Manifold (geometry): Difference between revisions
imported>Natalie Watson (Added a bit on differentiable manifolds) |
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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: an 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. | 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: an 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. | ||
Let M be a set. An | Let M be a set. An ''atlas'' of M is a collection of pairs <math>\scriptstyle \left(U_{\alpha}, \psi_{\alpha}\right)</math> for some <math>\scriptstyle \alpha</math> varying over an index [[set]] <math> A </math> such that | ||
*<math>\scriptstyle U_{\alpha} \in M, \quad M = \bigcup_{\alpha \in A} U_{\alpha} </math> | *<math>\scriptstyle U_{\alpha} \in M, \quad M = \bigcup_{\alpha \in A} U_{\alpha} </math> | ||
*<math> \psi_{\alpha} </math> maps <math>U_{\alpha}</math> [[bijectively]] to an open set <math> \scriptstyle V_{\alpha} \in \, \mathbb{R}^n</math>, and for <math>\scriptstyle \alpha,\,\beta \,\in A </math> the image <math>\scriptstyle \psi(U_{\alpha} \cap U_{\beta}) \, \in \, \mathbb{R}^n</math> is an open set. | *<math> \psi_{\alpha} </math> maps <math>U_{\alpha}</math> [[bijectively]] to an open set <math> \scriptstyle V_{\alpha} \in \, \mathbb{R}^n</math>, and for <math>\scriptstyle \alpha,\,\beta \,\in A </math> the image <math>\scriptstyle \psi(U_{\alpha} \cap U_{\beta}) \, \in \, \mathbb{R}^n</math> is an open set. The function <math>\scriptstyle \psi_{\alpha}: U_{\alpha} \rightarrow V_{\alpha} </math> is called a ''chart''. | ||
* For <math>\scriptstyle \alpha, \, \beta \, \in A</math>, the | * For <math>\scriptstyle \alpha, \, \beta \, \in A</math>, the ''coordinate change'' is a [[differentiable map]] between two open sets in <math>\scriptstyle \mathbb{R}^n </math> whereby | ||
:<math> \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta})</math> | :<math> \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta})</math> | ||
Then the set M is a differentiable manifold if and only if it comes equipped with a [[countable]] atlas and satisfies the Hausdorff property. | Then the set M is a differentiable manifold if and only if it comes equipped with a [[countable]] atlas and satisfies the Hausdorff property. |
Revision as of 18:09, 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
Topological Manifold
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 .
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: an 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.
Let M be a set. An atlas of M is a collection of pairs for some varying over an index set such that
- maps bijectively to an open set , and for the image is an open set. The function is called a chart.
- For , the coordinate change is a differentiable map between two open sets in whereby
Then the set M is a differentiable manifold if and only if it comes equipped with a countable atlas and satisfies the Hausdorff property.