The conjugate locus in convex 3-manifolds

TL;DR

This paper investigates the structure of conjugate loci in convex 3-manifolds using Jacobi fields, introducing a new coordinate system and analyzing a specific case of 3D ellipsoids to understand singularities.

Contribution

It presents a novel coordinate system based on Jacobi fields for classifying conjugate locus singularities in convex 3-manifolds, especially ellipsoids.

Findings

New coordinate system for tangent space analysis

Classification of conjugate locus singularities

Insights into conjugate points on 3D ellipsoids

Abstract

In this paper we study the conjugate locus in convex manifolds. Our main tool is Jacobi fields, which we use to define a special coordinate system on the unit sphere of the tangent space; this provides a natural coordinate system to study and classify the singularities of the conjugate locus. We pay particular attention to 3-dimensional manifolds, and describe a novel method for determining conjugate points. We then make a study of a special case: the 3-dimensional (quadraxial) ellipsoid. We emphasise the similarities with the focal sets of 2-dimensional ellipsoids.