Difference between revisions of "Differential manifold"

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(Definition)
(Definition)
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satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of <math>\R^n</math>.
 
satisfying the compatibility condition: the transition function between any two coordinate charts of the atlas is a diffeomomorphism of open subsets of <math>\R^n</math>.
  
By diffeomorphism, we here mean a <math>C^{\infty}</math> map.
+
By diffeomorphism, we here mean a <math>C^{\infty}</math> map with a <math>C^{\infty}</math> inverse.
  
 
upto the following equivalence:
 
upto the following equivalence:

Revision as of 16:47, 23 June 2007

Definition

A differential 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.

Relation with other structures

Weaker structures