Definition:Topological Manifold/Differentiable Manifold

Definition
A $C^k$ (resp. $C^\infty$, resp. complex analytic) manifold of dimension $d$ consists of


 * 1. A second countable locally Euclidean space $M$ of dimension $d$


 * 2. A differentiable structure $\mathscr F$ on $M$ of class $C^k$ (resp. $C^\infty$, resp. complex analytic).

When the differentiable structure is clear from the context then one often simply speaks of the manifold $M$.

A $C^\infty$ manifold is also called a smooth manifold.