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...

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...

{\displaystyle Y\in T_{p}M} . Given a metric tensor g on an n-dimensional real manifold, the quadratic form q(x) = g(x, x) associated with the metric tensor applied...

Riemannian manifold (M, g) is said to be homogeneous if for every pair of points x and y in M, there is some isometry f of the Riemannian manifold sending x to...

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

described from different points of view: A Kähler manifold is a symplectic manifold ( X , ω ) {\displaystyle (X,\omega )} equipped with an integrable almost-complex...

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...

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

Poisson manifold is a smooth manifold endowed with a Poisson structure. The notion of Poisson manifold generalises that of symplectic manifold, which in...

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

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

words, it is a ⁠ ( G , X ) {\displaystyle (G,X)} ⁠-manifold where X = C n {\displaystyle X=\mathbb {C} ^{n}} and G {\displaystyle G} is the group of complex...

the manifold can be classified into three disjoint types. A tangent vector X {\displaystyle X} is: timelike if g ( X , X ) < 0 {\displaystyle \,g(X,X)<0}...

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

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

x j ∂ x ~ b ∂ x k ) ( X ~ c ∂ x j ∂ x ~ c ) ( Y ~ d ∂ x k ∂ x ~ d ) = R ~ a b X ~ c Y ~ d ( ∂ x ~ a ∂ x j ∂ x j ∂ x ~ c ) ( ∂ x ~ b ∂ x k ∂ x k ∂ x ~...

equation. Let ( M , g ) {\displaystyle (M,g)} be a Riemannian or pseudo-Riemannian manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the space...

x , y ) = x y {\displaystyle \mu :G\times G\to G\quad \mu (x,y)=xy} means that μ is a smooth mapping of the product manifold G × G into G. The two requirements...

fiber X . {\displaystyle X.} Mapping tori of homeomorphisms of surfaces are of particular importance in 3-manifold topology. If G {\displaystyle G} is a...

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,...

and x {\displaystyle x} is a point on the manifold. More formally, let ( M , g ) {\displaystyle (M,g)} be a pseudo-Riemannian manifold. Then ( M , g ) {\displaystyle...

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...

also define principal G {\displaystyle G} -bundles in the category of smooth manifolds. Here π : P → X {\displaystyle \pi :P\to X} is required to be a...

b f ( x ) d x . {\displaystyle \int _{a}^{b}f(x)\,dx.} Similarly, the expression f ( x , y , z ) d x ∧ d y + g ( x , y , z ) d z ∧ d x + h ( x , y , z...

G is a Lie group, X is a smooth manifold and the action is smooth, G is an algebraic group, X is an algebraic variety and the action is regular. If G...

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

For instance, if (X, g) is a Riemannian manifold, one has the pseudogroup of its local isometries; if (X, ω) is a symplectic manifold, one has the pseudogroup...

compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same...

0–9 A B C D E F G H I J K L M N O P Q R S T U V W X Y Z Affine connection Alexandrov space a generalization of Riemannian manifolds with upper, lower...

manifold (named after Shigeo Sasaki) is a contact manifold ( M , θ ) {\displaystyle (M,\theta )} equipped with a special kind of Riemannian metric g {\displaystyle...