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...
8 KB (1,378 words) - 19:52, 7 September 2024
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...
59 KB (8,680 words) - 10:03, 21 October 2024
coordinates (e.g. CT scans). Manifolds can be equipped with additional structure. One important class of manifolds are differentiable manifolds; their differentiable...
69 KB (9,645 words) - 11:16, 30 October 2024
{\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...
9 KB (1,171 words) - 16:59, 28 September 2024
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...
20 KB (2,576 words) - 06:58, 26 June 2023
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...
60 KB (9,182 words) - 15:32, 16 October 2024
differential geometry, a Finsler manifold is a differentiable manifold M where a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent...
14 KB (1,942 words) - 10:32, 12 July 2024
described from different points of view: A Kähler manifold is a symplectic manifold ( X , ω ) {\displaystyle (X,\omega )} equipped with an integrable almost-complex...
33 KB (4,736 words) - 02:51, 10 August 2024
differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M {\displaystyle M} , equipped with a closed nondegenerate...
23 KB (3,630 words) - 23:54, 27 September 2024
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow...
67 KB (9,495 words) - 10:54, 26 October 2024
In differential geometry, a hyperkähler manifold is a Riemannian manifold ( M , g ) {\displaystyle (M,g)} endowed with three integrable almost complex...
13 KB (1,641 words) - 00:44, 7 November 2023
almost complex manifold is a smooth manifold equipped with a smooth linear complex structure on each tangent space. Every complex manifold is an almost...
16 KB (2,399 words) - 23:55, 6 March 2024
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...
24 KB (3,270 words) - 00:26, 22 October 2024
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...
11 KB (2,141 words) - 02:11, 28 September 2024
All manifolds are topological manifolds by definition. Other types of manifolds are formed by adding structure to a topological manifold (e.g. differentiable...
17 KB (2,037 words) - 04:42, 19 October 2024
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...
5 KB (885 words) - 08:37, 19 February 2024
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...
10 KB (1,311 words) - 18:37, 9 September 2024
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...
7 KB (926 words) - 15:50, 19 September 2024
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...
12 KB (1,358 words) - 22:02, 18 December 2023
Ricci curvature (category Riemannian manifolds)
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 ~...
34 KB (5,859 words) - 04:51, 6 July 2024
differential geometry, one can attach to every point x {\displaystyle x} of a differentiable manifold a tangent space—a real vector space that intuitively...
19 KB (3,167 words) - 23:13, 5 September 2024
Sectional curvature (redirect from Manifolds with constant sectional curvature)
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...
20 KB (3,103 words) - 21:48, 5 January 2024
Riemann curvature tensor (category Riemannian manifolds)
equation. Let ( M , g ) {\displaystyle (M,g)} be a Riemannian or pseudo-Riemannian manifold, and X ( M ) {\displaystyle {\mathfrak {X}}(M)} be the space...
19 KB (2,925 words) - 11:22, 13 October 2024
Differential form (redirect from Integration on manifolds)
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...
66 KB (9,952 words) - 00:22, 24 September 2024
(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...
7 KB (844 words) - 00:47, 22 August 2024
Volume form (category Integration on manifolds)
differential form of degree equal to the differentiable manifold dimension. Thus on a manifold M {\displaystyle M} of dimension n {\displaystyle n} , a...
14 KB (2,341 words) - 02:16, 8 May 2024
Causal structure (category Lorentzian manifolds)
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}...
22 KB (3,430 words) - 05:00, 7 April 2024
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...
25 KB (4,165 words) - 15:27, 23 February 2024
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}...
56 KB (8,866 words) - 08:52, 9 August 2024
homology G-manifold (without boundary) of dimension n over an abelian group G of coefficients is a locally compact topological space X with finite G-cohomological...
2 KB (253 words) - 09:35, 23 November 2023