Manifold Atlas:Definition of “manifold”

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{{beginthm|Definition|}}
{{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 $\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$.
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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 $U_x \subset \Rr^n$.
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* The '''interior''' of $M$, denoted $\mathop{\mathrm{int}}(M)$, is the subset of points for which $U_x$ is an open subset of $\Rr^n$.
* The '''boundary''' of $M$, written $\partial M$, is the complement of the interior of $M$.
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* The '''boundary''' of $M$, written $\partial M$, is the complement of $\mathop{\mathrm{int}}(M)$.
* $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty.
* $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty.
{{endthm}}
{{endthm}}
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== Atlases of charts ==
== Atlases of charts ==
<wikitex>;
<wikitex>;
We give a unified presentation of piecewise linear, smooth and complex manifolds. 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
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We give a unified presentation of the definition of piecewise linear, smooth and complex manifolds $M$. In the complex case, we assume that the dimension of $M$ is even and that the boundary of $M$ is empty.
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Recall that a chart on a topological manifold $M$ is a [[Wikipedia:Homeomorphism|homeomorphism]] $\phi_\alpha : U_\alpha \to V_\alpha$ from an open subset $U_\alpha$ of $M$ to an [[Wikipedia:Open_set|open subset]] $V_\alpha$ of $\Rr^n_+$. The transition function defined by two charts $\phi_\alpha$ and $\phi_\beta$ is the homeomorphism
<!-- -->
<!-- -->
$$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$
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$$ \phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_{\beta}(U_\alpha \cap U_\beta)} : \phi_\beta(U_\alpha \cap U_\beta) \longrightarrow \phi_\alpha(U_\alpha \cap U_\beta).$$
<!-- -->
<!-- -->
An atlas for $M$ is a collection of charts $A = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$ cover $M$.
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An [[Wikipedia:Differential_manifold#Atlases|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. That is we require the every $\phi_{\alpha, \beta}$ to be either [[Wikipedia:Piecewise_linear_function#Notation|piecewise linear]], [[Wikipedia:Smooth_function#Differentiability_classes_in_several_variables|smooth of class $C^\infty$]] or [[Wikipedia:Holomorphic_function|holomorphic]]. 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 [[Wikipedia:Zorn's_lemma|Zorn's Lemma]] each $\Cat$ atlas defines a unique maximal $\Cat$ atlas.
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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: that is, we require every $\phi_{\alpha, \beta}$ to be either [[Wikipedia:Piecewise_linear_function#Notation|piecewise linear]], [[Wikipedia:Smooth_function#Differentiability_classes_in_several_variables|smooth of class $C^\infty$]] or [[Wikipedia:Holomorphic_function|holomorphic]]. Two $\Cat$ atlases are compatible if their union again forms a $\Cat$ atlas and by [[Wikipedia:Zorn's_lemma|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 $A$.
{{beginthm|Definition}} A $\Cat$-manifold $(M, A)$ is a manifold $M$ together with a maximal $\Cat$ atlas $A$.
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</wikitex>
</wikitex>
<!-- == Piecewise-linear manifolds ==
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== Riemannian Manifolds ==
<wikitex>;
<wikitex>;
* A piecewise linear manifold $(M, A)$, PL-manifold, is a manifold $M$ together with a maximal atlas $\mathcal{A}$ of piecewise linear charts $A$.
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A [[Wikipedia:Riemannian_manifold#Riemannian metrics|Riemannian metric]] $g$ on a smooth manifold $M$ is a smooth family of scalar products
* 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 $B$.
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$$ g_x : T_xM \times T_xM \longmapsto \Rr$$
</wikitex>
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defined on the [[Wikipedia:Tangent_space|tangent spaces]] $T_xM$ for each $x$ in $M$. This means that for each pair of smooth [[Wikipedia:Vector_field|vector fields]] $v_1$ and $v_2$ on $M$ the map
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$$ M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x))$$
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is smooth.
== Smooth manifolds ==
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{{beginthm|Definition}}
<wikitex>;
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A [[Wikipedia:Riemannian_manifold|Riemannian manifold]] $(M, g)$ is a smooth manifold $M$ together with a Riemannian metric $g$.
We shall use the term smooth manifold to refer to $C^\infty$ smooth manifolds.
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* A smooth manifold $(M, \alpha)$ is a smooth manifold $M$ together with a maximal atlas of smooth charts $\alpha$.
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* 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 $\beta$.
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</wikitex>
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== Complex manifolds ==
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<wikitex>;
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* A complex manifold $(M, \gamma)$, is an even dimensional manifold $M$ together with a maximal atlas $\mathcal{A}$ of holomorphic charts $A$.
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* 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 $\delta$.
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</wikitex> -->
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<!-- == Riemannian manifolds ==
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<wikitex>;
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An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric $g$.
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{{endthm}}
</wikitex>
</wikitex>
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<!--
== References ==
== References ==
{{#RefList:}} -->
{{#RefList:}} -->
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<!--[[Category:Theory]]-->
[[Category:Theory]]
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<!-- {{Stub}} -->
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Latest revision as of 18:32, 24 November 2009

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 | v_1 \geq 0 \}.

  • The interior of M, denoted \mathop{\mathrm{int}}(M), is the subset of points for which U_x is an open subset of \Rr^n.
  • The boundary of M, written \partial M, is the complement of \mathop{\mathrm{int}}(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 piecewise-linear, smooth, complex, symplectic, contact, Riemannian, 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

We give a unified presentation of the definition of piecewise linear, smooth and complex manifolds M. In the complex case, we assume that the dimension of M is even and that the boundary of M is empty.

Recall that a chart on a topological manifold M is a homeomorphism \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 \circ \phi_\beta^{-1}|_{\phi_{\beta}(U_\alpha \cap U_\beta)} : \phi_\beta(U_\alpha \cap U_\beta) \longrightarrow \phi_\alpha(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: that is, we require every \phi_{\alpha, \beta} to be either piecewise linear, smooth of class C^\infty or holomorphic. Two \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

A Riemannian metric g on a smooth manifold M is a smooth family of scalar products

\displaystyle  g_x : T_xM \times T_xM \longmapsto \Rr

defined on the tangent spaces T_xM for each x in M. This means that for each pair of smooth vector fields v_1 and v_2 on M the map

\displaystyle  M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x))

is smooth.

Definition 3.1. A Riemannian manifold (M, g) is a smooth manifold M together with a Riemannian metric g.

An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric g.

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