On natural metrics on tangent bundles of Riemannian manifolds

Abstract

summary:There is a class of metrics on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$ (oriented , or non-oriented, respectively), which are ’naturally constructed’ from the base metric $g$ [Kow-Sek1]. We call them “$g$-natural metrics" on $TM$. To our knowledge, the geometric properties of these general metrics have not been studied yet. In this paper, generalizing a process of Musso-Tricerri (cf. [Mus-Tri]) of finding Riemannian metrics on $TM$ from some quadratic forms on $OM \times \mathbb {R}^m$ to find metrics (not necessary Riemannian) on $TM$, we prove that all $g$-natural metrics on $TM$ can be obtained by Musso-Tricerri’s generalized scheme. We calculate also the Levi-Civita connection of Riemannian $g$-natural metrics on $TM$. As application, we sort out all Riemannian $g$-natural metrics with the following properties, respectively: 1) The fibers of $TM$ are totally geodesic. 2) The geodesic flow on $TM$ is incompressible. We shall limit ourselves to the non-oriented situation.