Spaces of curves with constrained curvature on hyperbolic surfaces

TL;DR

This paper studies the topological structure of spaces of curves on hyperbolic surfaces with curvature constraints, classifying their homotopy types based on the curvature interval relative to [-1,1], and explores their behavior under coverings.

Contribution

It classifies the homotopy types of constrained curvature curve spaces on hyperbolic surfaces and analyzes their properties under covering maps, extending understanding of such geometric spaces.

Findings

Spaces fall into four classes based on curvature interval relation to [-1,1]

Homotopy types are explicitly computed in two classes

Spaces are nonempty for compact surfaces regardless of curvature constraints

Abstract

Let $ S $ be a hyperbolic surface. We investigate the topology of the space of all curves on $ S $ which start and end at given points in given directions, and whose curvatures are constrained to lie in a given interval $ (\kappa_1,\kappa_2) $. Such a space falls into one of four qualitatively distinct classes, according to whether $ (\kappa_1,\kappa_2) $ contains, overlaps, is disjoint from, or contained in the interval $ [-1,1] $. Its homotopy type is computed in the latter two cases. We also study the behavior of these spaces under covering maps when $ S $ is arbitrary (not necessarily hyperbolic nor orientable) and show that if $ S $ is compact then they are always nonempty.