Manifold

manifoldsboundarymanifold with boundarysmooth manifoldschartsmanifold theorymanifold with cornersreal manifoldwith boundary3-manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.wikipedia
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Topological space

topologytopological spacestopological structure
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
Other spaces, such as manifolds and metric spaces, are specializations of topological spaces with extra structures or constraints.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
One important class of manifolds is the class of differentiable manifolds; this differentiable structure allows calculus to be done on manifolds. In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, such as a differentiable structure. For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole.
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

Map projection

projectionequal-areacartographic projection
For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts).
Projections are a subject of several pure mathematical fields, including differential geometry, projective geometry, and manifolds.

Real projective plane

projective planeFlat projective planeprojective manifolds
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface.

Klein bottle

Klein bottlesbottleKlein
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
In topology, a branch of mathematics, the Klein bottle is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined.

Geometry

geometricgeometricalgeometries
The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described and understood in terms of the simpler local topological properties of Euclidean space.
These include the concepts of point, line, plane, distance, angle, surface, and curve, as well as the more advanced notions of topology and manifold.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
A Riemannian metric on a manifold allows distances and angles to be measured.
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M, g) is a real, smooth manifold M equipped with an inner product g p on the tangent space T p M at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g p (X is a smooth function.

Immersion (mathematics)

immersionimmersedimmersions
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
A regular homotopy between two immersions f and g from a manifold M to a manifold N is defined to be a differentiable function H : M × [0,1] → N such that for all t in [0, 1] the function H t : M → N defined by H t (x) = H(x, t) for all x ∈ M is an immersion, with H 0 = f, H 1 = g. A regular homotopy is thus a homotopy through immersions.

Atlas (topology)

atlascharttransition map
For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts).
In mathematics, particularly topology, one describes a manifold using an atlas.

Distance

distancesproximitydepth
A Riemannian metric on a manifold allows distances and angles to be measured.
In this way, many different types of "distances" can be calculated, such as for traversal of graphs, comparison of distributions and curves, and using unusual definitions of "space" (for example using a manifold or reflections).

Spacetime

space-timespace-time continuumspace and time
Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.

Complex manifold

complex structurecomplex manifoldscomplex structures
For topological or differentiable manifolds, one can also ask that every point have a neighborhood homeomorphic to all of Euclidean space (as this is diffeomorphic to the unit ball), but this cannot be done for complex manifolds, as the complex unit ball is not holomorphic to complex space.
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in C n, such that the transition maps are holomorphic.

Surface (topology)

surfaceclosed surfacesurfaces
Two-dimensional manifolds are also called surfaces.
In mathematics, a surface is a two-dimensional manifold.

Topological manifold

coordinate charttopological2-manifold
In geometry and topology, all manifolds are topological manifolds, possibly with additional structure, such as a differentiable structure.
All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure (e.g. differentiable manifolds are topological manifolds equipped with a differential structure).

Differential structure

differentiable structuresmooth structure
One important class of manifolds is the class of differentiable manifolds; this differentiable structure allows calculus to be done on manifolds.
The C k equivalence classes of such atlases are the distinct C k differential structures of the manifold.

Euclidean space

EuclideanspaceEuclidean vector space
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
This way of defining coordinates extends easily to other mathematical structures, and in particular to manifolds.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
For instance, differentiable manifolds have homeomorphisms on overlapping neighborhoods diffeomorphic with each other, so that the manifold has a well-defined set of functions which are differentiable in each neighborhood, and so differentiable on the manifold as a whole.
Given two manifolds M and N, a differentiable map f : M → N is called a diffeomorphism if it is a bijection and its inverse f −1 : N → M is differentiable as well.

Closed manifold

closedcompact manifoldcompact
Manifolds need not be closed; thus a line segment without its end points is a manifold.
In mathematics, a closed manifold is a type of topological space, namely a compact manifold without boundary.

Symplectic manifold

Lagrangian submanifoldsymplecticspecial Lagrangian submanifold
Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
A symplectic form on a smooth manifold M is a closed non-degenerate differential 2-form \omega.

Non-Hausdorff manifold

bug-eyed lineline with two originsnon-Hausdorff
Second countable and Hausdorff are point-set conditions; second countable excludes spaces which are in some sense 'too large' such as the long line, while Hausdorff excludes spaces such as "the line with two origins" (these generalizations of manifolds are discussed in non-Hausdorff manifolds).
In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space.

Orbifold

orbifoldsuniversal covering orbifoldapplication of orbifolds to music theory
Among the possible quotient spaces that are not necessarily manifolds, orbifolds and CW complexes are considered to be relatively well-behaved.
In the mathematical disciplines of topology, geometry, and geometric group theory, an orbifold (for "orbit-manifold") is a generalization of a manifold.

General relativity

general theory of relativitygeneral relativity theoryrelativity
Symplectic manifolds serve as the phase spaces in the Hamiltonian formalism of classical mechanics, while four-dimensional Lorentzian manifolds model spacetime in general relativity.
This formula, too, is readily generalized to curved spacetime by replacing partial derivatives with their curved-manifold counterparts, covariant derivatives studied in differential geometry.

Torus

toroidaltoriflat torus
Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
An n-torus in this sense is an example of an n-dimensional compact manifold.

Sphere

sphericalhemisphereglobose
For example, the surface of the sphere is not homeomorphic to the Euclidean plane, because (among other properties) it has the global topological property of compactness that Euclidean space lacks, but in a region it can be charted by means of map projections of the region into the Euclidean plane (in the context of manifolds they are called charts). Examples include the plane, the sphere, and the torus, which can all be embedded (formed without self-intersections) in three dimensional real space, but also the Klein bottle and real projective plane, which will always self-intersect when immersed in three-dimensional real space.
A sphere need not be smooth; if it is smooth, it need not be diffeomorphic to the Euclidean sphere (an exotic sphere).

Locally connected space

locally connectedlocally path-connectedlocally path connected
A "+" is not homeomorphic to a closed interval (line segment), since deleting the center point from the "+" gives a space with four components (i.e. pieces), whereas deleting a point from a closed interval gives a space with at most two pieces; topological operations always preserve the number of pieces.
In the latter part of the twentieth century, research trends shifted to more intense study of spaces like manifolds which are locally well understood (being locally homeomorphic to Euclidean space) but have complicated global behavior.