The conjugate locus in convex manifolds

Abstract

The caustic formed inside a manifold via the focusing of neighbouring geodesics emanating from a single base point is known as the conjugate locus. The geometry of the conjugate locus is governed entirely by the curvature of the parent manifold, and is often full of rich structure including points at which the conjugate locus fails to be regular. In this thesis we introduce a novel formalism for detecting points ofthe conjugate locus for convex manifolds in general. We then extend our results to construct a novel coordinate system that serves as a natural choice for studying the singular structure of the conjugate locus. We also present new images of the conjugate locus for 3-manifolds, including the quadraxial ellipsoid and various orders of3-dimensional spherical harmonic manifolds before analysing their singular structure using our constructed coordinate system

Date of Award	Apr 2022
Original language	English
Supervisor	Thomas Waters (Supervisor) & Andrew Harold Osbaldestin (Supervisor)