Differentiable Manifold
- Differential Geometry 1.3

08 Feb 2023 in Mathematics on Differential Geometry · 1 min

from 「Geometrical Methods of Mathematical Physics」: Bernard F. Schutz

• Differentiable manifold

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.