manifold

Summary.

A manifold is a space that islocally like ${ℝ}^n$, however lacking a preferred system ofcoordinates. Furthermore, a manifold can have global topologicalproperties, such as non-contractible loops (http://planetmath.org/Curve), that distinguish it fromthe topologically trivial ${ℝ}^n$.

Standard Definition.

An $n$-dimensional topological manifold $M$ is a second countable, Hausdorfftopological spacethat is locally homeomorphic to open subsets of${ℝ}^n$.

A differential manifold is a topological manifold with some additionalstructure information. A chart, also known as a systemof local coordinates, is a mapping $\alpha :U\to {ℝ}^n$, such that the domain $U\subset M$ is an open set, and such that $U$ is homeomorphic to the image $\alpha (U)$. Let$\alpha :U_{\alpha }\to {ℝ}^n,$ and$\beta :U_{\beta }\to {ℝ}^n$ be two charts with overlappingdomains (http://planetmath.org/Function). The continuous injection

$$\beta \circ {\alpha }^{-1}:\alpha (U_{\alpha }\cap U_{\beta })\to {ℝ}^n$$

is called a transition function,and also called a a change of coordinates. An atlas$𝒜$ is a collection of charts $\alpha :U_{\alpha }\to {ℝ}^n$whose domains cover $M$, i.e.

$$M=\bigcup _{\alpha }{U}_{\alpha }.$$

Note that each transition functionis really just $n$ real-valued functions of $n$ real variables, and sowe can ask whether these are continuously differentiable. The atlas$𝒜$ defines a differential structure on $M$, if every transition functionis continuously differentiable.

More generally, for $k=1,2,\mathrm{\dots },\mathrm{\infty },\omega$, the atlas $𝒜$ issaid to define a ${𝒞}^k$ differential structure, and $M$ is said to beof class ${𝒞}^k$, if all the transition functions are $k$-timescontinuously differentiable, or real analytic in the case of${𝒞}^{\omega }$. Two differential structures of class ${𝒞}^k$ on $M$ aresaid to be isomorphic if the union of the corresponding atlases isalso a ${𝒞}^k$ atlas, i.e. if all the new transition functions arisingfrom the merger of the two atlases remain of class ${𝒞}^k$. Moregenerally, two ${𝒞}^k$ manifolds $M$ and $N$ are said to bediffeomorphic, i.e. have equivalent differential structure, if thereexists a homeomorphism $\varphi :M\to N$ such that the atlas of $M$ isequivalent to the atlas obtained as $\varphi$-pullbacks of charts on $N$.

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

$$f:U\to ℝ,U\subset M$$

be a continuous function. For each $\alpha \in 𝒜$ we define

$$f_{\alpha }:V\to ℝ,V\subset {ℝ}^n,$$

called therepresentation of $f$ relative to chart $\alpha$, as the suitablyrestricted composition

$$f_{\alpha }=f\circ {\alpha }^{-1}.$$

We judge $f$ to be differentiable if allthe representations $f_{\alpha }$ are differentiable. A path

$$\gamma :I\to M,I\subset ℝ$$

is judged to be differentiable, if for all differentiablefunctions$f$, the suitably restricted composition $f\circ \gamma$ is adifferentiable function from $ℝ$ to $ℝ$. Finally, givenmanifolds $M,N$, we judge a continuous mapping $\varphi :M\to N$between them to be differentiable if for all differentiablefunctions$f$ on $N$, the suitably restricted composition $f\circ \varphi$ is adifferentiable function on $M$.

Title	manifold
Canonical name	Manifold
Date of creation	2013-03-22 12:20:22
Last modified on	2013-03-22 12:20:22
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	35
Author	matte (1858)
Entry type	Definition
Classification	msc 53-00
Classification	msc 57R50
Classification	msc 58A05
Classification	msc 58A07
Synonym	differentiable manifold
Synonym	differential manifold
Synonym	smooth manifold
Related topic	NotesOnTheClassicalDefinitionOfAManifold
Related topic	LocallyEuclidean
Related topic	3Manifolds
Related topic	Surface
Related topic	TopologicalManifold
Related topic	ProofOfLagrangeMultiplierMethodOnManifolds
Related topic	Submanifold
Defines	coordinate chart
Defines	chart
Defines	local coordinates
Defines	atlas
Defines	change of coordinates
Defines	differential structure
Defines	transition function
Defines	smooth structure
Defines	diffeomorphism
Defines	diffeomorphic
Defines	topological manifold
Defines	real-analytic manifold