The cut loci on ellipsoids and certain Liouville manifolds

TL;DR

This paper investigates the structure of cut loci on certain Riemannian manifolds, especially ellipsoids, revealing that they can be smoothly embedded disks, which enhances understanding of their geometric properties.

Contribution

It demonstrates that some manifolds, including ellipsoids with distinct axes, have cut loci that are smoothly embedded disks, a novel geometric insight.

Findings

Ellipsoids with distinct axes have cut loci that are smoothly embedded disks.

Certain Liouville manifolds exhibit similar cut locus properties.

The results extend understanding of cut locus topology on specific manifolds.

Abstract

We show that some riemannian manifolds diffeomorphic to the sphere have the property that the cut loci of general points are smoothly embedded closed disks of codimension one. Ellipsoids with distinct axes are typical examples of such manifolds.