Chapter VI Finslerian manifolds of constant sectional curvature [4

Abstract

This chapter is a study of isotropic and constant sectional curvature Finslerian manifolds. We first recall briefly the basics of Finslerian manifolds, define the isotropic manifolds and single out the properties of their curvature tensors. We then give a characterization of Finslerian manifolds with constant sectional curvature, generalizing Schur’s classical theorem. We next determine the necessary and sufficient conditions for an isotropic Finslerian manifold to be of constant sectional curvature. Our conditions bear on the Ricci directional curvature or on the second scalar curvature of Berwald. We show that the existence of normal geodesic coordinates of class C 2 on isotropic manifolds forces them to be Riemannian or locally Minkowskian. We also deal with the case of compact isotropic Finslerian manifolds with strictly negative curvatures. In chapter III we give a classification of complete Finslerian manifolds with constant sectional curvatures. We prove that all geodesically complete Finslerian manifolds of dimension n > 2 with negative constant sectional curvature (K < 0) and with bounded torsion vector are Riemannian. We show that all simply connected Finslerian manifolds of dimension n > 2 with strictly positive constant sectional curvature and whose indicatrix is symmetric and has a scalar curvature independent of the direction is homeomorphic to an n-sphere. In the case when the Berwald curvature H vanishes and torsion tensor as well as its covariant vertical derivative are bounded we prove that the manifold in question is Minkowskian. In the last chapter we establish the ‘axioms of the plane’. By defining the totally geodesic, semi-parallel and auto-parallel Finslerian submanifolds we establish the criteria that permit to identify if a Finslerian manifold is of constant sectional curvature in the Berwald connection (axiom 1), in the Finslerian connection (axiom 2) or is Riemannian (axiom 3).