Difference between revisions of "Differential manifold"

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(Definition in terms of sheaves)
(Definition)
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* A [[topological manifold]] <math>M</math> (in particular, <math>M</math> is Hausdorff and second-countable)
 
* A [[topological manifold]] <math>M</math> (in particular, <math>M</math> is Hausdorff and second-countable)
* An atlas of coordinate charts on <math>M</math> to <math>\R^n</math> (in other words an open cover of <math>M</math> with homeomorphisms from each member of the open cover to open sets in <math>\R^n</math>
+
* An atlas of coordinate charts <math>\varphi_i:U_i \to V_i, i \in I</math> on <math>M</math> to <math>\R^n</math> (in other words an open cover <math>U_i</math> of <math>M</math> with homeomorphisms from each member <math>U_i</math> of the open cover to open sets <math>V_i</math> in <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>.
+
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>. In symbols:
 +
 
 +
<math>\varphi_i \circ \varphi_j^{-1}</math>
 +
 
 +
is a homeomorphism when restricted to the set:
 +
 
 +
<math>(\varphi_j(U_i \cap U_j))</math>
  
 
By diffeomorphism, we here mean a <math>C^{\infty}</math> map with a <math>C^{\infty}</math> inverse.
 
By diffeomorphism, we here mean a <math>C^{\infty}</math> map with a <math>C^{\infty}</math> inverse.
  
upto the following equivalence:
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However, we need to quotient out this data by 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.
+
Two atlases of coordinate charts on a topological space define the same differential manifold structure if taking their union still gives an atlas of coordinate charts. In other words, given any coordinate chart in one atlas and any coordinate chart in the other atlas, the transition function between them is a diffeomorphism.
  
 
===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 infinitely 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 infinitely differentiable functions]]
  
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.
+
Such that every point has a neighborhood 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 20:57, 3 April 2008

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 \varphi_i:U_i \to V_i, i \in I on M to \R^n (in other words an open cover U_i of M with homeomorphisms from each member U_i of the open cover to open sets V_i 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. In symbols:

\varphi_i \circ \varphi_j^{-1}

is a homeomorphism when restricted to the set:

(\varphi_j(U_i \cap U_j))

By diffeomorphism, we here mean a C^{\infty} map with a C^{\infty} inverse.

However, we need to quotient out this data by the following equivalence:

Two atlases of coordinate charts on a topological space define the same differential manifold structure if taking their union still gives an atlas of coordinate charts. In other words, given any coordinate chart in one atlas and any coordinate chart in the other atlas, 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 neighborhood 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