Projective vs metric structures

TL;DR

This paper develops conditions and an algorithm to determine when a projective structure on an n-dimensional manifold includes a Levi-Civita connection of some metric, linking projective invariants with metric structures.

Contribution

It introduces an effective algorithm to check and find Levi-Civita connections within projective structures, and explores invariants via Cartan's normal projective connection.

Findings

Provided necessary conditions for metric inclusion in projective structures

Developed an algorithm to identify Levi-Civita connections

Linked projective invariants with conformal structures

Abstract

We present a number of conditions which are necessary for an n-dimensional projective structure (M,[nabla]) to include the Levi-Civita connection nabla of some metric on M. We provide an algorithm, which effectively checks if a Levi-Civita connection is in the projective class and, in the positive, which finds this connection and the metric. The article also provides a basic information on invariants of projective structures, including the treatment via Cartan's normal projective connection. In particular we show that there is a number of Fefferman-like conformal structures, defined on a subbundle of the Cartan bundle of the projective structure, which encode the projectively invariant information about (M,[nabla]).