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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.

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Abstract. We consider a certain pseudo-Riemannian metric G on the tangent bundle T M of a Riemannian manifold (M, g) and obtain necessary and sufficient conditions for the pseudo-Riemannian manifold (T M, G) to be Ricci flat (see Theorem 2). A Riemannian metric G on T M has been defined by using the Levi-Civita connection of g and two smooth real-valued functions u(t), v(t) depending on the energy density only and such that u(t) > 0 and u(t)+2tv(t) > 0 for all t ∈ [0, ∞). He has also considered an almost complex structure J on T M , related to the metric G and has studied the conditions under which (T M, G, J) is a Kähler Einstein manifold. Note that in [10], the author excludes some important cases which appeared, in a certain sense, as singular cases. These singular cases have been studied by V. Oproiu and the present author in [11], [9], [12]. Note also that one of the important cases studied in [11] is when the Riemannian metric G on T M is defined by using a certain Lagrangian L on the base manifold (M, g) depending on the energy density only (i.e. the case when v(t) = u ′ (t)). On the other hand, in [8], V. Oproiu has studied a pseudo-Riemannian structure on the tangent bundle of a Lagrange manifold M , considering the pseudo-Riemannian metric G on T M as being the complete lift of a quadratic form defined by the Lagrangian L considered.