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manifold (Definition)

Summary.

A manifold is a space that is locally like $ \mathbb{R}^n$, however lacking a preferred system of coordinates. Furthermore, a manifold can have global topological properties, such as non-contractible loops, that distinguish it from the topologically trivial $ \mathbb{R}^n$.

Standard Definition.

An $ n$-dimensional topological manifold $ M$ is a second countable, Hausdorff topological space 1that is locally homeomorphic to open subsets of $ \mathbb{R}^n$.

A differential manifold is a topological manifold with some additional structure information. A chart, also known as a system of local coordinates, is a mapping $ \alpha: U \to \mathbb{R}^n$, such that the domain $ U\subset M$ is an open set, and such that $ U$ is homeomorphic to the image $ \alpha(U)$. Let $ \alpha: U_\alpha \rightarrow \mathbb{R}^n,$ and $ \beta:U_\beta\rightarrow\mathbb{R}^n$ be two charts with overlapping domains. The continuous injection

$\displaystyle \beta\circ\alpha^{-1}: \alpha(U_\alpha\cap U_\beta)\rightarrow\mathbb{R}^n$
is called a transition function, and also called a a change of coordinates. An atlas $ \mathcal{A}$ is a collection of charts $ \alpha:U_\alpha\rightarrow\mathbb{R}^n$ whose domains cover $ M$, i.e.
$\displaystyle M = \bigcup_{\alpha} U_\alpha.$
Note that each transition function is really just $ n$ real-valued functions of $ n$ real variables, and so we can ask whether these are continuously differentiable. The atlas $ \mathcal{A}$ defines a differential structure on $ M$, if every transition function is continuously differentiable.

More generally, for $ k=1,2,\ldots,\infty,\omega$, the atlas $ \mathcal{A}$ is said to define a $ \mathcal{C}^k$ differential structure, and $ M$ is said to be of class $ \mathcal{C}^k$, if all the transition functions are $ k$-times continuously differentiable, or real analytic in the case of $ \mathcal{C}^\omega$. Two differential structures of class $ \mathcal{C}^k$ on $ M$ are said to be isomorphic if the union of the corresponding atlases is also a $ \mathcal{C}^k$ atlas, i.e. if all the new transition functions arising from the merger of the two atlases remain of class $ \mathcal{C}^k$. More generally, two $ \mathcal{C}^k$ manifolds $ M$ and $ N$ are said to be diffeomorphic, i.e. have equivalent differential structure, if there exists a homeomorphism $ \phi:M\to N$ such that the atlas of $ M$ is equivalent to the atlas obtained as $ \phi$-pullbacks of charts on $ N$.

The atlas allows us to define differentiable mappings to and from a manifold. Let

$\displaystyle f:U\rightarrow\mathbb{R},\quad U\subset M$
be a continuous function. For each $ \alpha\in \mathcal{A}$ we define
$\displaystyle f_\alpha: V \rightarrow \mathbb{R},\quad V\subset\mathbb{R}^n,$
called the representation of $ f$ relative to chart $ \alpha$, as the suitably restricted composition
$\displaystyle f_\alpha = f\circ \alpha^{-1}.$
We judge $ f$ to be differentiable if all the representations $ f_\alpha$ are differentiable. A path
$\displaystyle \gamma: I\rightarrow M,\quad I\subset\mathbb{R}$
is judged to be differentiable, if for all differentiable functions $ f$, the suitably restricted composition $ f\circ\gamma$ is a differentiable function from $ \mathbb{R}$ to $ \mathbb{R}$. Finally, given manifolds $ M, N$, we judge a continuous mapping $ \phi:M\rightarrow N$ between them to be differentiable if for all differentiable functions $ f$ on $ N$, the suitably restricted composition $ f\circ\phi$ is a differentiable function on $ M$.


Footnotes

...1
For connected manifolds, the assumption that $ M$ is second-countable is logically equivalent to $ M$ being paracompact, or equivalently to $ M$ being metrizable. The topological hypotheses in the definition of a manifold are needed to exclude certain counter-intuitive pathologies. Standard illustrations of these pathologies are given by the long line (lack of paracompactness) and the forked line (points cannot be separated). These pathologies are fully described in Spivak. See this page.


"manifold" is owned by matte. [ full author list (2) | owner history (1) ]
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See Also: notes on the classical definition of a manifold, locally Euclidean, 3-manifold, surface, topological manifold, Lagrange multipliers on manifolds, submanifold

Other names:  differentiable manifold, differential manifold, smooth manifold
Also defines:  coordinate chart, chart, local coordinates, atlas, change of coordinates, differential structure, transition function, smooth structure, diffeomorphism, diffeomorphic, topological manifold, real-analytic manifold

Attachments:
notes on the classical definition of a manifold (Topic) by rmilson
topological manifold (Definition) by juanman
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Cross-references: path, differentiable, composition, restricted, representation, homeomorphism, equivalent, union, isomorphic, real analytic, class, continuously differentiable, variables, real, functions, cover, collection, injection, continuous, image, homeomorphic, domain, mapping, information, structure, open subsets, locally homeomorphic, separated, points, line, paracompactness, long line, metrizable, paracompact, logically equivalent, connected, Hausdorff topological space, second countable, properties, coordinates
There are 273 references to this entry.

This is version 32 of manifold, born on 2002-02-15, modified 2007-10-14.
Object id is 1981, canonical name is Manifold.
Accessed 77296 times total.

Classification:
AMS MSC: 53-00 (Differential geometry :: General reference works )
 57R50 (Manifolds and cell complexes :: Differential topology :: Diffeomorphisms)
 58A05 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Differentiable manifolds, foundations)
 58A07 (Global analysis, analysis on manifolds :: General theory of differentiable manifolds :: Real-analytic and Nash manifolds)
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Closed manifolds vs boundaries? by Schneemann on 2007-04-07 18:48:03
Does the above definition say anything about the manifold being closed or not, or containing its boundary?

What about the open disk, the closed disk, and so on? Will they (with their natural topology) be regarded as 2-manifolds in the above sense? And what extra rule would we need to define the difference of being closed or not?
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