Manifold Atlas:Definition of “manifold”
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== Atlases of charts == | == Atlases of charts == | ||
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Recall that a chart on a topological manifold $M$ is a homeomporphism $\phi_\alpha : U_\alpha \to V_\alpha M$ 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 | Recall that a chart on a topological manifold $M$ is a homeomporphism $\phi_\alpha : U_\alpha \to V_\alpha M$ 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 | ||
− | $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ An atlas for $M$ is a collection of charts $ | + | $$ \phi_{\alpha, \beta} : \phi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \phi_\beta(U_\alpha \cap U_\beta).$$ 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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+ | Let $\Cat$ denote either the piecewise linear, smooth or complex categories. 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 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 $A$ of $\Cat$ charts $A$. | ||
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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 $B$. | ||
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− | == Piecewise-linear manifolds == | + | <!-- == Piecewise-linear manifolds == |
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* A piecewise linear manifold $(M, A)$, PL-manifold, is a manifold $M$ together with a maximal atlas $\mathcal{A}$ of piecewise linear charts $A$. | * 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 complex manifold $(M, \gamma)$, is an even dimensional manifold $M$ together with a maximal atlas $\mathcal{A}$ of holomorphic charts $A$. | * A complex manifold $(M, \gamma)$, is an even dimensional manifold $M$ together with a maximal atlas $\mathcal{A}$ of holomorphic charts $A$. | ||
* 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 $\delta$. | * 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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== Riemannian manifolds == | == Riemannian manifolds == | ||
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== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} |
Revision as of 21:02, 16 September 2009
Contents |
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 .
- The boundary of , written , is the complement of the interior 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 smooth, Riemannian, complex, etc. The extra structure will be emphasised or suppressed in notation and vocabulary. We briefly review the some common categories of manifolds below.
2 Atlases of charts
Recall that a chart on a topological manifold is a homeomporphism from an open subset of to an open subset of . The transition function defined by two charts and is
Let denote either the piecewise linear, smooth or complex categories. 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.
Definition 2.1. A -manifold is a manifold together with a maximal atlas of charts .
A -isomorphism is a homeomorphism which is a morphism when viewed in every pair of charts in and .
3 Riemannian manifolds
4 References
This page has not been refereed. The information given here might be incomplete or provisional. |