Manifold Atlas:Definition of “manifold”

From Manifold Atlas
Revision as of 20:52, 16 September 2009 by Diarmuid Crowley (Talk | contribs)
Jump to: navigation, search

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 M is a second countable Hausdorff space for which every point x \in M has a neighbourhood U_x homeomorphic to an open subset of \Rr^n_+ := \{ v \in \Rr^n | x_1 \geq 0 \}.

  • The interior of M, denoted \mathrm{int}(M), is the subset of points for which U_x \subset \Rr^n.
  • The boundary of M, written \partial M, is the complement of the interior of M.
  • M is called closed if M is compact and \partial M is empty.

A manifold M 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 M is a homeomporphism \phi_\alpha : U_\alpha \to V_\alpha M from an open subset U_\alpha of M to an open subset V_\alpha of \Rr^n. The transition function defined by two charts \phi_\alpha and \phi_\beta is

\displaystyle  \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).
An atlas for M is a collection of charts \mathcal{A} = \{ (U_\alpha, \phi_\alpha)\} such that the U_\alpha cover M. An atlas is called resp. piecwise linear, smooth or complex, etc if every transition function is resp. piecewise linear, smooth or complex, etc. Atlases of a given category are compatible if their union again forms an Atlas of that category any by Zorn's Lemma each atlas of a given category defines a unique maximal atlas in that category.

3 Piecewise-linear manifolds

  • A piecewise linear manifold (M, A), PL-manifold, is a manifold M together with a maximal atlas \mathcal{A} of piecewise linear charts A.
  • A PL-homeomorphism (M, A) \cong (N, B) is a homeomorphism f: M \cong N which is piecewise linear when viewed in every pair of charts in A and B.

4 Smooth manifolds

We shall use the term smooth manifold to refer to C^\infty smooth manifolds.

  • A smooth manifold (M, \alpha) is a smooth manifold M together with a maximal atlas of smooth charts \alpha.
  • A diffeomorphism between smooth manfiolds (M, \alpha) \cong (N, \beta) is a homeomorphism f : N \to M such that f is C^\infty when viewed in every pair of charts in \alpha and \beta.

5 Complex manifolds

  • A complex manifold (M, \gamma), is an even dimensional manifold M together with a maximal atlas \mathcal{A} of holomorphic charts A.
  • A complex diffeomorphism (M, \gamma) \cong (N, \delta) is a homeomorphism f: M \cong N which is piecewise linear when viewed in ever pair of charts in \gamma and \delta.

6 Riemannian manifolds


7 References

8 References

This page has not been refereed. The information given here might be incomplete or provisional.

Personal tools
Variants
Actions
Navigation
Interaction
Toolbox