- Fields of mathematics
- >
- Geometry
- >
- Topology
- >
- Manifolds

- Fields of mathematics
- >
- Topology
- >
- Differential topology
- >
- Manifolds

- Fields of mathematics
- >
- Topology
- >
- Geometric topology
- >
- Manifolds

- Fields of mathematics
- >
- Topology
- >
- Topological spaces
- >
- Manifolds

- Geometry
- >
- Fields of geometry
- >
- Differential geometry
- >
- Manifolds

- Geometry
- >
- Topology
- >
- Differential topology
- >
- Manifolds

- Geometry
- >
- Topology
- >
- Geometric topology
- >
- Manifolds

- Geometry
- >
- Topology
- >
- Topological spaces
- >
- Manifolds

- Mathematical concepts
- >
- Mathematical structures
- >
- Topological spaces
- >
- Manifolds

- Mathematical objects
- >
- Mathematical structures
- >
- Topological spaces
- >
- Manifolds

- Mathematics
- >
- Fields of mathematics
- >
- Topology
- >
- Manifolds

- Topology
- >
- General topology
- >
- Topological spaces
- >
- Manifolds

Busemann G-space

In mathematics, a Busemann G-space is a type of metric space first described by Herbert Busemann in 1942. If is a metric space such that 1.
* for every two distinct there exists such that 2.
* every

History of manifolds and varieties

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of m

Dold manifold

In mathematics, a Dold manifold is one of the manifolds , where is the involution that acts as −1 on the m-sphere and as complex conjugation on the complex projective space . These manifolds were cons

Uniformization theorem

In mathematics, the uniformization theorem says that every simply connected Riemann surface is conformally equivalent to one of three Riemann surfaces: the open unit disk, the complex plane, or the Ri

Tame manifold

In geometry, a tame manifold is a manifold with a well-behaved compactification. More precisely, a manifold is called tame if it is homeomorphic to a compact manifold with a closed subset of the bound

Stiefel manifold

In mathematics, the Stiefel manifold is the set of all orthonormal k-frames in That is, it is the set of ordered orthonormal k-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel

Nearly Kähler manifold

In mathematics, a nearly Kähler manifold is an almost Hermitian manifold , with almost complex structure ,such that the (2,1)-tensor is skew-symmetric. So, for every vector field on . In particular, a

Complex Lie group

In geometry, a complex Lie group is a Lie group over the complex numbers; i.e., it is a complex-analytic manifold that is also a group in such a way is holomorphic. Basic examples are , the general li

Piecewise linear manifold

In mathematics, a piecewise linear (PL) manifold is a topological manifold together with a piecewise linear structure on it. Such a structure can be defined by means of an atlas, such that one can pas

Double (manifold)

In the subject of manifold theory in mathematics, if is a manifold with boundary, its double is obtained by gluing two copies of together along their common boundary. Precisely, the double is where fo

Banach bundle

In mathematics, a Banach bundle is a vector bundle each of whose fibres is a Banach space, i.e. a complete normed vector space, possibly of infinite dimension.

Closed manifold

In mathematics, a closed manifold is a manifold without boundary that is compact. In comparison, an open manifold is a manifold without boundary that has only non-compact components.

Toric manifold

In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an -dimensional compact torus w

Fréchet manifold

In mathematics, in particular in nonlinear analysis, a Fréchet manifold is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space. More preci

Lie group–Lie algebra correspondence

In mathematics, Lie group–Lie algebra correspondence allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorph

Steenrod problem

In mathematics, and particularly homology theory, Steenrod's Problem (named after mathematician Norman Steenrod) is a problem concerning the realisation of homology classes by singular manifolds.

Quaternionic manifold

In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the

General covariant transformations

In physics, general covariant transformations are symmetries of gravitation theory on a world manifold . They are gauge transformations whose parameter functions are vector fields on . From the physic

Plumbing (mathematics)

In the mathematical field of geometric topology, among the techniques known as surgery theory, the process of plumbing is a way to create new manifolds out of . It was first described by John Milnor a

Hilbert manifold

In mathematics, a Hilbert manifold is a manifold modeled on Hilbert spaces. Thus it is a separable Hausdorff space in which each point has a neighbourhood homeomorphic to an infinite dimensional Hilbe

Lens space

A lens space is an example of a topological space, considered in mathematics. The term often refers to a specific class of 3-manifolds, but in general can be defined for higher dimensions. In the 3-ma

Haken manifold

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 incompressible surface. Sometimes one consid

Weyl's tube formula

Weyl's tube formula gives the volume of an object defined as the set of all points within a small distance of a manifold. Let be an oriented, closed, two-dimensional surface, and let denote the set of

Formal manifold

In geometry and topology, a formal manifold can mean one of a number of related concepts:
* In the sense of Dennis Sullivan, a formal manifold is one whose real homotopy type is a formal consequence

Eells–Kuiper manifold

In mathematics, an Eells–Kuiper manifold is a compactification of by a sphere of dimension , where , or . It is named after James Eells and Nicolaas Kuiper. If , the Eells–Kuiper manifold is diffeomor

Essential manifold

In geometry, an essential manifold is a special type of closed manifold. The notion was first introduced explicitly by Mikhail Gromov.

Algebraic manifold

In mathematics, an algebraic manifold is an algebraic variety which is also a manifold. As such, algebraic manifolds are a generalisation of the concept of smooth curves and surfaces defined by polyno

Tubular neighborhood

In mathematics, a tubular neighborhood of a submanifold of a smooth manifold is an open set around it resembling the normal bundle. The idea behind a tubular neighborhood can be explained in a simple

Classification of manifolds

In mathematics, specifically geometry and topology, the classification of manifolds is a basic question, about which much is known, and many open questions remain.

Global analysis

In mathematics, global analysis, also called analysis on manifolds, is the study of the global and topological properties of differential equations on manifolds and vector bundles. Global analysis use

Pseudomanifold

In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of forms a

Kervaire manifold

In mathematics, specifically in differential topology, a Kervaire manifold is a piecewise-linear manifold of dimension constructed by Michel Kervaire by plumbing together the tangent bundles of two -s

Link (knot theory)

In mathematical knot theory, a link is a collection of knots which do not intersect, but which may be linked (or knotted) together. A knot can be described as a link with one component. Links and knot

Hyperbolic manifold

In mathematics, a hyperbolic manifold is a space where every point looks locally like hyperbolic space of some dimension. They are especially studied in dimensions 2 and 3, where they are called hyper

Statistical manifold

In mathematics, a statistical manifold is a Riemannian manifold, each of whose points is a probability distribution. Statistical manifolds provide a setting for the field of information geometry. The

Banach manifold

In mathematics, a Banach manifold is a manifold modeled on Banach spaces. Thus it is a topological space in which each point has a neighbourhood homeomorphic to an open set in a Banach space (a more i

Non-Hausdorff manifold

In geometry and topology, it is a usual axiom of a manifold to be a Hausdorff space. In general topology, this axiom is relaxed, and one studies non-Hausdorff manifolds: spaces locally homeomorphic to

Affine manifold

In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected) covered by an open subset

Link concordance

In mathematics, two links and are concordant if there exists an embedding such that and . By its nature, link concordance is an equivalence relation. It is weaker than isotopy, and stronger than homot

Stable manifold

In mathematics, and in particular the study of dynamical systems, the idea of stable and unstable sets or stable and unstable manifolds give a formal mathematical definition to the general notions emb

Prime manifold

In topology, a branch of mathematics, a prime manifold is an n-manifold that cannot be expressed as a non-trivial connected sum of two n-manifolds. Non-trivial means that neither of the two is an n-sp

Category of manifolds

In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This

Real projective line

In geometry, a real projective line is a projective line over the real numbers. It is an extension of the usual concept of a line that has been historically introduced to solve a problem set by visual

Analytic manifold

In mathematics, an analytic manifold, also known as a manifold, is a differentiable manifold with analytic transition maps. The term usually refers to real analytic manifolds, although complex manifol

Fibered manifold

In differential geometry, in the category of differentiable manifolds, a fibered manifold is a surjective submersion that is, a surjective differentiable mapping such that at each point the tangent ma

Prime decomposition of 3-manifolds

In mathematics, the prime decomposition theorem for 3-manifolds states that every compact, orientable 3-manifold is the connected sum of a unique (up to homeomorphism) finite collection of prime 3-man

Sobolev mapping

In mathematics, a Sobolev mapping is a mapping between manifolds which has smoothness in some sense. Sobolev mappings appear naturally in manifold-constrained problems in the calculus of variations an

Geodesic manifold

In mathematics, a complete manifold (or geodesically complete manifold) M is a (pseudo-) Riemannian manifold for which, starting at any point p, you can follow a "straight" line indefinitely along any

Crumpling

In geometry and topology, crumpling is the process whereby a sheet of paper or other two-dimensional manifold undergoes disordered deformation to yield a three-dimensional structure comprising a rando

Isoparametric manifold

In Riemannian geometry, an isoparametric manifold is a type of (immersed) submanifold of Euclidean space whose normal bundle is flat and whose principal curvatures are constant along any parallel norm

Semi-s-cobordism

In mathematics, a cobordism (W, M, M−) of an (n + 1)-dimensional manifold (with boundary) W between its boundary components, two n-manifolds M and M−, is called a semi-s-cobordism if (and only if) the

Acceleration (differential geometry)

In mathematics and physics, acceleration is the rate of change of velocity of a curve with respect to a given linear connection. This operation provides us with a measure of the rate and direction of

Mazur manifold

In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth four-dimensional manifold (with boundary) which is not diffeomorphic to the standard 4-ball. The

Tameness theorem

In mathematics, the tameness theorem states that every complete hyperbolic 3-manifold with finitely generated fundamental group is topologically tame, in other words homeomorphic to the interior of a

Lie group

In mathematics, a Lie group (pronounced /liː/ LEE) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract c

Topological manifold

In topology, a branch of mathematics, a topological manifold is a topological space that locally resembles real n-dimensional Euclidean space. Topological manifolds are an important class of topologic

Whitehead manifold

In mathematics, the Whitehead manifold is an open 3-manifold that is contractible, but not homeomorphic to J. H. C. Whitehead discovered this puzzling object while he was trying to prove the Poincaré

Parallelizable manifold

In mathematics, a differentiable manifold of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at every point of the tangent vectorsprovide a basis of

Geodesic curvature

In Riemannian geometry, the geodesic curvature of a curve measures how far the curve is from being a geodesic. For example, for 1D curves on a 2D surface embedded in 3D space, it is the curvature of t

Hyperbolic 3-manifold

In mathematics, more precisely in topology and differential geometry, a hyperbolic 3–manifold is a manifold of dimension 3 equipped with a hyperbolic metric, that is a Riemannian metric which has all

Diffiety

In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equation

Configuration space (mathematics)

In mathematics, a configuration space is a construction closely related to state spaces or phase spaces in physics. In physics, these are used to describe the state of a whole system as a single point

Poincaré duality

In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional

Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties. There are different types

Collapsing manifold

In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as i goes to infinity the manifold is close to a k-

Quasitoric manifold

In mathematics, a quasitoric manifold is a topological analogue of the nonsingular projective toric variety of algebraic geometry. A smooth -dimensional manifold is a quasitoric manifold if it admits

Density on a manifold

In mathematics, and specifically differential geometry, a density is a spatially varying quantity on a differentiable manifold that can be integrated in an intrinsic manner. Abstractly, a density is a

Natural bundle

In mathematics, a natural bundle is any fiber bundle associated to the s-frame bundle for some . It turns out that its transition functions depend functionally on local changes of coordinates in the b

Manifold

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

Quaternion-Kähler manifold

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is a subgroup of Sp(n)·Sp(1) for some . Here Sp(n)

Web (differential geometry)

In mathematics, a web permits an intrinsic characterization in terms of Riemannian geometry of the additive separation of variables in the Hamilton–Jacobi equation.

Dynamic fluid film equations

Fluid films, such as soap films, are commonly encountered in everyday experience. A soap film can be formed by dipping a closed contour wire into a soapy solution as in the figure on the right. Altern

Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual charts that, roughly speaking, describe individual regions of the manifold. If the manif

Lefschetz duality

In mathematics, Lefschetz duality is a version of Poincaré duality in geometric topology, applying to a manifold with boundary. Such a formulation was introduced by Solomon Lefschetz, at the same time

A Guide to the Classification Theorem for Compact Surfaces

A Guide to the Classification Theorem for Compact Surfaces is a textbook in topology, on the classification of two-dimensional surfaces. It was written by Jean Gallier and Dianna Xu, and published in

5-manifold

In mathematics, a 5-manifold is a 5-dimensional topological manifold, possibly with a piecewise linear or smooth structure. Non-simply connected 5-manifolds are impossible to classify, as this is hard

Almost flat manifold

In mathematics, a smooth compact manifold M is called almost flat if for any there is a Riemannian metric on M such that and is -flat, i.e. for the sectional curvature of we have . Given n, there is a

Timeline of manifolds

This is a timeline of manifolds, one of the major geometric concepts of mathematics. For further background see history of manifolds and varieties.

Conformally flat manifold

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation. In practice, the metric of the manifold has to be co

P-compact group

In mathematics, in particular algebraic topology, a p-compact group is a homotopical version of a compact Lie group, but with all the local structure concentrated at a single prime p. This concept was

Local tangent space alignment

Local tangent space alignment (LTSA) is a method for manifold learning, which can efficiently learn a nonlinear embedding into low-dimensional coordinates from high-dimensional data, and can also reco

List of manifolds

This is a list of particular manifolds, by Wikipedia page. See also list of geometric topology topics. For categorical listings see Category:Manifolds and its subcategories.

Klein bottle

In topology, a branch of mathematics, the Klein bottle (/ˈklaɪn/) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector can

Lamination (topology)

In topology, a branch of mathematics, a lamination is a :
* "topological space partitioned into subsets"
* decoration (a structure or property at a point) of a manifold in which some subset of the m

Signature cocycle

In mathematics, the Meyer signature cocycle, introduced by Meyer. is an integer-valued 2-cocyle on a symplectic group that describes the signature of a fiber bundle whose base and fiber are both Riema

Twisted Poincaré duality

In mathematics, the twisted Poincaré duality is a theorem removing the restriction on Poincaré duality to oriented manifolds. The existence of a global orientation is replaced by carrying along local

Dupin hypersurface

In differential geometry, a Dupin hypersurface is a submanifold in a space form, whose principal curvatures have globally constant multiplicities.

Bony–Brezis theorem

In mathematics, the Bony–Brezis theorem, due to the French mathematicians Jean-Michel Bony and Haïm Brezis, gives necessary and sufficient conditions for a closed subset of a manifold to be invariant

Bing–Borsuk conjecture

In mathematics, the Bing–Borsuk conjecture states that every -dimensional homogeneous absolute neighborhood retract space is a topological manifold. The conjecture has been proved for dimensions 1 and

Lie groupoid

In mathematics, a Lie groupoid is a groupoid where the set of objects and the set of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map a

Boundary-incompressible surface

In low-dimensional topology, a boundary-incompressible surface is a two-dimensional surface within a three-dimensional manifold whose topology cannot be made simpler by a certain type of operation kno

© 2023 Useful Links.