Singularities of solutions of time dependent Hamilton-Jacobi equations. Applications to Riemannian geometry

If $U:[0,+\mathrm{\infty }[\times M$ is a uniformly continuous viscosity solution of the evolution Hamilton-Jacobi equation $${\mathrm{\partial }}_{t}U+H(x,{\mathrm{\partial }}_{x}U)=0,$$ where $M$ is a not necessarily compact manifold, and $H$ is a Tonelli Hamiltonian, we prove the set $\mathrm{\Sigma }(U)$, of points where $U$ is not differentiable, is locally contractible. Moreover, we study the homotopy type of $\mathrm{\Sigma }(U)$. We also give an application to the singularities of a distance function to a closed subset of a complete Riemannian manifold.