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Manifold - Wikipedia, the free encyclopedia
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Manifold
From Wikipedia, the free encyclopedia
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For other uses, see
Manifold (disambiguation).
The
sphere (surface of a
ball) is a two-dimensional manifold since it can be represented by a collection of two-dimensional maps.
In
mathematics (specifically in
differential geometry and
topology), a manifold is a
topological space that on a small enough scale resembles the
Euclidean space of a specific dimension, called the
dimension of the manifold. Thus, a
line and a
circle are one-dimensional manifolds, a
plane and
sphere (the surface of a
ball) are two-dimensional manifolds, and so on into
high-dimensional space. More formally, every point of an n-dimensional manifold has a
neighborhood homeomorphic to an
open subset of the n-dimensional space Rn.
Although manifolds resemble Euclidean spaces near each point ("locally"), the global structure of a manifold may be more complicated. For example, any point on the usual two-dimensional surface of a
sphere is surrounded by a circular region that can be flattened to a circular region of the
plane, as in a geographical map. However, the sphere differs from the plane "in the large": in the language of
topology, they are not homeomorphic. The structure of a manifold is encoded by a collection of charts that form an
atlas, in analogy with an atlas consisting of charts of the surface of the Earth.
The concept of manifolds is central to many parts of
geometry and modern
mathematical physics because it allows more complicated structures to be expressed and understood in terms of the relatively well-understood properties of simpler spaces. For example, a manifold is typically endowed with a
differentiable structure that allows one to do
calculus and a
Riemannian metric that allows one to measure
distances and
angles.
Symplectic manifolds serve as the
phase spaces in the
Hamiltonian formalism of
classical mechanics, while four-dimensional
Lorentzian manifolds model
space-time in
general relativity.
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