On some differential operators on natural Riemann extensions

Abstract

Natural Riemann extensions are pseudo-Riemannian metrics (introduced by Sekizawa and studied then by Kowalski–Sekizawa), which generalize the classical Riemann extension defined by Patterson–Walker. Let \(M\) be a manifold with an affine connection and let \(T^{*}M\) be the total space of its cotangent bundle. On \(T^{*}M\) endowed with a natural Riemann extension, we study here the Laplacian and give necessary and sufficient conditions for the harmonicity of a certain family of (local) functions. We also prove a gradient formula for natural Riemann extensions.