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

TL;DR

This paper explores the relationship between the cut locus, Thom space, and Morse-Bott functions in Riemannian geometry, revealing new topological insights and applications related to the structure of manifolds and their submanifolds.

Contribution

It establishes that the square of the distance function is Morse-Bott outside the cut locus and relates the Thom space of the normal bundle to a quotient space, providing new geometric and topological connections.

Findings

Gradient flow lines give a deformation retraction to the submanifold.

Thom space of the normal bundle is homeomorphic to a quotient involving the cut locus.

Several applications include homology computations and homotopy classifications.

Abstract

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.