Infinitesimal Projective Transformations on the Tangent Bundle of a Riemannian Manifold with a Class of Lift Metrics

Abstract

Let ‎$‎ (M,g) $ be a Riemannian manifold and ‎$‎ TM $ be its tangent bundle. ‎In the present paper‎, ‎we study infinitesimal projective transformations on $ TM $ with respect to the Levi-Civita connection of a class of (pseudo-)Riemannian metrics ‎$ ‎‎\tilde{g}‎ $ which is a generalization of the three classical lifts of the metric ‎$‎g$. We characterized this type of transformations and then ‎we prove that if $ (TM,‎\tilde{g}‎) $ admits a non-affine infinitesimal projective transformation‎, ‎then $ M $ and $ TM‎‎ $ are locally flat.