Manifold (geometry): Difference between revisions
imported>Michael Hardy (Linking to "open" or to "neighbourhood" seems very silly. Obviously an article titled "neighbourhood" is not likely to be about the concept in topology.) |
imported>Michael Hardy m (→See also) |
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===Topological manifold=== | ===Topological manifold=== | ||
In [[topology]], a manifold of dimension <math>n</math>, or an '''n-manifold''', is defined as a [[Hausdorff space]] where an [[open set|open]] [[neighbourhood (topology)|neighbourhood]] of each point is [[homeomorphic]] (i.e. there exists a smooth bijective map from the manifold with a smooth inverse | In [[topology]], a manifold of dimension <math>n</math>, or an '''''n''-manifold''', is defined as a [[Hausdorff space]] where an [[open set|open]] [[neighbourhood (topology)|neighbourhood]] of each point is [[homeomorphic]] to <math>\scriptstyle \mathbb{R}^n </math> (i.e. there exists a smooth bijective map from the manifold with a smooth inverse to <math>\scriptstyle \mathbb{R}^n </math>). | ||
===Differentiable manifold=== | ===Differentiable manifold=== | ||
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==See also== | ==See also== | ||
* [[Tangent vector]] | * [[Tangent vector]] | ||
* [[Differential geometry]] | * [[Differential geometry]] | ||
* [[Riemannian geometry]] | * [[Riemannian geometry]] |
Revision as of 19:04, 12 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 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
- 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
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 .