Difference between revisions of "Differential manifold"

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(Definition in terms of sheaves)
(Definition in terms of sheaves)
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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 infintely differentiable functions]]
 
* A subsheaf of the [[sheaf of continuous functions]] from <math>M</math> to <math>\R</math>, 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 <math>\R^n</math>, such that the ring associated to that open set inthe sheaf, corresponds to the sheaf of infinitely differentiable functions on that open set.
+
Such that every point has a neighbourhood with a homeomorphism to an open set in <math>\R^n</math>, 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==
 
==Relation with other structures==

Revision as of 20:56, 26 December 2007

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 \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:

Such that every point has a neighbourhood with a homeomorphism to an open set in \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