论文标题
切割基因座，汤姆空间和摩尔斯 - 摩特功能之间的连接
A connection between cut locus, Thom space and Morse-Bott functions
论文作者
论文摘要
与连接的Riemannian歧管$ M $中的每个封闭的，嵌入式的Submanifold $ n $相关联，有距离函数$ d_n $，它可以测量$ m $与$ n $相关的点。我们分析了此功能的正方形，并表明它是$ n $的切割基因座$ \ mathrm {cu}（n）$的补充，只要$ m $完成。此外，梯度流线提供$ M- \ Mathrm {Cu}（n）$的变形回收。如果$ m $是一个封闭的多种流形，那么我们证明$ n $的普通捆绑包的汤姆空间是同型至$ m/\ mathrm {cu}（n）$。我们还讨论了几个有趣的结果，这些结果要么是有关切割基因座理论的相关观察结果。这些结果包括但不限于单个矩阵的局部同源性的计算，对球体内部同源球的剪切基因座的分类，无限元组$ u（p，q）$ u（p，q）$ to $ u（p）$ u（p）\ u（p）\ times u（q）$（q）$（q）$（q）$（q）$ gl（q）$ gl（n） $ o（n，\ mathbb {r}）$，它与革兰氏schmidt撤回不同。
Associated to every closed, embedded submanifold $N$ in a connected Riemannian manifold $M$, there is the distance function $d_N$ which measures the distance of a point in $M$ from $N$. We analyze the square of this function and show that it is Morse-Bott on the complement of the cut locus $\mathrm{Cu}(N)$ of $N$, provided $M$ is complete. Moreover, the gradient flow lines provide a deformation retraction of $M-\mathrm{Cu}(N)$ to $N$. If $M$ is a closed manifold, then we prove that the Thom space of the normal bundle of $N$ is homeomorphic to $M/\mathrm{Cu}(N)$. We also discuss several interesting results which are either applications of these or related observations regarding the theory of cut locus. These results include, but are not limited to, a computation of the local homology of singular matrices, a classification of the homotopy type of the cut locus of a homology sphere inside a sphere, a deformation of the indefinite unitary group $U(p,q)$ to $U(p)\times U(q)$ and a geometric deformation of $GL(n,\mathbb{R})$ to $O(n,\mathbb{R})$ which is different from the Gram-Schmidt retraction.
