端点映射的同伦性质与次黎曼几何中的Serre定理
Homotopy properties of endpoint maps and a theorem of Serre in subriemannian geometry
摘要：
讨论了仿射控制系统端点映射的同伦性质。我们证明了对于一定的$p>1$，这些映射是关于轨迹空间上的某种$W^{1,p}$拓扑的Hurewicz纤维化。我们研究了这些仿射控制系统的几何代价临界点，证明了如果基流形是紧的，那么它们的临界点的个数是无穷多的（我们使用Lusternik-Schnirelmann范畴结合Hurewicz性质）.在控制系统是次黎曼的特殊情况下，这一结果可以理解为塞尔定理的相应版本，即紧致黎曼流形上两点之间存在无穷多条测地线。在次黎曼情形下，我们证明了Hurewicz性质对所有的$p\geq1$成立，并且具有$W^{1,2}$拓扑的水平闭路空间具有CW-复形的同伦类型（只要端点映射至少有一个正则值）;特别地，包含在普通的水平环空间中的水平环空间是一个同伦等价。
英文摘要：
We discuss homotopy properties of endpoint maps for affine control systems. We prove that these maps are Hurewicz fibrations with respect to some $W^{1,p}$ topology on the space of trajectories, for a certain $p>1$. We study critical points of geometric costs for these affine control systems, proving that if the base manifold is compact then the number of their critical points is infinite (we use Lusternik-Schnirelmann category combined with the Hurewicz property). In the special case where the control system is subriemannian this result can be read as the corresponding version of Serre's theorem, on the existence of infinitely many geodesics between two points on a compact riemannian manifold. In the subriemannian case we show that the Hurewicz property holds for all $p\geq1$ and the horizontal-loop space with the $W^{1,2}$ topology has the homotopy type of a CW-complex (as long as the endpoint map has at least one regular value); in particular the inclusion of the horizontal-loop space in the ordinary one is a homotopy equivalence.
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