A Note on Some Metrics on Tangent Bundles and Unit Tangent Sphere Bundles

Abstract:

      In this paper we study a class of metrics with some compatible almost complex structures on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$, which are parallel to those in [10]. These metrics generalize the classical Sasaki metric and Cheeger-Gromoll metric. We prove that the tangent bundle $TM$ endowed with each pair of the above metrics and the corresponding almost complex structures is a locally conformal almost \kr manifold. We also find that, when restricted to the unit tangent sphere bundle, these metrics and corresponding almost complex structures define new examples of contact metric structures.