On curvature in Finsler geometry

Abstract

Introduction. H. Busemann in [2] generalizes to Finsler geometry most of those global theorems in Riemannian geometry which relate the nonpositive curvature properties of the Riemannian metric to the topological properties of the manifold. This is carried out by means of his synthetic or axiomatic approach. We have approached the problem of generalizing theorems concerning positive as well as nonpositive curvature from the point of view of the theory of connections. We take as our starting point the connection for a Finsler manifold as calculated by Cartan in [3]. Using as a guide those theorems in Riemannian geometry which relate the position of conjugate points on a geodesic to the sectional curvature along the geodesic, we have given an analytic definition of sectional curvature which we-believe is natural and which further permits us to generalize such theorems as: Myers' Theorem [5], which states that any complete Riemann manifold with positive mean curvature is compact; Synge's Theorem [8 ], which states that if an orientable even dimensional Riemann manifold is complete and has positive sectional curvature then it is simply connected; and that if M is a simply connected, complete Riemann manifold that has nonpositive curvature, then it is homeomorphic to the Euclidean space of the same dimension. We shall assume in presenting the details of this paper that the reader is familiar with E. Cartan's book [3 ] and with S. S. Chern's paper [4]. We wish to express our appreciation to Professor Chern without whose guidance and encouragement this work would not have been possible. 1. On the reduction of the group. Let V be an n dimensional vector space and let D be the n-1 dimensional space of directions in V. D may be defined as the equivalence classes of V, with the origin deleted, under the following equivalence relation: vi, v2E V will be said to be equivalent if there exists a constant k, greater than zero, such that v1 = kv2. It is clear that D may be considered topologically as an n -1 dimensional sphere. Now let M be a differentiable manifold. Let T be the tangent bundle to M and let S be the associated sphere bundle. Further, let p: S->M be the projection map. Then to each point m E M we may associate V(m) and D(m) as the fiber over the point m in the bundles T and S respectively. We shall denote an element of S by (m, d) and an element of T by (m, v). We then have the well defined concept of the vector v lying in the direction d at m, provided v is not the zero vector.