Differential manifold

Definition

Definition in terms of atlases

A differential manifold or smooth manifold is the following data:

• A topological manifold $M$ (in particular, $M$ is Hausdorff and second-countable)
• An atlas of coordinate charts on $M$ to ${\mathbb{R}}^n$ (in other words an open cover of $M$ with homeomorphisms from each member of the open cover to open sets in ${\mathbb{R}}^n$

satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of ${\mathbb{R}}^n$.

By diffeomorphism, we here mean a $C^{\mathrm{\infty }}$ map with a $C^{\mathrm{\infty }}$ inverse.

upto the following equivalence:

Two atlases of coordinate charts on a topological space define the same differential manifold structure if given any coordinate chart in one and any coordinate chart in the other, the transition function between them is a diffeomorphism.

Definition in terms of sheaves

A differential manifold or smooth manifold is the following data:

• A topological manifold $M$
• A subsheaf of the sheaf of continuous functions from $M$ to $\mathbb{R}$, which plays the role of the sheaf of infintely differentiable functions

Such that every point has a neighbourhood with a homeomorphism to an open set in ${\mathbb{R}}^n$, such that the ring associated to that open set inthe sheaf, corresponds to the sheaf of infinitely differentiable functions on that open set.

Relation with other structures

Weaker structures

• Topological manifold