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...

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...

{\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...

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...

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

g } = ω ( X f , X g ) . {\displaystyle \{f,g\}=\omega (X_{f},X_{g}).} A standard example of a symplectic manifold, and thus of a Poisson manifold, is...

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

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

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

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

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...

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

In mathematics, the Stiefel manifold V k ( R n ) {\displaystyle V_{k}(\mathbb {R} ^{n})} is the set of all orthonormal k-frames in R n . {\displaystyle...

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...

sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are...

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 ~...

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...

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...

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

Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional linear subspace σp of the tangent space at a point p of the manifold. It can...

differential geometry, one can attach to every point x {\displaystyle x} of a differentiable manifold a tangent space—a real vector space that intuitively...

S ( f ( x , y , z ) d x ∧ d y + g ( x , y , z ) d z ∧ d x + h ( x , y , z ) d y ∧ d z ) . {\displaystyle \int _{S}(f(x,y,z)\,dx\wedge dy+g(x,y,z)\,dz\wedge...

(G,X)-manifold where X = R n {\displaystyle X=\mathbb {R} ^{n}} and G {\displaystyle G} is the group of affine transformations. An affine manifold is...

differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M {\displaystyle M} of dimension n {\displaystyle n} , a...

subset X {\displaystyle X} of a manifold M {\displaystyle M} and a subset Y {\displaystyle Y} of a manifold N {\displaystyle N} , a function f : X → Y {\displaystyle...

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}...

of manifold M and any (smooth) vector fields X and Y on U, the real function g ( X , Y ) ( p ) = g p ( X p , Y p ) {\displaystyle g(X,Y)(p)=g_{p}(X_{p}...

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...