科研之友
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摘要
We prove two theorems about homotopies of curves on two-dimensional Riemannian manifolds. We show that, for any , if two simple closed curves are homotopic through curves of bounded length L, then they are also isotopic through curves of length bounded by . If the manifold is orientable, then for any we show that, if we can contract a curve traversed twice through curves of length bounded by L, then we can also contract through curves bounded in length by . Our method involves cutting curves at their self-intersection points and reconnecting them in a prescribed way. We consider the space of all curves obtained in this way from the original homotopy, and use a novel approach to show that this space contains a path which yields the desired homotopy.
- 出版日期2014-8
