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

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