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
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