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.
Although near each point, a manifold resembles Euclidean space, globally a manifold might not. For example, the surface of the sphere is not a Euclidean space, but in a region it can be charted by means of geographic maps: map projections of the region into the Euclidean plane. When a region appears in two neighbouring maps (in the context of manifolds they are called charts), the two representations do not coincide exactly and a transformation is needed to pass from one to the other, called a transition map.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows more complicated structures to be described and understood in terms of the relatively well-understood properties of Euclidean space. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. Manifolds may have additional features. One important class of manifolds is the class of differentiable manifolds. This differentiable structure allows calculus to be done on manifolds. A Riemannian metric on a manifold allows distances and angles to be measured. Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
Other articles related to "manifold, manifolds":
... 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 ...
... up the single point in phase space where 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 ...
... Suppose is a Riemannian manifold and ... that is, there exists 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 ...
... 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 ... 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 ...
... 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 ...
Famous quotes containing the word manifold:
“A large city cannot be experientially known; its life is too manifold for any individual to be able to participate in it.”
—Aldous Huxley (18941963)
“There must be no cessation
Of motion, or of the noise of motion,
The renewal of noise
And manifold continuation....”
—Wallace Stevens (18791955)
“Before abstraction everything is one, but one like chaos; after abstraction everything is united again, but this union is a free binding of autonomous, self-determined beings. Out of a mob a society has developed, chaos has been transformed into a manifold world.”
—Novalis [Friedrich Von Hardenberg] (17721801)