Manifold Atlas:Definition of “manifold”
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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 $\Rr^n_+ := \{ v \in \Rr^n | v_1 \geq 0 \}$. | + | 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 | + | * 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 | + | * 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. | ||
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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, | + | 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. |
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== Atlases of charts == | == Atlases of charts == | ||
<wikitex>; | <wikitex>; | ||
− | Recall that a chart on a topological manifold $M$ is a | + | 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 | ||
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− | $$ \phi_{\alpha | + | $$ \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).$$ |
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− | An atlas for $M$ is a collection of charts $A = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$ cover $M$. | + | An [[Wikipedia:Differential_manifold#Atlases|atlas]] for $M$ is a collection of charts $A = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$ cover $M$. |
− | Let $\Cat$ denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate $C^\infty$ maps. An atlas is a $\Cat$ Atlas if every transition function defined by the that atlas is a $\Cat$ function. | + | Let $\Cat$ denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate $C^\infty$ maps. An atlas is a $\Cat$ Atlas if every transition function defined by the that atlas is a $\Cat$ function: that is, we require every $\phi_{\alpha, \beta}$ to be either [[Wikipedia:Piecewise_linear_function#Notation|piecewise linear]], [[Wikipedia:Smooth_function#Differentiability_classes_in_several_variables|smooth of class $C^\infty$]] or [[Wikipedia:Holomorphic_function|holomorphic]]. Two $\Cat$ atlases are compatible if their union again forms a $\Cat$ atlas and by [[Wikipedia:Zorn's_lemma|Zorn's Lemma]] each $\Cat$ atlas defines a unique maximal $\Cat$ atlas. |
− | $\Cat$ atlases are compatible if their union again forms a $\Cat$ atlas and by Zorn's Lemma each $\Cat$ atlas defines a unique maximal $\Cat$ atlas. | + | |
{{beginthm|Definition}} A $\Cat$-manifold $(M, A)$ is a manifold $M$ together with a maximal $\Cat$ atlas $A$. | {{beginthm|Definition}} A $\Cat$-manifold $(M, A)$ is a manifold $M$ together with a maximal $\Cat$ atlas $A$. | ||
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− | + | == Riemannian Manifolds == | |
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− | + | A [[Wikipedia:Riemannian_manifold#Riemannian metrics|Riemannian metric]] $g$ on a smooth manifold $M$ is a smooth family of scalar products | |
− | + | $$ 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 [[Wikipedia:Riemannian_manifold|Riemannian manifold]] $(M, g)$ is a smooth manifold $M$ together with a Riemannian metric $g$. | |
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+ | An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric $g$. | ||
+ | {{endthm}} | ||
</wikitex> | </wikitex> | ||
+ | <!-- | ||
== References == | == References == | ||
− | {{#RefList:}} | + | {{#RefList:}} --> |
− | + | <!--[[Category:Theory]]--> | |
− | [[Category:Theory]] | + | <!-- {{Stub}} --> |
− | {{Stub}} | + |
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 manifoldTex syntax erroris a second countable, Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- The interior of
Tex syntax error
, denoted , is the subset of points for which is an open subset of . - The boundary of
Tex syntax error
, written , is the complement of . -
Tex syntax error
is called closed ifTex syntax error
is compact and is empty.
Tex syntax erroras 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
Tex syntax error. In the complex case, we assume that the dimension of
Tex syntax erroris even and that the boundary of
Tex syntax erroris empty. Recall that a chart on a topological manifold
Tex syntax erroris a homeomorphism from an open subset of
Tex syntax errorto an open subset of . The transition function defined by two charts and is the homeomorphism
Tex syntax erroris a collection of charts such that the cover
Tex syntax error.
Let denote either the piecewise linear, smooth or complex categories where by “smooth" we indicate maps. An atlas is a Atlas if every transition function defined by the that atlas is a function: that is, we require every to be either piecewise linear, smooth of class or holomorphic. Two atlases are compatible if their union again forms a atlas and by Zorn's Lemma each atlas defines a unique maximal atlas.
Tex syntax errortogether with a maximal atlas . A -isomorphism is a homeomorphism which is a morphism when viewed in every pair of charts in and .
3 Riemannian Manifolds
Tex syntax erroris a smooth family of scalar products
Tex syntax error. This means that for each pair of smooth vector fields and on
Tex syntax errorthe map
is smooth.
Definition 3.1.
A Riemannian manifold is a smooth manifoldTex syntax errortogether with a Riemannian metric .
An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric .