Abstract
In this paper, we prove that the \(C^0\) and \(C^1\) topologies are the same on the set of \(C^1\) regular curves in the 2-sphere whose tangent vectors are Lipschitz continuous, and the a.e. existing geodesic curvatures are essentially bounded in an open interval. Besides, we study the subset consisting of curves that start and end at given points with given directions, and prove that this subset is a Banach manifold.
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Acknowledgements
I would like to thank my Ph.D. advisor Nicolau Saldanha for his guidance and great support during my Ph.D. studies. I would like to thank the reviewer for the very valuable comments on revising this article. I also would like to thank Capes and Faperj for financial support during my graduate studies at PUC-Rio.
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The author received Ph.D. scholarship from Capes and Faperj, Brazil during the preparation of this paper.
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Zhou, C. On the Space of \(C^1\) Regular Curves on Sphere with Constrained Curvature. Results Math 76, 223 (2021). https://doi.org/10.1007/s00025-021-01534-y
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DOI: https://doi.org/10.1007/s00025-021-01534-y