# Differential manifold

## Contents

## Definition

### Definition in terms of atlases

A **differential manifold** or **smooth manifold** is the following data:

- A topological manifold (in particular, is Hausdorff and second-countable)
- An atlas of coordinate charts on to (in other words an open cover of with homeomorphisms from each member of the open cover to open sets in

satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of .

By diffeomorphism, we here mean a map with a 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
- A subsheaf of the sheaf of continuous functions from to , 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 , such that the sheaf restriction to that open set, corresponds to the sheaf of infinitely differentiable functions on that open set.