# Regular curves on Riemannian manifolds

@article{Smale1958RegularCO, title={Regular curves on Riemannian manifolds}, author={Stephen Smale}, journal={Transactions of the American Mathematical Society}, year={1958}, volume={87}, pages={492-512} }

Introduction. A regular curve on a Riemannian manifold is a curve with a continuously turning nontrivial tangent vector.(2) A regular homotopy is a homotopy which at every stage is a regular curve, keeps end points and directions fixed and such that the tangent vector moves continuously with the homotopy. A regular curve is closed if its initial point and tangent coincides with its end point and tangent. In 1937 Hassler Whitney [17] classified the closed regular curves in the plane according to…

## 101 Citations

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While the topology of the space of all smooth immersed curves on the $2$-sphere $\mathbb{S}^2$ that start and end at given points in given directions is well known, it is an open problem to…

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Let $S$ be a complete flat surface, such as the Euclidean plane. We determine the homeomorphism class of the space of all curves on $S$ which start and end at given points in given directions and…

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The use of rotation numbers in the classification of regular closed curves in the plane up to regular homotopy sparked the investigation of winding numbers to classify regular closed curves on other…

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In his thesis '[6], Smale has found the regular homotopy classes of regular closed curves (i. e., immersed circles) on a Riemannian manifold M. His work leaves unanswered the question: Which homotopy…

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A natural curve similarity measure that can be easily extended and computed for curves on general orientable 2-manifolds is developed based on how hard it is to deform one curve into the other one continuously, and defines this "hardness" as the minimum possible surface area swept by a homotopy between the curves.

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