In mathematics, a manifold of dimension n is a topological space that near each point resembles n-dimensional Euclidean space. More precisely, each point of an n-dimensional manifold has a neighbourhood that is homeomorphic to the Euclidean space of dimension n. Lines and circles, but not figure eights, are one-dimensional manifolds. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, which can all be realized in three dimensions, but also the Klein bottle and real projective plane which cannot.
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Some articles on manifold:
... The Whitehead manifold is an example of a 3-manifold that is contractible but not simply connected at infinity ... property is invariant under homeomorphism, this proves that the Whitehead manifold is not homeomorphic to R3 ... However, it is a theorem that any contractible n-manifold which is also simply connected at infinity is homeomorphic to Rn ...
... Suppose is a Riemannian manifold and ... some function such that for all and all then is constant, and the manifold has constant sectional curvature (also known as a space form when is complete) the Ricci ...
... In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gn, such that as n goes to infinity the manifold is ... The simplest example is a flat manifold, whose metric can be rescaled by 1/n, so that the manifold is close to a point, but its curvature remains 0 for all n ...
... Why does one study manifolds? Manifolds, and generalized spaces composed of manifolds such as stratified spaces, occupy a central role in topology ... can be stratified into manifold pieces), and that they are the space "modeled on" Euclidean space (a space that looks locally like Euclidean space) – i.e ... Manifolds are homogeneous and tame (locally isomorphic to Euclidean space) in this manner, and one may ask if all "tame" homogeneous spaces are manifolds, or whether there is a natural class where more ...
... the collision occurs into a collision manifold, the phase space point is cut out and in its place a smooth manifold is pasted ... McGehee then went on to study the flow on the collision manifold ...
More definitions of "manifold":
- (noun): A pipe that has several lateral outlets to or from other pipes.
- (adj): Many and varied; having many features or forms.
Example: "Manifold reasons"; "our manifold failings"; "manifold intelligence"
- (noun): A lightweight paper used with carbon paper to make multiple copies.
Synonyms: manifold paper
- (noun): A set of points such as those of a closed surface or and analogue in three or more dimensions.
- (verb): Combine or increase by multiplication.
Famous quotes containing the word manifold:
“Thy love is such I can no way repay,
The heavens reward thee manifold I pray.
Then while we live, in love lets so persever,
That when we live no more, we may live ever.”
—Anne Bradstreet (c. 16121672)
“The Lord wrote it all down on the little slate
Of the baby tortoise.
Outward and visible indication of the plan within,
The complex, manifold involvedness of an individual creature”
—D.H. (David Herbert)
“Odysseus saw the sirens; they were charming,
Blonde, with snub breasts and little neat posteriors,”
—John Streeter Manifold (b. 1915)