Remarks on some almost Hermitian structure on the tangent bundle

Abstract

In [5], M. Tahara and Y. Watanabe constructed a family of almost Hermitian structures (J,G) on the tangent bundle TM of a Riemannian manifold and constructed a family of Hermitian and Kähler structure on the tangent bundle on a space form. It is well-known that there are sixteen classes of almost Hermitian manifolds ([3]). In this paper, we give the conditions for (J,G) such that TM belongs to each of these sixteen classes.