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Differential manifold

4 bytes added, 19:04, 5 April 2008
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* A [[topological manifold]] <math>M</math> (in particular, <math>M</math> is Hausdorff and second-countable)
* An [[atlas ]] of coordinate charts <math>\varphi_i:U_i \to V_i, i \in I</math> from <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>. In symbols:
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