Old and New Structures on the Tangent Bundle

Abstract

In this paper we study a Riemanian metric on the tangent bundle $T(M)$ of a Riemannian manifold $M$ which generalizes Sasakian metric and Cheeger–Gromoll metric along a compatible almost complex structure which together with the metric confers to $T(M)$ a structure of locally conformal almost Kählerian manifold. This is the natural generalization of the well known almost Kählerian structure on $T(M)$. We found conditions under which $T(M)$ is almost Kählerian, locally conformal Kählerian or Kählerian or when $T(M)$ has constant sectional curvature or constant scalar curvature.