Old and New Structures on the Tangent Bundle

In this paper we study a Riemanian metric on the tangent bundle T (M) of a Riemannian manifoldM 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.