Definition:Topological Manifold/Differentiable Manifold

Definition
Let $M$ be a second countable locally Euclidean space of dimension $d$.

Let $\mathscr F$ be a $d$-dimensional differentiable structure on $M$ of class $\mathcal C^k$, where $k \ge 1$.

Then $\left({M, \mathscr F}\right)$ is a differentiable manifold of class $C^k$ and dimension $d$.

Also see

 * Smooth Manifold