Chapter V Properties of compact finslerian manifolds of non-negative curvature

Abstract

(Abstract) The objective of this chapter is to obtain a classification of Finslerian manifolds. Let (M, g) be a Finslerian manifold of dimension n, and W(M) the fibre bundle of unit tangent vectors to M. The curvature form of the Finslerian connection (Cartan) associated to (M, g) is a two from on W(M) with values in the space of skew-symmetric endomorphisms of the tangent space to M. It is the sum of three two forms of type (2, 0), (1, 1) and (0, 2) whose coefficients R, P and Q constitute the three curvature tensors of the given connection. In the first part we study the Landsberg manifolds, manifolds with minimal fibration and Berwald manifolds. The manifold M is called a Landsberg manifold if P vanishes everywhere. This condition is equivalent to the vanishing of the covariant derivative in the direction of the canonical section v: M → V(M) of the torsion tensor. For a Riemannian metric (0,2) on V(M) this condition means that for every x ∈ M the fibre p− 1(x) becomes a totally geodesic manifold where p: V(M) → M (see [5 and §7]). We examine the case when V(M) → M is of minimal fibration as well as when M is a Berwald manifold. When M is compact and without boundary we put some global conditions on the first curvature tensor R or flag curvature of the Cartan connection. In the second part we study by deformations the metric of compact Finslerian manifolds in order that their indicatrix become Einstein manifolds.