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}}
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 as is appropriate. We briefly review 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 [[Wikipedia:Piecewise_linear_manifold|piecewise-linear]], [[Wikipedia:Differential_manifold|smooth]], [[Wikipedia:Complex_manifold|complex]], [[Wikipedia:Symplectic_manifold|symplectic]], [[Wikipedia:Contact_manifold|contact]], [[Wikipedia:Riemannian_manifold|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.
</wikitex>
</wikitex>
== Atlases of charts ==
== Atlases of charts ==
<wikitex>;
<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
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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. An atlas is a $\Cat$ Atlas if every transition function defined by the that atlas is a $\Cat$ function.
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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.
$\Cat$ 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.
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{{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:}}
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{{#RefList:}} -->
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<!--[[Category:Theory]]-->
[[Category:Theory]]
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<!-- {{Stub}} -->
{{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
Tex syntax error
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
    Tex syntax error
    , denoted \mathop{\mathrm{int}}(M), is the subset of points for which U_x is an open subset of \Rr^n.
  • The boundary of
    Tex syntax error
    , written \partial M, is the complement of \mathop{\mathrm{int}}(M).
  • Tex syntax error
    is called closed if
    Tex syntax error
    is compact and \partial M is empty.
A manifold
Tex syntax error
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
Tex syntax error
. In the complex case, we assume that the dimension of
Tex syntax error
is even and that the boundary of
Tex syntax error
is empty. Recall that a chart on a topological manifold
Tex syntax error
is a homeomorphism \phi_\alpha : U_\alpha \to V_\alpha from an open subset U_\alpha of
Tex syntax error
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
Tex syntax error
is a collection of charts A = \{ (U_\alpha, \phi_\alpha)\} such that the U_\alpha cover
Tex syntax error
.

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
Tex syntax error
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
Tex syntax error
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
Tex syntax error
. This means that for each pair of smooth vector fields v_1 and v_2 on
Tex syntax error
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
Tex syntax error
together with a Riemannian metric g.

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

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