  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.
