Differentiable Manifold

Definition

Let $\mathcal{M}$ be a Topological Manifold. It is $k$-times differentiable if the following condition holds: $\forall (U_1,{\phi }_1),(U_2,{\phi }_2),U_1\cap U_2\ne \varnothing ,{\phi }_2\circ {\phi }_1^{-1}\in C^k$ where $C^k$ is Differentiability Class.