Difference between revisions of "Manifold"
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[[Image:Mobius.png|right|thumb|The [[Möbius strip]] is an example of a 2-manifold.]] | [[Image:Mobius.png|right|thumb|The [[Möbius strip]] is an example of a 2-manifold.]] | ||
| − | An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. | + | An ''n''-dimensional manifold (or ''n''-manifold) ''M'' is a [[topological space]] such that every point in ''M'' has a neighbourhood that is [[homeomorphism|homeomorphic]] to ''R''<sup>''n''</sup>. Additionally, the change of coordinate functions for ''M'' must be continuous. |
| − | 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 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]]. | ||
| + | ==Constructing new manifolds from old== | ||
| + | |||
| + | *Suppose that <math>f:R^n \rightarrow R^m</math> is a differentiable function. Then <math>f^{-1}(y)</math> is a smooth manifold if ''y'' is a regular value of ''f''. | ||
[[Category:Mathematics]] | [[Category:Mathematics]] | ||
[[category: Topology]] | [[category: Topology]] | ||
Revision as of 23:04, July 3, 2008
The Möbius strip is an example of a 2-manifold.
An n-dimensional manifold (or n-manifold) M is a topological space such that every point in M has a neighbourhood that is homeomorphic to Rn. Additionally, the change of coordinate functions for M must be continuous.
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.
Constructing new manifolds from old
- Suppose that is a differentiable function. Then is a smooth manifold if y is a regular value of f.
