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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.

## Contents

1.1 CirclemehrManifold aus Wikipedia.

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