# Category: Structures on manifolds

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