Randers Spaces Of Constant S-curvature Induced By Almost Contact Metric Structures

In Finsler geometry, there is an important class of Finsler metrics—Randers metric which was introduced by G. Randers in 1941 to discuss the asymmetrical metric in the four dimensional space of general relativity. It has been studied later by many physicists and mathematicians as the simplest Finslerian deformation of a Riemannian space (M,α) by a linear 1-formβ.The S-curvature is one of the most important non-Riemannian quantities in Finsler geometry which was first introduced by Zhongmin Shen when he studied volume comparison in Riemann-Finsler geometry. The study of Finsler manifolds with constant S-curvature plays a very important role in Finsler geometry.In this thesis, we construct a positive definite Randers metric F_ε=α+εη, 0 < |ε| < 1 induced by an almost contact metric structure M(φ,ξ,η,α) and compute the S-curvature. We show that a connected almost contact Riemannian manifold of odd dimension n > 3 induces a natural structure of positive definite Randers space. We conclude that: if M(φ,ξ,η,α) is a Sasakian manifold, then (M,F_ε) has vanishing S-curvature S = 0, and F_εis not a Berwald metric; if M(φ,ξ,η,α) is a cosymplectic manifold, then (M,F_ε) has vanishing S-curvature S = 0; and if M(φ,ξ,η,α) is a Kenmotsu manifold, then (M, F_ε) can not be of almost isotropic S-curvature.This paper is organized as follows. In Chapter 1, we recall the basics about Randers space and S-curvature. In Chapter 2, we recall some results on almost contact metric structures. And Chapter 3 is the section where we prove our main results, i.e.: A connected almost contact Riemannian manifold M (φ,ξ,η,α) of odd dimension n≥3 naturally induces a positive definite Randers metric F_ε=α+εη, 0 < |ε| < 1. If M(φ,ξ,η,α) is a Sasakian manifold, then (M,F_ε) has vanishing S-curvature 5 = 0, and F_εis not a Berwald metric; if M(φ,ξ,η,α) is a cosymplectic manifold, then (M, F_ε) has vanishing S-curvature 5 = 0; and if M(φ,ξ,η,α) is a Kenmotsu manifold, then (M, F_ε) can not be of almost isotropic S-curvature.