On Projective Transformations and Conformal Transformations of the Tangent Bundles of Riemannian Manifolds

Abstract

In the present paper everything will be always discussed in the C ^∞ category, and Riemannian manifolds will be assumed to be connected and dimension > 1. Let M be a Riemannian manifold with Riemannian metric g, and T(M) be the tangent bundle of M. There are many Riemannian or pseudo-Riemannian metrics in T(M) which are defined by g, for example, the Sasaki metric, the complete lift metric, the Cheeger-Gromoll metric, etc. In this paper we consider infinitesimal projective transformations and infinitesimal conformal transformations of T(M) with the Sasaki metric, or the complete lift metric, and we shall prove the following theorems.