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Hyperbolic space

In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stro

Hermitian symmetric space

In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural

Weakly symmetric space

In mathematics, a weakly symmetric space is a notion introduced by the Norwegian mathematician Atle Selberg in the 1950s as a generalisation of symmetric space, due to Élie Cartan. Geometrically the s

Kostant's convexity theorem

In mathematics, Kostant's convexity theorem, introduced by Bertram Kostant, states that the projection of every coadjoint orbit of a connected compact Lie group into the dual of a Cartan subalgebra is

Symmetric space

In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be stu

Iwasawa manifold

In mathematics, in the field of differential geometry, an Iwasawa manifold is a compact quotient of a 3-dimensional complex Heisenberg group by a cocompact, discrete subgroup. An Iwasawa manifold is a

Clifford–Klein form

In mathematics, a Clifford–Klein form is a double coset space Γ\G/H, where G is a reductive Lie group, H a closed subgroup of G, and Γ a discrete subgroup of G that acts properly discontinuously on th

Homogeneous space

In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts tr

Nilmanifold

In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it. As such, a nilmanifold is an example of a homogeneous space and is di

Stiefel manifold

In mathematics, the Stiefel manifold is the set of all orthonormal k-frames in That is, it is the set of ordered orthonormal k-tuples of vectors in It is named after Swiss mathematician Eduard Stiefel

Klein geometry

In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program. More specifically, it is a homogeneous space X together with a transitive action on

Parabolic geometry (differential geometry)

In differential geometry and the study of Lie groups, a parabolic geometry is a homogeneous space G/P which is the quotient of a semisimple Lie group G by a parabolic subgroup P. More generally, the c

Quaternionic projective space

In mathematics, quaternionic projective space is an extension of the ideas of real projective space and complex projective space, to the case where coordinates lie in the ring of quaternions Quaternio

Erlangen program

In mathematics, the Erlangen program is a method of characterizing geometries based on group theory and projective geometry. It was published by Felix Klein in 1872 as Vergleichende Betrachtungen über

Sphere

A sphere (from Ancient Greek σφαῖρα (sphaîra) 'globe, ball') is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the

Complex hyperbolic space

In mathematics, hyperbolic complex space is a Hermitian manifold which is the equivalent of the real hyperbolic space in the context of complex manifolds. The complex hyperbolic space is a Kähler mani

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

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