Killing vector fields on tangent sphere bundles

Abstract

The tangent bundle TM of a Riemannian manifold (M, g) admits a Riemannian metric G called the Sasaki metric. The general forms of Killing vector fields on (TM, G) are determined by Tanno [4]. The total space of the tangent sphere bundle T^λM is the set of all tangent vectors of (M, g) whose lengths are all equal to λ(≠0), and it is a hypersurface of (TM, G). In the present paper we study Killing vector fields on T^λM which are fiber preserving. The main theorem of this paper shows that any fiber preserving Killing vector field on (T^λM, G^λ) is extended to a Killing vector field on (TM, G). Moreover, we will find a Riemannian manifold (M, g) such that any Killing vector fields on T¹M is fiber preserving.