Let $ S $ be a hyperbolic surface. We investigate the topology of the spaceof all curves on $ S $ which start and end at given points in given directions,and whose curvatures are constrained to lie in a given interval $(kappa_1,kappa_2) $. Such a space falls into one of four qualitativelydistinct classes, according to whether $ (kappa_1,kappa_2) $ contains,overlaps, is disjoint from, or contained in the interval $ [-1,1] $. Itshomotopy type is computed in the latter two cases. We also study the behaviorof these spaces under covering maps when $ S $ is arbitrary (not necessarilyhyperbolic nor orientable) and show that if $ S $ is compact then they arealways nonempty.
展开▼
机译：让$ S $是双曲线表面。我们调查$ S $的SPACEOF的拓扑，该曲线在给定方向上的给定点处启动和结束，并且其曲率被约束为位于给定的间隔$（ Kappa_1， kappa_2）$。根据$（ kappa_1， kappa_2）$包含，重叠，或包含在时间间隔$ [-1,1] $的情况下，这样的空间落入四个定性的类中的四个类中的一个。在后两种情况下，ithomotopy类型被计算。我们还在覆盖地图下讨论这些空间的行为，当$ S $是任意的（不必要的手工性也不是可倾向），并表明，如果$ S $ CAMPACT，那么他们是巨大的。
展开▼
