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.

Keywords

• Riemannian manifold
• Cotangent bundle
• Natural Riemann extension
• Laplacian operator
• Harmonic function