The Holomorphic Sectional Curvature of General Natural K\"Ahler Structures on Cotangent Bundles

TL;DR

This paper investigates the conditions for a natural K"ahler structure on the cotangent bundle of a Riemannian manifold to have constant holomorphic sectional curvature, deriving relations among parameters and curvatures.

Contribution

It provides explicit conditions and formulas relating the parameters of the K"ahler structure to the base manifold's curvature and the holomorphic sectional curvature.

Findings

Parameter expressed as a rational function of other parameters and curvatures.

Conditions for constant holomorphic sectional curvature derived.

Relations between base manifold curvature and cotangent bundle structure established.

Abstract

We study the conditions under which a K\"ahlerian 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 in the condition for $(T^*M,G,J)$ to be a K\"ahlerian 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 K\"ahlerian structure $(G,J)$.