Manifold Atlas:Definition of “manifold”
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+ | == Introduction == | ||
+ | <wikitex>; | ||
This page defines the term “manifold” as used in the Manifold Atlas. | This page defines the term “manifold” as used in the Manifold Atlas. | ||
− | <!-- and summarises a few key properties of all manifolds --> | + | <!-- 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 $ | + | * 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 | + | * 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}} | ||
− | + | A manifold $M$ as above is often called a topological manifold for emphasis or clarity. | |
− | Typically, but not necessarly, the word | + | 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 == | ||
+ | <wikitex>; | ||
+ | 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 [[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 | ||
+ | <!-- --> | ||
+ | $$ \phi_{\alpha\beta} := \phi_\alpha \circ \phi_\beta^{-1}|_{\phi_{\beta}(U_\alpha \cap U_\beta)} : \phi_\beta(U_\alpha \cap U_\beta) \longrightarrow \phi_\alpha(U_\alpha \cap U_\beta).$$ | ||
+ | <!-- --> | ||
+ | An [[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: 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. | |
− | + | ||
− | [[Category:Theory]] | + | {{beginthm|Definition}} A $\Cat$-manifold $(M, A)$ is a manifold $M$ together with a maximal $\Cat$ atlas $A$. |
− | {{Stub}} | + | |
+ | 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 [[Wikipedia:Riemannian_manifold|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}} --> |
Latest revision as of 19:32, 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 is a second countable, Hausdorff space for which every point has a neighbourhood homeomorphic to an open subset of .
- The interior of , denoted , is the subset of points for which is an open subset of .
- The boundary of , written , is the complement of .
- is called closed if is compact and is empty.
A manifold 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 of is even and that the boundary of is empty.
Recall that a chart on a topological manifold is a homeomorphism from an open subset of to an open subset of . The transition function defined by two charts and is the homeomorphism
An atlas for is a collection of charts such that the cover .
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.
Definition 2.1. A -manifold is a manifold together 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
A Riemannian metric on a smooth manifold is a smooth family of scalar products
defined on the tangent spaces for each in . This means that for each pair of smooth vector fields and on the map
is smooth.
Definition 3.1. A Riemannian manifold is a smooth manifold together with a Riemannian metric .
An isometry between Riemannian manifolds is a diffeomorphism whose differential preserves the metric .