Manifold Atlas:Definition of “manifold”

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A manifold $M$ as above is often called a topological manifold for emphasis or clarity.
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.
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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 as is appropriate. We briefly review the some common categories of manifolds below.
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Revision as of 20:07, 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 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 as is appropriate. 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 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 the homeomorphism

\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 A = \{ (U_\alpha, \phi_\alpha)\} such that the U_\alpha cover M.

Let \Cat denote either the piecewise linear, smooth or complex categories. An atlas is a \Cat Atlas if every transition function defined by the that atlas is a \Cat function. Cat Atlases are compatible if their union again forms a \Cat Atlas and by Zorn's Lemma each \Cat atlas defines a unique maximal \Cat atlas.

Definition 2.1. A \Cat-manifold (M, A) is a manifold M together with a maximal \Cat atlas A.

A \Cat-isomorphism (M, A) \cong (N, B) is a homeomorphism f: M \cong N which is a \Cat morphism when viewed in every pair of charts in A and B.


3 Riemannian manifolds


4 References

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

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