A Cauchy surface can possess corners, and thereby need not be a differentiable submanifold of the spacetime; it is however always continuous (and even Lipschitz continuous). By using the flow of a vector field chosen to be complete, smooth, and timelike, it is elementary to prove that if a Cauchy surface S is C -smooth then the spacetime is C -diffeomorphic to the product S × R, and that any two such Cauchy surfaces are C -diffeomorphic.