Manifold (geometry): Difference between revisions
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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 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> | 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> | ||
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 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]]. | ||
Revision as of 12:34, 11 July 2007
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
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 differentiable manifolds and Riemannian manifolds.
Mathematical Definition
In topology, a manifold of dimension , or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic to .
