Remarks on some almost Hermitian structure on the tangent bundle, II

Abstract

Tahara, Vanhecke and Watanabe constructed a family of almost Her- mitian structure $(J_1;G_1)$ and two almost complex structures $J_2$, $J_3$ on the tangent bundle $TM$ over an almost Hermitian manifold $M$. In this paper, we define Riemannian metrics $G_2$ and $G_3$ on $TM$ which make $TM$ an almost Hermitian manifold and determine the conditions for $(J_i, G_i) (i = 1; 2; 3)$ so that $(TM; J_i, G_i)$ belongs to each of the sixteen classes established by Gray and Hervella.