Manifold Atlas:Definition of “manifold”

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

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$== Introduction == <wikitex>; This page defines the term “manifold” as used in the Manifold Atlas. <!-- and summarises a few key properties of all manifolds --> We assume that all manifolds are of a fixed dimension n. {{beginthm|Definition|}} An '''n-dimensional manifold''' $M$ is a [[Wikipedia:Second_countable|second countable]] [[Wikipedia:Hausdorff_space|Hausdorff space]] for which every point $x \in M$ has a neighbourhood $U_x$ homeomorphic to an open subset of $\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$. * 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. {{endthm}} 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 some common categories of manifolds below. </wikitex> == Atlases of charts == <wikitex>; 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 <!-- --> $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ <!-- --> 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 where by “smooth" we indicate $C^\infty$ maps. 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. {{beginthm|Definition}} 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 $B$. </wikitex> <!-- == Piecewise-linear manifolds == <wikitex>; * 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 $B$. </wikitex> == Smooth manifolds == <wikitex>; 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 $\beta$. </wikitex> == Complex manifolds == <wikitex>; * 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 $\delta$. </wikitex> --> == Riemannian manifolds == <wikitex>; </wikitex> == References == {{#RefList:}} [[Category:Theory]] {{Stub}}M$ is a second countable Hausdorff space for which every point $x \in M$$x \in M$ has a neighbourhood $U_x$$U_x$ homeomorphic to an open subset of $\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$$\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$.

• The interior of

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  $M$, denoted $\mathrm{int}(M)$$\mathrm{int}(M)$, is the subset of points for which $U_x \subset \Rr^n$$U_x \subset \Rr^n$.
• The boundary of

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  $M$, written $\partial M$$\partial M$, is the complement of the interior of

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  $M$.

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  $M$ is called closed if

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  $M$ is compact and $\partial M$$\partial M$ is empty.

A manifold

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$M$ 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 some common categories of manifolds below.

2 Atlases of charts

Recall that a chart on a topological manifold

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$M$ is a homeomporphism $\phi_\alpha : U_\alpha \to V_\alpha$$\phi_\alpha : U_\alpha \to V_\alpha$ from an open subset $U_\alpha$$U_\alpha$ of

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$M$ to an open subset $V_\alpha$$V_\alpha$ of $\Rr^n_+$$\Rr^n_+$. The transition function defined by two charts $\phi_\alpha$$\phi_\alpha$ and $\phi_\beta$$\phi_\beta$ is the homeomorphism

$\displaystyle \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$ $$\displaystyle \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$

An atlas for

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$M$ is a collection of charts $A = \{ (U_\alpha, \phi_\alpha)\}$$A = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$$U_\alpha$ cover

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$M$.

Let $\Cat$$\Cat$ denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate $C^\infty$$C^\infty$ maps. An atlas is a $\Cat$$\Cat$ Atlas if every transition function defined by the that atlas is a $\Cat$$\Cat$ function. $\Cat$$\Cat$ atlases are compatible if their union again forms a $\Cat$$\Cat$ atlas and by Zorn's Lemma each $\Cat$$\Cat$ atlas defines a unique maximal $\Cat$$\Cat$ atlas.

Definition 2.1. A $\Cat$$\Cat$-manifold $(M, A)$$(M, A)$ is a manifold

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

4 References

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