# Vertices of closed curves in Riemannian surfaces

### Mohammad Ghomi

Georgia Institute of Technology, Atlanta, United States

## 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.

## Cite this article

Mohammad Ghomi, Vertices of closed curves in Riemannian surfaces. Comment. Math. Helv. 88 (2013), no. 2, pp. 427–448

DOI 10.4171/CMH/290