Cotangent Bundles with General Natural Kahler Structures

TL;DR

This paper investigates the conditions for cotangent bundles of Riemannian manifolds to admit Kähler structures, deriving algebraic and integrability conditions for the almost Hermitian structures of natural lift type.

Contribution

It provides new algebraic and integrability conditions characterizing when cotangent bundles have Kähler structures of natural lift type, depending on three key parameters.

Findings

Manifold is almost Hermitian under certain algebraic conditions.

Integrability of the almost complex structure is characterized.

Kählerian structures depend on three essential parameters.

Abstract

We study the conditions under which an almost Hermitian structure $(G,J)$ of general natural lift type on the cotangent bundle $T^*M$ of a Riemannian manifold $(M,g)$ is K\" ahlerian. First, we obtain the algebraic conditions under which the manifold $(T^*M,G,J)$ 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 $(T^*M,G,J)$ 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 $T^*M$ depends on three essential parameters (one is a certain proportionality factor, the other two are parameters involved in the definition of $J$).