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.

Read more about Manifold: History, Mathematical Definition, Charts, Atlases, and Transition Maps, Construction, Classification and Invariants, Maps of Manifolds, Generalizations of Manifolds

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### 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 (1894–1963)

“There must be no cessation

Of motion, or of the noise of motion,

The renewal of noise

And *manifold* continuation....”

—Wallace Stevens (1879–1955)

“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] (1772–1801)