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Solvmanifold

In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group. It may also be characterized as a quotient of a connected solvable Lie group by a closed subgroup. (Some author

Supermanifold

In physics and mathematics, supermanifolds are generalizations of the manifold concept based on ideas coming from supersymmetry. Several definitions are in use, some of which are described below.

Categories of manifolds

No description available.

Foliation

In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p

Simplicial manifold

In physics, the term simplicial manifold commonly refers to one of several loosely defined objects, commonly appearing in the study of Regge calculus. These objects combine attributes of a simplex wit

Poisson–Lie group

In mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure on the manifold. The infinitesimal

Hermitian connection

In mathematics, a Hermitian connection is a connection on a Hermitian vector bundle over a smooth manifold which is compatible with the Hermitian metric on , meaning that for all smooth vector fields

Kosmann lift

In differential geometry, the Kosmann lift, named after Yvette Kosmann-Schwarzbach, of a vector field on a Riemannian manifold is the canonical projection on the orthonormal frame bundle of its natura

Spinor bundle

In differential geometry, given a spin structure on an -dimensional orientable Riemannian manifold one defines the spinor bundle to be the complex vector bundle associated to the corresponding princip

Generalized complex structure

In the field of mathematics known as differential geometry, a generalized complex structure is a property of a differential manifold that includes as special cases a complex structure and a symplectic

Symplectic spinor bundle

In differential geometry, given a metaplectic structure on a -dimensional symplectic manifold the symplectic spinor bundle is the Hilbert space bundle associated to the metaplectic structure via the m

Symplectic frame bundle

In symplectic geometry, the symplectic frame bundle of a given symplectic manifold is the canonical principal -subbundle of the tangent frame bundle consisting of linear frames which are symplectic wi

Haefliger structure

In mathematics, a Haefliger structure on a topological space is a generalization of a foliation of a manifold, introduced by André Haefliger in 1970. Any foliation on a manifold induces a special kind

Hyperkähler manifold

In differential geometry, a hyperkähler manifold is a Riemannian manifold endowed with three integrable almost complex structures that are Kähler with respect to the Riemannian metric and satisfy the

Poisson manifold

In differential geometry, a Poisson structure on a smooth manifold is a Lie bracket (called a Poisson bracket in this special case) on the algebra of smooth functions on , subject to the Leibniz rule

Calibrated geometry

In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibratio

Pachner moves

In topology, a branch of mathematics, Pachner moves, named after Udo Pachner, are ways of replacing a triangulation of a piecewise linear manifold by a different triangulation of a homeomorphic manifo

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

Sasakian manifold

In differential geometry, a Sasakian manifold (named after Shigeo Sasaki) is a contact manifold equipped with a special kind of Riemannian metric , called a Sasakian metric.

Hodge structure

In mathematics, a Hodge structure, named after W. V. D. Hodge, is an algebraic structure at the level of linear algebra, similar to the one that Hodge theory gives to the cohomology groups of a smooth

Hermitian manifold

In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold. More precisely, a Hermitian manifold is a complex manifold with a

Real structure

In mathematics, a real structure on a complex vector space is a way to decompose the complex vector space in the direct sum of two real vector spaces. The prototype of such a structure is the field of

Hilbert–Smith conjecture

In mathematics, the Hilbert–Smith conjecture is concerned with the transformation groups of manifolds; and in particular with the limitations on topological groups G that can act effectively (faithful

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.

Open book decomposition

In mathematics, an open book decomposition (or simply an open book) is a decomposition of a closed oriented 3-manifold M into a union of surfaces (necessarily with boundary) and solid tori. Open books

Quaternion-Kähler symmetric space

In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space. Any quaternion-Kähler symm

Symplectization

In mathematics, the symplectization of a contact manifold is a symplectic manifold which naturally corresponds to it.

Fubini–Study metric

In mathematics, the Fubini–Study metric is a Kähler metric on projective Hilbert space, that is, on a complex projective space CPn endowed with a Hermitian form. This metric was originally described i

Hauptvermutung

The Hauptvermutung of geometric topology is a now refuted conjecture asking whether any two triangulations of a triangulable space have subdivisions that are combinatorially equivalent, i.e. the subdi

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

Spin structure

In differential geometry, a spin structure on an orientable Riemannian manifold (M, g) allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry. S

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

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

Hypercomplex manifold

In differential geometry, a hypercomplex manifold is a manifold with the tangent bundleequipped with an action by the algebra of quaternionsin such a way that the quaternions define integrable almost

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

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

G2 manifold

In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group is one of the five exceptional simple Lie groups. It can be described

Metaplectic structure

In differential geometry, a metaplectic structure is the symplectic analog of spin structure on orientable Riemannian manifolds. A metaplectic structure on a symplectic manifold allows one to define t

(G,X)-manifold

In geometry, if X is a manifold with an action of a topological group G by analytical diffeomorphisms, the notion of a (G, X)-structure on a topological space is a way to formalise it being locally is

Clifford module bundle

In differential geometry, a Clifford module bundle, a bundle of Clifford modules or just Clifford module is a vector bundle whose fibers are Clifford modules, the representations of Clifford algebras.

Linear complex structure

In mathematics, a complex structure on a real vector space V is an automorphism of V that squares to the minus identity, −I. Such a structure on V allows one to define multiplication by complex scalar

Lipschitz continuity

In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is

Canonical ring

In mathematics, the pluricanonical ring of an algebraic variety V (which is non-singular), or of a complex manifold, is the graded ring of sections of powers of the canonical bundle K. Its nth graded

Triangulation (topology)

In mathematics, triangulation describes the replacement of topological spaces by piecewise linear spaces, i.e. the choice of a homeomorphism in a suitable simplicial complex. Spaces being homeomorphic

Smooth structure

In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold.

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

Killing spinor

Killing spinor is a term used in mathematics and physics. By the more narrow definition, commonly used in mathematics, the term Killing spinor indicates those twistorspinors which are also eigenspinor

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)

Gibbons–Hawking space

In mathematical physics, a Gibbons–Hawking space, named after Gary Gibbons and Stephen Hawking, is essentially a hyperkähler manifold with an extra U(1) symmetry. (In general, Gibbons–Hawking metrics

G2-structure

In differential geometry, a -structure is an important type of G-structure that can be defined on a smooth manifold. If M is a smooth manifold of dimension seven, then a G2-structure is a reduction of

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

Frölicher space

In mathematics, Frölicher spaces extend the notions of calculus and smooth manifolds. They were introduced in 1982 by the mathematician Alfred Frölicher.

Kirby–Siebenmann class

In mathematics, more specifically in geometric topology, the Kirby–Siebenmann class is an obstruction for topological manifolds to allow a PL-structure.

G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures include

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