# The Bourguignon Laplacian and harmonic symmetric bilinear forms

## TL;DR

This paper explores the properties of harmonic symmetric bilinear forms on Riemannian manifolds, focusing on the Bourguignon Laplacian, and establishes their finite-dimensionality and invariance under certain curvature conditions.

## Contribution

It proves the finite-dimensionality of the kernel of the Bourguignon Laplacian and demonstrates invariance of harmonic forms under parallel translation on manifolds with non-negative curvature.

## Key findings

- Kernel of Bourguignon Laplacian is finite-dimensional.
- Harmonic symmetric bilinear forms are invariant under parallel translation with non-negative curvature.
- Spectral properties of the Bourguignon Laplacian are investigated.

## Abstract

The theory of harmonic symmetric bilinear forms on a Riemannian manifold is an analogue of the theory of harmonic exterior differential forms on this manifold. To show this, we must consider every symmetric bilinear form on a Riemannian manifold as a one-form with values in the cotangent bundle of this manifold. In this case, there are the exterior differential and codifferential defined on the vector space of these differential one-forms. Then a symmetric bilinear form is said to be harmonic if it is closed and coclosed as a one-form with values in the cotangent bundle of a Riemannian manifold. In the present paper we prove that the kernel of the little known Bourguignon Laplacian is a finite-dimensional vector space of harmonic symmetric bilinear forms on a compact Riemannian manifold. We also prove that every harmonic symmetric bilinear form on a compact Riemannian manifold with non-negative sectional curvature is invariant under parallel translations. In addition, we investigate the spectral properties of the little studied Bourguignon Laplacian.