to Professor Z. Shen at Indianan University-Purdue University Indianapolis for providing them with valuable materials.

References [1] D. Amir, Characterizations of product spaces, Birh¨auser Verlag, Basel. Boston. Stuggart, 1986. [2] P. L. Antonelli, R. S. Ingarden and M. Matsumato, The theory of sprays and Finsler spaces with applications in Physics and Biology, FTPH 58, Kluwer Academic Publishers, 1993. [3] D. Bao, S. S. Chern and Z. Shen, An Introduction to Riemann-Finsler Geometry, Springer-Verlag, New York, 2000. [4] S. S. Chern and Z. Shen, Riemann-Finsler Geometry, World Scientific Publishers, 2004. [5] S. Deng and Z. Hou, The group of isometries of a Finsler space, Pacific J. Math. 207 (2002), 149-155. 5

[6] S. Helgason, Differential Geometry, Lie groups and Symmetric Spaces, 2nd ed., Academic Press, 1978. [7] Mathematical Reviews, MR1974469 (Reviewer: Z. Shen), American Math. Soc. [8] S. Mazur, Quelques proprietes characteristiques des espaces euclideans, Comptes Rendues Acad. Sci. Paris, 2007 (1938), 761-764. [9] Z. Shen, Differntial Geometry of Spray and Finsler spaces, Kluwer Academic Publishers, Dordrecht, 2001. [10] Z. I. Szab´o, Positive Definite Berwald Spaces, Tensor, N. S., 38 (1981), 25-39. [11] A. C. Thompson, Minkowski Geometry, Cambridge University Press, 1996. [12] J. Tits, Sur certaine d’espaces homog`enes de groupes de Lie, Acad. Roy. Belg. Cl. Sci. M´em. Coll. 29 (1955), No. 3. [13] H. C. Wang, Two point Homogeneous Spaces, Ann. of Math. 55 (1952), 177-191.

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