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

Let $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ 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 $\boldsymbol{P}$ and $\boldsymbol{Q}$, 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 $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$. 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 $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ 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 $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ are the same. Further, we show that a curve in $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ can be determined by the solutions of differential equation $\Phi'(t) = \Phi(t)\Lambda(t)$ with $\Phi(t)\in \textrm{SO}_3(\mathbb{R})$ with special constraints to $\Lambda(t)\in\mathfrak{so}_3(\mathbb{R})$ and give a complete metric on $\mathcal{P}_{\kappa_1}^{\kappa_2}(\boldsymbol{P}, \boldsymbol{Q})$ such that it becomes a (trivial) Banach manifold.