The width of embedded circles

TL;DR

This paper introduces a Morse-Lusternik-Schnirelmann theory for the width of embedded circles in Riemannian manifolds, generalizing classical plane curve width and analyzing geodesic configurations.

Contribution

It develops a new theoretical framework for understanding the width of embedded circles in Riemannian geometry, extending classical concepts and classifying special curve configurations.

Findings

Defines a generalized width for embedded circles in Riemannian manifolds.

Classifies geodesic configurations for circles bounding convex discs.

Characterizes Riemannian analogues of constant width curves.

Abstract

We develop a Morse-Lusternik-Schnirelmann theory for the distance between two points of a smoothly embedded circle in a complete Riemannian manifold. This theory suggests very naturally a definition of width that generalises the classical definition of the width of plane curves. Pairs of points of the circle realising the width bound one or more minimising geodesics that intersect the curve in special configurations. When the circle bounds a totally convex disc, we classify the possible configurations under a further geometric condition. We also investigate properties and characterisations of curves that can be regarded as the Riemannian analogues of plane curves of constant width.