A Kaehler Einstein structure on the cotangent bundle of a Riemannian manifold

Abstract

We use the natural lifts of the fundamental tensor field g to the cotangent bundle T*M of a Riemannian manifold (M,g), in order to construct an almost Hermitian structure (G,J) of diagonal type on T*M. The obtained almost complex structure J on T*M is integrable if and only if the base manifold has constant sectional curvature and the second coefficient, involved in its definition is expressed as a rational function of the first coefficient and its first order derivative. Next one shows that the obtained almost Hermitian structure is almost Kaehlerian. Combining the obtained results we get a family of Kaehlerian structures on T*M, depending on one essential parameter. Next we study the conditions under which the considered Kaehlerian structure is Einstein. In this case (T*M,G,J) has constant holomorphic curvature.