An anti-Kählerian Einstein structure on the tangent bundle of a space form
In  we have considered a family of almost anti-Hermitian structures $(G,J)$ on the tangent bundle $TM$ of a Riemannian manifold $(M,g)$, where the almost complex structure $J$ is a natural lift of $g$ to $TM$ interchanging the vertical and horizontal distributions $VTM$ and $HTM$ and the metric $G$ is a natural lift of $g$ of Sasaki type, with the property of being anti-Hermitian with respect to $J$. Next, we have studied the conditions under which $(TM, G, J)$ belongs to one of the eight classes of anti-Hermitian structures obtained in the classification in . In this paper, we study some geometric properties of the anti-Kählerian structure obtained in . In fact we prove that it is Einstein. This result offers nice examples of anti-Kählerian Einstein manifolds studied in .