Difference between revisions of "Differential manifold"

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(Definition)
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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.
  
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===Definition in terms of sheaves===
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A '''differential manifold''' or '''smooth manifold''' is the following data:
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* A [[topological space]] <math>M</math>
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* 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
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{{fillin}}
 
==Relation with other structures==
 
==Relation with other structures==
  

Revision as of 20:57, 11 December 2007

Definition

Definition in terms of atlases

A differential manifold or smooth manifold is the following data:

  • A topological space M
  • An atlas of coordinate charts on M to \R^n (in other words an open cover of M with homeomorphisms from each member of the open cover to open sets in \R^n

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

Fill this in later

Relation with other structures

Weaker structures