The Bourguignon Laplacian and Harmonic Symmetric Bilinear Forms

Abstract

In this paper, we study the kernel and spectral properties of the Bourguignon Laplacian on a closed Riemannian manifold, which acts on the space of symmetric bilinear forms (considered as one-forms with values in the cotangent bundle of this manifold). We prove that the kernel of this Laplacian is an infinite-dimensional vector space of harmonic symmetric bilinear forms, in particular, such forms on a closed manifold with quasi-negative sectional curvature are zero. We apply these results to the description of surface geometry.

Keywords

• riemannian manifold
• bourguignon laplacian
• symmetric bilinear form
• harmonic
• curvature
• spectral theory
• vanishing theorem