A Harmonic Endomorphism in a Semi-Riemannian Context

Abstract

On the total space of the cotangent bundle $T^{*}M$ of a Riemannian manifold $(M,h)$ we consider the natural Riemann extension $\bar{g}$ with respect to the Levi-Civita connection of $h$. In this setting, we construct on $T^{*}M$ a new para-complex structure $P$, whose harmonicity with respect to $\bar{g}$ is characterized here by using the reduction of $\bar{g}$ to the (classical) Riemann extension.