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The Geometry of $$C^1$$ C 1 Regular Curves in Sphere with Constrained Curvature
The Journal of Geometric Analysis ( IF 1.1 ) Pub Date : 2020-09-15 , DOI: 10.1007/s12220-020-00511-1
Cong Zhou

In this article, we study \(C^1\) regular curves in the 2-sphere that start and end at given points with given directions, whose tangent vectors are Lipschitz continuous, and their a.e. existing geodesic curvatures have essentially bounds in an open interval. Especially, we show that a \(C^1\) regular curve is such a curve if and only if the infimum of its lower curvature and the supremum of its upper curvature are constrained in the same interval.



中文翻译:

约束曲率的球面$$ C ^ 1 $$ C 1正则曲线的几何

在本文中,我们研究2球面中的((C ^ 1 \)正则曲线,该曲线在给定点处以给定方向开始和结束,其切向量为Lipschitz连续,并且它们的现有测地曲率在一个开放区域中基本上具有边界间隔。尤其是,我们证明\(C ^ 1 \)正则曲线是这样的曲线,当且仅当其下曲率的最小值和上曲率的最大值被限制在相同的间隔内时。

更新日期:2020-09-15
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