Manifold Atlas:Definition of “manifold”
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Revision as of 21:01, 16 September 2009
Contents |
1 Introduction
This page defines the term “manifold” as used in the Manifold Atlas. We assume that all manifolds are of a fixed dimension n.
Definition 1.1. An n-dimensional manifold is a second countable Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- The interior of , denoted , is the subset of points for which .
- The boundary of , written , is the complement of the interior of .
- is called closed if is compact and is empty.
A manifold as above is often called a topological manifold for emphasis or clarity. Typically, but not necessarly, the word “manifold” will mean "topological manifold with extra structure", be it smooth, Riemannian, complex, etc. The extra structure will be emphasised or suppressed in notation and vocabulary. We briefly review the some common categories of manifolds below.
2 Atlases of charts
Recall that a chart on a topological manifold is a homeomporphism from an open subset of to an open subset of . The transition function defined by two charts and is
3 Piecewise-linear manifolds
- A piecewise linear manifold , PL-manifold, is a manifold together with a maximal atlas of piecewise linear charts .
- A PL-homeomorphism is a homeomorphism which is piecewise linear when viewed in every pair of charts in and .
4 Smooth manifolds
We shall use the term smooth manifold to refer to smooth manifolds.
- A smooth manifold is a smooth manifold together with a maximal atlas of smooth charts .
- A diffeomorphism between smooth manfiolds is a homeomorphism such that is when viewed in every pair of charts in and .
5 Complex manifolds
- A complex manifold , is an even dimensional manifold together with a maximal atlas of holomorphic charts .
- A complex diffeomorphism is a homeomorphism which is piecewise linear when viewed in ever pair of charts in and .
6 Riemannian manifolds
7 References
8 References
This page has not been refereed. The information given here might be incomplete or provisional. |