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

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