3-symmetric Finsler Manifolds

Abstract

symmetric Finsler manifold is the generalization of 3-symmetric Riemannian manifold. In this paper we give the definition of 3-symmetric Finsler manifold and characterize 3- symmetric Finsler manifold as homogeneous space. We also give the conditions for the existence of 3-symmetric Finsler metrics on homogeneous spaces and the relation between 3-symmetric Finsler manifold and 3-symmetric Riemannian manifold. In the end, we give the flag curvature and curvature tensor of naturally reductive 3-symmetric Finsler manifold. Keywords: 3-symmetric Finsler manifold; local cubic diffeomorphism; flag curvature MR(2010) Subject Classification: 53C30; 53C60 / CLC number: O186.14 Document code: A Article ID: 1000-0917(2015)04-0607-07 3-symmetric Finsler manifold is the generalization of 3-symmetric Riemannian manifold. It has close relation to complex Finsler manifold (6) . In 1972, Gray gave the definition of pseudo-Riemannian 3-symmetric space and the classification of pseudo-Riemannian 3-symmetric spaces (9) . The theory of pseudo-Riemannian 3-symmetric spaces parallels that of ordinary sym- metric spaces (7,10) to a great extent. However, there are important exceptions. For example, 3-symmetric spaces are automatically almost complex manifolds. In this paper we generalize 3-symmetric Riemannian manifolds to 3-symmetric Finsler man- ifolds. This paper is organized as follows. In Section 1 we give some basic definitions of Finsler manifold. In Section 2 we give the definition of 3-symmetric Finsler manifold. In Section 3 we give the algebraic structure of 3-symmetric Finsler manifold and the condition for the exis- tence of invariant Finsler metric on a homogeneous space which makes the homogeneous space a 3-symmetric Finsler manifold. In Section 4 we give the curvature tensor and flag curvature of naturally reductive 3-symmetric Finsler manifold. In Section 5 we give some examples of 3-symmetric Finsler manifolds. 1 Preliminaries