Manifold Atlas:Definition of “manifold”
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== Atlases of charts == | == Atlases of charts == | ||
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− | We give a unified presentation of piecewise linear, smooth and complex manifolds. In the complex case, we assume that the dimension is even and that the dimension is even. | + | 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 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 | + | 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 |
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$$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ | $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ | ||
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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_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. | 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. | ||
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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 $B$. | 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 $B$. | ||
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== Riemannian Manifolds == | == Riemannian Manifolds == | ||
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Revision as of 16:35, 21 September 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 the interior ofTex syntax error
. -
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
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 manifoldTex syntax erroris a homeomporphism 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. That is we require every to be either piecewise linear, smooth of class or holomorphic. An atlas is a Atlas if every transition function defined by the that atlas is a function. 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 .