Hello, I notice that most books on differential geometry introduce the definition of differentiable manifold by describing what I would regard as a differentiable manifold of class C∞ (i.e. a smooth manifold). Why so? Don't we simply need a class C1 differentiable manifold in order to have tangent spaces and do differential geometry? What do we need the partial derivatives of all orders, in particular of third, fourth, fifth order for? The Jacobian is made of only first-order partial derivatives after all.