Manifold Atlas:Definition of “manifold”
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− | An '''n-dimensional | + | 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 $\Rr^n_+ := \{ v \in \Rr^n | x_1 \geq 0 \}$. |
− | * The '''interior''' of $M$, denoted $\mathrm{int}(M)$, is the subset of points for which $ | + | * The '''interior''' of $M$, denoted $\mathrm{int}(M)$, is the subset of points for which $U_x \subset \Rr^n$. |
* The '''boundary''' of $M$, written $\partial M$, is the complement of the interior of $M$. | * 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. | * $M$ is called '''closed''' if $M$ is compact and $\partial M$ is empty. | ||
{{endthm}} | {{endthm}} | ||
− | Typically, but not necessarly, the word “manifold” will mean | + | 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, 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. | |
</wikitex> | </wikitex> | ||
+ | == Atlases of charts == | ||
+ | <wikitex> | ||
+ | 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 $\mathcal{A} = \{ (U_\alpha, \phi_\alpha)\}$ such that the $U_\alpha$ cover $M$. An atlas is called resp. piecwise linear, smooth or complex, etc if every transition function is resp. piecewise linear, smooth or complex, etc. Atlases of a given category are compatible if their union again forms an Atlas of that category any by Zorn's Lemma each atlas of a given category defines a unique maximal atlas in that category. | ||
+ | </wikitex> | ||
+ | == Piecewise-linear manifolds == | ||
+ | <wikitex>; | ||
+ | * 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 PL-homeomorphism $(M, A) \cong (N, B)$ is a homeomorphism $f: M \cong N$ which is piecewise linear when viewed in every pair of charts in $A$ and $B$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Smooth manifolds == | ||
+ | <wikitex>; | ||
+ | We shall use the term smooth manifold to refer to $C^\infty$ smooth manifolds. | ||
+ | * A smooth manifold $(M, \alpha)$ is a smooth manifold $M$ together with a maximal atlas of smooth charts $\alpha$. | ||
+ | * A diffeomorphism between smooth manfiolds $(M, \alpha) \cong (N, \beta)$ is a homeomorphism $f : N \to M$ such that $f$ is $C^\infty$ when viewed in every pair of charts in $\alpha$ and $\beta$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Complex manifolds == | ||
+ | <wikitex>; | ||
+ | * 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 $\delta$. | ||
+ | </wikitex> | ||
+ | |||
+ | == Riemannian manifolds == | ||
+ | <wikitex>; | ||
+ | |||
+ | </wikitex> | ||
+ | == References == | ||
== References == | == References == |
Revision as of 20:52, 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 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 . - 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 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
Tex 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
An atlas for
Tex syntax erroris a collection of charts such that the cover
Tex syntax error. An atlas is called resp. piecwise linear, smooth or complex, etc if every transition function is resp. piecewise linear, smooth or complex, etc. Atlases of a given category are compatible if their union again forms an Atlas of that category any by Zorn's Lemma each atlas of a given category defines a unique maximal atlas in that category.
3 Piecewise-linear manifolds
- A piecewise linear manifold , PL-manifold, is a manifold
Tex syntax error
together with a maximal atlas of piecewise linear charts . - A PL-homeomorphism is a homeomorphism which is piecewise linear when viewed in every pair of charts in and .
4 Smooth manifolds
We shall use the term smooth manifold to refer to smooth manifolds.
- A smooth manifold is a smooth manifold
Tex syntax error
together with a maximal atlas of smooth charts . - A diffeomorphism between smooth manfiolds is a homeomorphism such that is when viewed in every pair of charts in and .
5 Complex manifolds
- A complex manifold , is an even dimensional manifold
Tex syntax error
together with a maximal atlas of holomorphic charts . - A complex diffeomorphism is a homeomorphism which is piecewise linear when viewed in ever pair of charts in and .
6 Riemannian manifolds
7 References
8 References
This page has not been refereed. The information given here might be incomplete or provisional. |