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...

geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex structures I , J ...

the causal structure of a Lorentzian manifold describes the possible causal relationships between points in the manifold. Lorentzian manifolds can be classified...

coordinates (e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable...

unitary structure (U(n) structure) on the manifold. By dropping this condition, we get an almost Hermitian manifold. On any almost Hermitian manifold, we...

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...

terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential...

almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...

geometry, a field in mathematics, a Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises...

a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The...

mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is...

In differential geometry and complex geometry, a complex manifold is a manifold with a complex structure, that is an atlas of charts to the open unit disc...

All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...

geometry, a G2 manifold or Joyce manifold is a seven-dimensional Riemannian manifold with holonomy group contained in G2. The group G 2 {\displaystyle G_{2}}...

In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature...

differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate...

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 mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four,...

geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection. Equivalently, it is a manifold that is (if connected)...

language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free...

of a manifold and related structures such as vector bundles and other fiber bundles. The definition of an atlas depends on the notion of a chart. A chart...

dt\wedge \theta \,} on its cone is symplectic (this is one of the possible definitions of a contact structure). A contact Riemannian manifold is Sasakian, if...

coordinates the condition that ( M , g ) {\displaystyle (M,g)} be an Einstein manifold is simply R a b = k g a b . {\displaystyle R_{ab}=kg_{ab}.} Taking...

subgroup of G acting freely on X ; this is a special case of a complete (G,X)-structure. If a given manifold admits a geometric structure, then it admits...

differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent space...

algebraic and differential geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has certain properties,...

geometry, a G 2 {\displaystyle G_{2}} -structure is an important type of G-structure that can be defined on a smooth manifold. If M is a smooth manifold of dimension...

geometry is the study of a geometric structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete...

mathematics, a 3-manifold is a topological space that locally looks like a three-dimensional Euclidean space. A 3-manifold can be thought of as a possible...

bundle Connection (mathematics) G-structure Bishop, Richard L.; Goldberg, Samuel I. (1968), Tensor Analysis on Manifolds, New York: Macmillan, p. 160 Milnor...