Cotangent Bundles with General Natural Kahler Structures

Abstract

We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^{\ast }M$ of a Riemannian manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^{\ast }M,G,J)$ is almost Hermitian. Next we get the integrability conditions for the almost complex structure $J$, then the conditions under which the associated 2-form is closed. The manifold $(T^{\ast }M,G,J)$ is K\" ahlerian iff it is almost Kahlerian and the almost complex structure $J$ is integrable. It follows that the family of Kahlerian structures of above type on $T^{\ast }M$ depends on three essential parameters (one is a certain proportionality factor, the other two are parameters involved in the definition of $J$).

9 pages

References in corpus (1)

• A locally symmetric Kaehler Einstein structure on a tube in the nonzero cotangent bundle of a space form