On the components of spaces of curves on the 2-sphere with geodesic curvature in a prescribed interval

TL;DR

This paper characterizes the connected components of the space of closed curves on the 2-sphere with geodesic curvatures within a specific interval, revealing their topological structure and homeomorphism classes.

Contribution

It provides simple criteria to identify connected components of such curve spaces and determines their topological types based on curvature bounds.

Findings

Connected components characterized by curvature bounds.

Homeomorphism classes of certain components determined.

Topological structure of curve spaces elucidated.

Abstract

We obtain simple characterizations of the connected components of the space of closed curves on the 2-sphere whose geodesic curvatures are constrained to lie in an open interval $(\kappa_1,\kappa_2)$, in terms of $\kappa_1$ and $\kappa_2$. Many results concerning the topology of these spaces are established. In particular, we determine the homeomorphism class of some of these components.