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

Abstract

Let ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$ denote the set of $C^1$ regular curves in the $2$-sphere ${\mathbb{S}}^2$ that start and end at given points with the corresponding Frenet frames $\mathbf{P}$ and $\mathbf{Q}$, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in $({\kappa }_1,{\kappa }_2)$, $-\infty <{\kappa }_1<{\kappa }_2<\infty$. In this article, firstly we study the geometric property of the curves in ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$. We introduce the concepts of the lower and upper curvatures at any point of a $C^1$ regular curve and prove that a $C^1$ regular curve is in ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$ if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in $({\kappa }_1,{\kappa }_2)$. Secondly we prove that the $C^0$ and $C^1$ topologies on ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$ are the same. Further, we show that a curve in ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$ can be determined by the solutions of differential equation ${\Phi }^{\prime }(t)=\Phi (t)\Lambda (t)$ with $\Phi (t)\in {\text{SO}}_3(\mathbb{R})$ with special constraints to $\Lambda (t)\in {\mathfrak{so}}_3(\mathbb{R})$ and give a complete metric on ${\mathcal{P}}_{{\kappa }_1}^{{\kappa }_2}(\mathbf{P},\mathbf{Q})$ such that it becomes a (trivial) Banach manifold.

25 pages, 4 figures