We study the conditions under which the cotangent bundle $T^*M$ of a Riemaannian manifold $(M,g)$, endowed with a Kahlerian structure $(G,J)$ of general natural lift type (see cite{Druta1}), is Einstein. We first obtain a general natural Kahler-Einst
ein 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 Kahlerian 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 Kahlerian manifold to have constant holomorphic sectional curvature (see cite{Druta2}). In the second case, we obtain a general natural Kahler-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 $rho$.
We study the conditions under which a Kahlerian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ has constant holomorphic sectional curvature. We obtain that a certain parameter involved i
n the condition for $(T^*M,G,J)$ to be a Kahlerian manifold, is expressed as a rational function of the other two, their derivatives, the constant sectional curvature of the base manifold $(M,g)$, and the constant holomorphic sectional curvature of the general natural Kahlerian structure $(G,J)$.
We study the conditions under which the tangent bundle $(TM,G)$ of an $n$-dimensional Riemannian manifold $(M,g)$ is conformally flat, where $G$ is a general natural lifted metric of $g$. We prove that the base manifold must have constant sectional c
urvature and we find some expressions for the natural lifted metric $G$, such that the tangent bundle $(TM,G)$ become conformally flat.
