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If $U:[0,+\infty[\times M$ is a uniformly continuous viscosity solution ofthe evolution Hamilton-Jacobi equation $$\partial_tU+ H(x,\partial_xU)=0,$$where $M$ is a not necessarily compact manifold, and $H$ is a TonelliHamiltonian, we prove the set $\Sigma(U)$, of points where $U$ is notdifferentiable, is locally contractible. Moreover, we study the homotopy typeof $\Sigma(U)$. We also give an application to the singularities of a distancefunction to a closed subset of a complete Riemannian manifold.

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