Manifold (disambiguation): Difference between revisions

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The word '''manifold''' refers to:
*[[Manifold (geometric)]], a euclidian topological space
*[[manifold (automotive)]], a fitting on an internal combustion engine
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{{r|Manifold (geometry)|In geometry}}
A '''manifold''' is an abstract mathematical space that looks locally like [[Euclidean]] space, but globally may have a very different structure. An example of this is a [[sphere]]: if one is very close to the surface of the sphere, it looks like a flat [[plane]], but globally the sphere and plane are very different. Other examples of manifolds include [[lines]] and [[circles]], and more abstract spaces such as the [[orthogonal group]] <math>O(n)</math>
{{r|Manifold (automotive)|In automotive engineering}}
 
The concept of a manifold is very important within [[mathematics]] and [[physics]], and is fundamental to certain fields such as [[differential geometry]], [[Riemannian geometry]] and [[General Relativity]].
 
The most basic manifold is a topological manifold, but additional structures can be defined on the manifold to create objects such as differentiable manifolds and Riemannian manifolds.
 
== Mathematical Definition ==
 
===Topological Manifold===
 
In [[topology]], a manifold of dimension <math>n</math>, or an '''n-manifold''', is defined as a [[Hausdorff]] space where an [[open]] [[neighbourhood]] of each point is [[homeomorphic]] (i.e. there exists a smooth bijective map from the manifold with a smooth inverse) to <math>\scriptstyle \mathbb{R}^n </math>.
 
===Differentiable Manifold===
 
To define differentiable manifolds, the concept of an '''atlas''', '''chart''' and a '''coordinate change''' need to be introduced. An atlas of the Earth uses these concepts: the atlas is a collection of different overlapping patches of small parts of a spherical object onto a plane. The way in which these different patches overlap is defined by the coordinate change.

Latest revision as of 09:56, 31 May 2009

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