Differentiable manifold
Given a manifold $M$ and its atlas $\mathcal{A}$, let’s think of transition maps. If all transition maps are differentiable, $M$ is a differentiable manifold. Furthermore, if all transition maps are $C^k$, $M$ is a $C^k$-manifold.
We can think of following concepts, too.
- $C^1$-manifold : differentiable manifold
- $C^\infty$-manifold : smooth manifold
- $C^\omega$-manifold : analytic manifold
Since differentiable manifolds are the main objects of differential geometry, in many cases, unless otherwise noted, manifold refer to a differentiable manifold.