The Riemannian curvature tensor (also holding for manifolds with torsion) is for the vector fields $X,Y,Z$ formally given by: $R(X,Y)Z = (\nabla_X \nabla_Y - \nabla_Y \nabla_X - \nabla_{[X,Y]})Z$.
This tensor clearly exist for smooth manifolds with connection. But what is if the manifold is non-smooth and one cannot find a connection because it is not defined everywhere on the manifold? Are there some generalizations of this curvature tensor (and also generalizations of the connection)?