Projective Geometry and PDE Prolongation

TL;DR

This dissertation explores the prolongation of overdetermined projectively invariant PDEs using differential geometry and tractor calculus, addressing complex open problems in the field.

Contribution

It introduces a geometric approach to prolongation of PDEs, including the projective metrisability equation, advancing understanding of invariant equations in projective geometry.

Findings

Recovered projective tractor and cotractor connections via PDE prolongation

Studied prolongation of projectively invariant equations

Analyzed the projective metrisability equation

Abstract

In this dissertation we study basic local differential geometry, projective differential geometry, and prolongations of overdetermined geometric partial differential equations. It is simple to prolong an n-th order linear ordinary differential equation into n first order equations. For partial differential equations there is a related process but it is far more subtle and complex. Considerable work has been done in this area but much of this is very abstract and there are many open problems even at a relatively elementary level. We introduce the reader to differential geometry and tractor calculus before recovering the projective tractor and cotractor connections via the prolongation of appropriate partial differential equations. Following this, we study prolongation of other projectively invariant equations, a particular focus is an equation known as the projective metrisability equation.