Geometria Complessa e Geometria Differenziale
Geometria Complessa e Geometria Differenziale
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S. Basu - S. Prasad

A connection between cut locus, Thom space and Morse-Bott functions

created by prasad on 01 Dec 2023



Inserted: 1 dec 2023
Last Updated: 1 dec 2023

Year: 2020

ArXiv: 2011.02972 PDF


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.

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