论文标题
在$ c^1 $的空间上，曲率有限
On the Space of $C^1$ Regular Curves on Sphere with Constrained Curvature
论文作者
论文摘要
令$ \ Mathcal {p} _ {κ_1}^{κ_2}（\ boldsymbol {p}，\ boldsymbol {q}）$表示$ c^1 $常规曲线的集合中的$ 2 $ 2 $ -sphere $ \ sphere $ \ mathbb { $ \ boldsymbol {p} $和$ \ boldsymbol {q} $，其切线向量是Lipschitz的连续，并且其A.E.现有的测量曲线基本上具有$（κ_1，κ_2）$，$ - \ infty <κ__1<κ__2<\ infty $。在本文中，首先，我们研究了$ \ Mathcal {p} _ {κ_1}^{κ_2}（\ boldsymbol {p}，\ boldsymbol {q}）$中的曲线的几何特性。 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 $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ if and only if the infimum of its lower curvature and the supremum其上部曲率的限制在$（κ_1，κ_2）$中。其次，我们证明了$ c^0 $和$ c^1 $ topologies在$ \ mathcal {p} _ {κ_1}^{κ_2}（\ boldsymbol {p}，\ boldsymbol {q}）$相同。此外，我们表明，$ \ Mathcal {p} _ {κ_1}^{κ_2}（\ boldsymbol {p}，\ boldsymbol {q}）$可以通过差分方程$φ'（t）=φ（t）up $（t）$（T）确定\ textrm {so} _3（\ Mathbb {r}）$具有特殊约束至$λ（t）\ in \ mathfrak {so} _3（\ so} _3（\ Mathbb {r}）$ \ boldsymbol {q}）$，使其成为（琐碎的）Banach歧管。
Let $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ 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 $(κ_1, κ_2)$, $-\infty<κ_1<κ_2<\infty$. In this article, firstly we study the geometric property of the curves in $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$. 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 $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in $(κ_1,κ_2)$. Secondly we prove that the $C^0$ and $C^1$ topologies on $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ are the same. Further, we show that a curve in $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ can be determined by the solutions of differential equation $Φ'(t) = Φ(t)Λ(t)$ with $Φ(t)\in \textrm{SO}_3(\mathbb{R})$ with special constraints to $Λ(t)\in\mathfrak{so}_3(\mathbb{R})$ and give a complete metric on $\mathcal{P}_{κ_1}^{κ_2}(\boldsymbol{P}, \boldsymbol{Q})$ such that it becomes a (trivial) Banach manifold.
