Some results on the geometry of tangent bundle of Finsler manifolds

Abstract

Abstract. In this paper we study the geometry of tangent bundle of a Finslermanifold endowed with Sasaki metric and obtain some results. We characterize theRiemannian manifold as the Finsler manifold such that the vertical lift of any vectorfield is divergence-free or equivalently, such that the horizontal distribution is minimalin the tangent bundle of the slit tangent bundle, and prove that the almost complexstructure on the slit tangent bundle is integrable if and only if the base manifold haszero flag curvature. In that case, the slit tangent bundle is K¨ahlerian. We also provethat the slit tangent bundle is locally symmetric if and only if the base manifold islocally Euclidean. Our results generalize the corresponding results for the Riemanniansetting in the literature. 1. IntroductionThe geometry of tangent bundle or tangent sphere bundle of a Riemannianmanifold has been well developed. In the general case, however, the geometry oftangent bundle or the indicatrix bundle of a Finsler manifold has not been studiedat the same pace. Several people have made some fundamental contributions tothis subject from various points of view. For instance, Hasegawa, Yamauchi,and Shimada proved that the indicatrix bundle of a Finsler manifold (M,F) withthe induced almost contact metric structure is Sasakian if and only if (M,F) is ofconstant flag curvature 1 [8], while Bejancu and Farran obtained that (M,F)