Kahler-Einstein Structures of General Natural Lifted Type on the Cotangent Bundles

Abstract

We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a K\"ahlerian structure $(G,J)$ of general natural lift type (see \cite{Druta1}), is Einstein. We first obtain a general natural K\"ahler-Einstein structure on the cotangent bundle $T^*M$. In this case, a certain parameter, $\lambda$ involved in the condition for $(T^*M,G,J)$ to be a K\"ahlerian manifold, is expressed as a rational function of the other two, the value of the constant sectional curvature, $c$, of the base manifold $(M,g)$ and the constant $\rho$ involved in the condition for the structure of being Einstein. This expression of $\lambda$ is just that involved in the condition for the K\"ahlerian manifold to have constant holomorphic sectional curvature (see \cite{Druta2}). In the second case, we obtain a general natural K\"ahler-Einstein structure only on $T_0M$, the bundle of nonzero cotangent vectors to $M$. For this structure, $\lambda$ is expressed as another function of the other two parameters, their derivatives, $c$ and $\rho$.