In [11] we have considered a family of natural almost anti-Hermitian structures (G, J) on the tangent bundle T M of a Riemannian manifold (M, g), where the semi-Riemannian metric G is a lift of natural type of g to T M, such that the vertical and horizontal distributions V T M, HT M are maximally isotropic and the almost complex structure J is a usual natural lift of g of diagonal type interchanging V T M, HT M (see [5], [15]). We have obtained the conditions under which this almost anti-Hermitian structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification given in [1]. In this paper we consider another semi-Riemannian metric G on T M such that the vertical and horizontal distributions are orthogonal to each other. We study the conditions under which the above almost complex structure J defines, together with G, an almost anti-Hermitian structure on T M. Next, we obtain the conditions under which this structure belongs to one of the eight classes of anti-Hermitian manifolds obtained in the classification in [1]. Mathematics Subject Classification (2000). Primary 53C55, 53C15, 53C05.

Abstract. We obtain a Kähler Einstein structure on the tangent bundle of a Riemannian manifold of constant negative curvature. Moreover, the holomorphic sectional curvature of this Kähler Einstein structure is constant. Similar results are obtained for a tube around zero section in the tangent bundle, in the case of the Riemannian manifolds of constant positive curvature.

No abstract