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Homotopy type of spaces of curves with constrained curvature on flat surfaces

Nicolau C. SaldanhaPedro Zühlke
arXiv: Geometric Topology
Oct 2014
摘要
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 whose curvatures are constrained to lie in a given open interval, in terms of all parameters involved. Any connected component of such a space is either contractible or homotopy equivalent to an $n$-sphere, and every $n\geq 1$ is realizable. Explicit homotopy equivalences between the components and the corresponding spheres are constructed.
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