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 $U_x$ is an open subset of $Rr^n$. * 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 [[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> == Atlases of charts == <wikitex>; We give a unified presentation of the definition of piecewise linear, smooth and complex manifolds. In the complex case, we assume that the dimension is even and that the dimension is even. Recall that a chart on a topological manifold $M$ is a [[Wikipedia:Homeomorphism|homeomporphism]] $\phi_\alpha : U_\alpha \to V_\alpha$ 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 <!-- --> $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ <!-- --> An [[Wikipedia:Differential_manfiold#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 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. {{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> == Riemannian Manifolds == <wikitex>; A [[Wikipedia:Riemannian_manifold#Riemannian metrics|Riemannian metric]] $g$ on a smooth manifold $M$ is a smooth family of scalar products $$ g_x : T_xM \times T_xM \longmapsto \Rr$$ 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 $$ M \to \Rr, ~~~ x \longmapsto g_x(v_1(x),v_2(x))$$ is smooth. {{beginthm|Definition}} 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 $g$. {{endthm}} </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)$$\mathrm{int}(M)$, is the subset of points for which $U_x$$U_x$ is an open subset of $Rr^n$$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 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. In the complex case, we assume that the dimension is even and that the dimension is even.

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. That is we require every $\phi_{\alpha, \beta}$$\phi_{\alpha, \beta}$ to be either piecewise linear, smooth of class $C^\infty$$C^\infty$ or holomorphic. 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$.

3 Riemannian Manifolds

A Riemannian metric $g$$g$ on a smooth manifold

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$M$ is a smooth family of scalar products

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

defined on the tangent spaces $T_xM$$T_xM$ for each $x$$x$ in

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$M$. This means that for each pair of smooth vector fields $v_1$$v_1$ and $v_2$$v_2$ on

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$M$ the map

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

is smooth.

Definition 3.1.

A Riemannian manifold $(M, g)$$(M, g)$ is a smooth manifold

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$M$ together with a Riemannian metric $g$$g$.

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