Cut Locus of Submanifolds: A Geometric and Topological Viewpoint

Abstract

Associated to every closed, embedded submanifold $N$ of 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. \bigskip \noindent If a compact Lie group $G$ acts on a Riemannian manifold $M$ freely then $M/G$ is a manifold. In addition, if the action is isometric, then the metric of $M$ induces a metric on $M/G$. We show that if $N$ is a $G$-invariant submanifold of $M$, then the cut locus $\mathrm{Cu}(N)$ is $G$-invariant, and $\mathrm{Cu}(N)/G = \mathrm{Cu}\left( N/G \right) $ in $M/G$. An application of this result to complex projective hypersurfaces has been provided.