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 $C^{\infty }$ map with a $C^{\infty }$ 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 infinitely differentiable functions

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

Relation with other structures

Weaker structures

• Topological manifold