锥流形上测地线的莫尔斯理论
Morse Theory for Geodesics in Conical Manifolds
摘要：
本文的目的是将测地线的莫尔斯理论推广到锥流形。我们将这些流形定义为具有有限个圆锥奇点的$\R^n$的子流形。为了建立一个好的莫尔斯理论，我们必须使用一个适当的测地线定义。本文的主要定理是，虽然能量是非光滑的，但在没有临界点的情况下，我们可以发现它的子能级的连续收缩。这样，我们就可以很好地定义孤立临界值和孤立临界点的指数。证明了Morse关系成立，并给出了测地线重数的几何意义。
英文摘要：
The aim of this paper is to extend the Morse theory for geodesics to the conical manifolds. We define these manifolds as submanifolds of $\R^n$ with a finite number of conical singularities. To formulate a good Morse theory we must use an appropriate definition of geodesic. The main theorem of this paper claims that, although the energy is nonsmooth, we can find a continuous retraction of its sublevels in absence of critical points. So, we can give a good definition of index for isolated critical values and for isolated critical points. We prove that Morse relations hold and, at last, we give a definition of multiplicity of geodesics which is geometrical meaningful.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
