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 of the sphere, it looks like a flat [[plane]], but globally the sphere and plane are very different. Other examples of manifolds include [[line|lines]] and [[circle|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 [[line]]s and [[circle]]s, and more abstract spaces such as the [[orthogonal group]] [[orthogonal_group|O(n)]].


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]].  
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# 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>\qquad \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta}).</math>
# 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>\qquad \psi_{\beta} \circ \psi_{\alpha}^{-1}: \psi_{\alpha} \left(U_{\alpha} \cap U_{\beta} \right) \rightarrow \psi_{\beta} (U_{\alpha} \cap U_{\beta}).</math>


The set M is a '''differentiable manifold''' if and only if it comes equipped with a [[countable]] atlas and satisfies the Hausdorff property. The important difference between a differentiable manifold and a topological manifold is that the charts are [[diffeomorphisms]] (a differentiable function with a differentiable inverse) rather than homeomorphisms.
The set M is a '''differentiable manifold''' if and only if it comes equipped with a [[countable]] atlas and satisfies the Hausdorff property. The important difference between a differentiable manifold and a topological manifold is that the charts are [[diffeomorphism]]s (a differentiable function with a differentiable inverse) rather than homeomorphisms.


Differentiable manifolds have a [[tangent space]] <math>T_p M</math>, the space of all [[tangent vectors]], associated with each point <math>p</math> on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within <math>\scriptstyle \mathbb{R}^{n+1}</math>, it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible.
Differentiable manifolds have a [[tangent space]] <math>T_p M</math>, the space of all [[tangent vector]]s, associated with each point <math>p</math> on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within <math>\scriptstyle \mathbb{R}^{n+1}</math>, it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible.


===Riemannian manifolds===
===Riemannian manifolds===


To define distances and angles on a differentiable manifold, it is necessary to define a '''[[metric]]'''. A differentiable manifold equipped with a metric is called a '''Riemannian manifold'''. A Riemannian metric is a generalisation of the usual idea of the scalar or [[dot product]] to a manifold. In other words, a Riemannian metric <math> g = \{g_p\}_{p \in M} </math> is a set of symmetric [[inner products]]
To define distances and angles on a differentiable manifold, it is necessary to define a '''[[metric]]'''. A differentiable manifold equipped with a metric is called a '''Riemannian manifold'''. A Riemannian metric is a generalisation of the usual idea of the scalar or [[dot product]] to a manifold. In other words, a Riemannian metric <math> g = \{g_p\}_{p \in M} </math> is a set of symmetric [[inner product]]s
:<math> g_p : T_pM \times T_pM \rightarrow \mathbb{R} </math>
:<math> g_p : T_pM \times T_pM \rightarrow \mathbb{R} </math>
which depend smoothly on <math>p</math>.
which depend smoothly on <math>p</math>.
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* [[Real projective space]] <math>\scriptstyle R \mathbb{P}^2</math> -- the set of all lines through the origin in <math>\scriptstyle \mathbb{R}^n </math>
* [[Real projective space]] <math>\scriptstyle R \mathbb{P}^2</math> -- the set of all lines through the origin in <math>\scriptstyle \mathbb{R}^n </math>
* The [[Grassmannian]] <math>\scriptstyle G_k(\mathbb{R}^n)</math> -- The set of all k-dimensional subspaces of <math>\scriptstyle \mathbb{R}^n</math>.
* The [[Grassmannian]] <math>\scriptstyle G_k(\mathbb{R}^n)</math> -- The set of all k-dimensional subspaces of <math>\scriptstyle \mathbb{R}^n</math>.
* [[Lie groups]] e.g. <math>\scriptstyle O(n), \, U(n),\, GL(n),</math> and the [[Lorentz group]],
* [[Lie group]]s e.g. [[orthogonal_group|O(n)]], [[unitary_group|U(n)]], [[general_linear_group|GL(n)]],</math> and the [[Lorentz group]],


==See also==
==See also==

Revision as of 06:21, 22 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 O(n).

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 , or an n-manifold, is defined as a Hausdorff space where an open neighbourhood of each point is homeomorphic to (i.e. there exists a smooth bijective map from the manifold with a smooth inverse 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: 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.

Let M be a set. An atlas of M is a collection of pairs for some varying over an index set such that

  1. maps bijectively to an open set , and for the image is an open set. The function is called a chart.
  2. For , the coordinate change is a differentiable map between two open sets in whereby

The set M is a differentiable manifold if and only if it comes equipped with a countable atlas and satisfies the Hausdorff property. The important difference between a differentiable manifold and a topological manifold is that the charts are diffeomorphisms (a differentiable function with a differentiable inverse) rather than homeomorphisms.

Differentiable manifolds have a tangent space , the space of all tangent vectors, associated with each point on the manifold. This tangent space is also n-dimensional. Although it is normal to visualise a tangent space being embedded within , it can be defined without such an embedding, and in the case of abstract manifolds this visualisation is impossible.

Riemannian manifolds

To define distances and angles on a differentiable manifold, it is necessary to define a metric. A differentiable manifold equipped with a metric is called a Riemannian manifold. A Riemannian metric is a generalisation of the usual idea of the scalar or dot product to a manifold. In other words, a Riemannian metric is a set of symmetric inner products

which depend smoothly on .

Examples of manifolds

This list is not exhaustive.

  • Circles, lines, planes etc.
  • Paraboloid of revolution
  • Real projective space -- the set of all lines through the origin in
  • The Grassmannian -- The set of all k-dimensional subspaces of .
  • Lie groups e.g. O(n), U(n), GL(n),</math> and the Lorentz group,

See also