Topological space that locally resembles Euclidean space
For other uses, see Manifold (disambiguation).
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The Klein bottle immersed in three-dimensional spaceThe surface of the Earth requires (at least) two charts to include every point. Here the globe is decomposed into charts around the North and South Poles.
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an -dimensional manifold, or -manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of -dimensional Euclidean space.
One-dimensional manifolds include lines and circles, but not self-crossing curves such as a figure 8. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane.
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g. CT scans).
Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable structure allows calculus to be done. 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.
The study of manifolds requires working knowledge of calculus and topology.
manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n {\displaystyle n} -dimensional manifold,...
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...
a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with...
differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate...
mathematics. All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...
as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "anticlockwise"...
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geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear...
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known as manifold learning, is any of various related techniques that aim to project high-dimensional data onto lower-dimensional latent manifolds, with...
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all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the...
almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...
manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice. The E 8 {\displaystyle E_{8}} manifold...
A hydraulic manifold is a component that regulates fluid flow between pumps and actuators and other components in a hydraulic system. It is like a switchboard...
In mathematics, a Haken manifold is a compact, P²-irreducible 3-manifold that is sufficiently large, meaning that it contains a properly embedded two-sided...
In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to R 3 . {\displaystyle \mathbb {R} ^{3}.} J. H...
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