# Difference between revisions of "Differential manifold"

From Diffgeom

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==Definition== | ==Definition== | ||

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+ | ===Definition in terms of atlases=== | ||

A '''differential manifold''' or '''smooth manifold''' is the following data: | A '''differential manifold''' or '''smooth manifold''' is the following data: | ||

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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. | 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 space]] <math>M</math> | ||

+ | * A subsheaf of the [[sheaf of continuous functions]] from <math>M</math> to <math>\R</math>, which plays the role of the sheaf of differentiable functions | ||

+ | {{fillin}} | ||

==Relation with other structures== | ==Relation with other structures== | ||

## Revision as of 20:57, 11 December 2007

## Contents

## Definition

### Definition in terms of atlases

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

- A topological space
- 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 space
- A subsheaf of the sheaf of continuous functions from to , which plays the role of the sheaf of differentiable functions

*Fill this in later*