Difference between revisions of "Manifold"
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An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood ''U'' that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. The homeomorphisms <math>\phi_U : U\rightarrow \mathbf{R}^n</math> should be thought of as providing local coordinates in the neighborhood ''U''. Whenever two such coordinate neighborhoods ''U'' and ''V'' intersect, we also require that the change of coordinate maps <math>\phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V)</math> be homeomorphisms. | An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood ''U'' that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. The homeomorphisms <math>\phi_U : U\rightarrow \mathbf{R}^n</math> should be thought of as providing local coordinates in the neighborhood ''U''. Whenever two such coordinate neighborhoods ''U'' and ''V'' intersect, we also require that the change of coordinate maps <math>\phi_U\circ\phi_V^{-1}:\phi_V(U\cap V)\rightarrow \phi_U(U\cap V)</math> be homeomorphisms. | ||
| − | An ''n''-dimensional complex manifold ''N'' is a [[topological space]] such that every point in ''N'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to '''C'''<sup>''n''</sup>. | + | An ''n''-dimensional differentiable manifold is an ''n''-manifold with the property that the change of coordinate maps are differentiable functions. |
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| + | An ''n''-dimensional complex manifold ''N'' is a [[topological space]] such that every point in ''N'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to '''C'''<sup>''n''</sup>, and such that the change of coordinate functions are holomorphic. | ||
Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]]. | Manifolds are [[Hausdorff space|Hausdorff]] and [[2nd-countable space|2nd-countable]]. | ||
Revision as of 22:15, June 30, 2008
An n-dimensional manifold (or n-manifold) M is a topological space such that every point in M has a neighbourhood U that is homeomorphic to Rn. The homeomorphisms should be thought of as providing local coordinates in the neighborhood U. Whenever two such coordinate neighborhoods U and V intersect, we also require that the change of coordinate maps be homeomorphisms.
An n-dimensional differentiable manifold is an n-manifold with the property that the change of coordinate maps are differentiable functions.
An n-dimensional complex manifold N is a topological space such that every point in N has a neighbourhood that is homeomorphic to Cn, and such that the change of coordinate functions are holomorphic.
Manifolds are Hausdorff and 2nd-countable.