On the Space of $$C^1$$ Regular Curves on Sphere with Constrained Curvature

Abstract

In this paper, we prove that the \(C^0\) and \(C^1\) topologies are the same on the set of \(C^1\) regular curves in the 2-sphere whose tangent vectors are Lipschitz continuous, and the a.e. existing geodesic curvatures are essentially bounded in an open interval. Besides, we study the subset consisting of curves that start and end at given points with given directions, and prove that this subset is a Banach manifold.