# A Canonical Laplacian on the Algebra of Densities on a Projectively Connected Manifold

## TL;DR

This paper introduces a canonical second order differential operator on the algebra of densities on a projectively connected manifold, extending brackets from functions to densities in a natural way.

## Contribution

It constructs a canonical Laplacian-like operator on densities that generalizes classical brackets to the algebra of all densities on such manifolds.

## Key findings

- Defines a second order differential operator on densities
- Extends brackets from functions to densities canonically
- Applies to manifolds with a projective connection

## Abstract

On a manifold with a projective connection we canonically assign a second order differential operator acting on the algebra of all densities to any tensor density $S^{ij}$ of fixed weight $\lambda$. In particular, this implies that on any projectively connected manifold, a `bracket' (symmetric biderivation) on the algebra of functions extends canonically to the algebra of densities.