Projective Connections and the Algebra of Densities

TL;DR

This paper explores the relationship between projective connections and the algebra of densities on manifolds, introducing a Laplace-type operator derived from a Thomas projective connection and a symmetric tensor.

Contribution

It establishes a new connection between projective geometry and the algebra of densities, and constructs a Laplace-type operator using these concepts.

Findings

Relation between projective connections and algebra of densities clarified

Constructed a Laplace-type operator from a Thomas projective connection

Demonstrated applications in integrable systems and quantisation

Abstract

Projective connections first appeared in Cartan's papers in the 1920's. Since then they have resurfaced periodically in, for example, integrable systems and perhaps most recently in the context of so called projectively equivariant quantisation. We recall the notion of projective connection and describe its relation with the algebra of densities on a manifold. In particular, we construct a Laplace-type operator on functions using a Thomas projective connection and a symmetric contravariant tensor of rank 2 (`upper metric').