Vertices of closed curves in Riemannian surfaces

TL;DR

This paper explores the relationship between the topology of complete Riemannian surfaces and the minimum number of vertices of closed curves, revealing unique properties of space forms with finite fundamental groups.

Contribution

It characterizes surfaces where all closed curves have more than two vertices, especially identifying space forms with finite fundamental groups as unique cases.

Findings

Space forms with finite fundamental group have all simple closed curves with more than two vertices.

Simply connected space forms are the only surfaces where all closed curves bounding a compact surface have more than two vertices.

The study links topological properties of surfaces to the critical points of geodesic curvature.

Abstract

We uncover some connections between the topology of a complete Riemannian surface M and the minimum number of vertices, i.e., critical points of geodesic curvature, of closed curves in M. In particular we show that the space forms with finite fundamental group are the only surfaces in which every simple closed curve has more than two vertices. Further we characterize the simply connected space forms as the only surfaces in which every closed curve bounding a compact immersed surface has more than two vertices.