scispace - formally typeset
Search or ask a question
Topic

Symmetric space

About: Symmetric space is a research topic. Over the lifetime, 2988 publications have been published within this topic receiving 63913 citations.


Papers
More filters
Book
01 Jan 1978
TL;DR: In this article, the structure of semisimplepleasure Lie groups and Lie algebras is studied. But the classification of simple Lie algesbras and of symmetric spaces is left open.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces Structure of semisimple Lie groups The classification of simple Lie algebras and of symmetric spaces Solutions to exercises Some details Bibliography List of notational conventions Symbols frequently used Index Reviews for the first edition.

6,321 citations

Book
01 Jan 1962
TL;DR: In this article, the classification of symmetric spaces has been studied in the context of Lie groups and Lie algebras, and a list of notational conventions has been proposed.
Abstract: Elementary differential geometry Lie groups and Lie algebras Structure of semisimple Lie algebras Symmetric spaces Decomposition of symmetric spaces Symmetric spaces of the noncompact type Symmetric spaces of the compact type Hermitian symmetric spaces On the classification of symmetric spaces Functions on symmetric spaces Bibliography List of notational conventions Symbols frequently used Author index Subject index Reviews for the first edition.

3,013 citations

Journal ArticleDOI
TL;DR: In this paper, the authors present a maximally supersymmetric IIB string background with a conformally flat lorentzian symmetric space G/K with solvable G, with a homogeneous five-form flux.
Abstract: We present a maximally supersymmetric IIB string background. The geometry is that of a conformally flat lorentzian symmetric space G/K with solvable G, with a homogeneous five-form flux. We give the explicit supergravity solution, compute the isometries, the 32 Killing spinors, and the symmetry superalgebra, and then discuss T-duality and the relation to M-theory.

822 citations

Journal ArticleDOI
TL;DR: This work defines the Log‐Euclidean mean from a Riemannian point of view, based on a lie group structure which is compatible with the usual algebraic properties of this matrix space and a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure.
Abstract: In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive‐definite matrices, called Log‐Euclidean. The approach is based on two novel algebraic structures on symmetric positive‐definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From bi‐invariant metrics on the Lie group structure, we define the Log‐Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine‐invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in g...

791 citations

Book ChapterDOI
TL;DR: In this article, it was shown that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-Invariant.
Abstract: Given a discrete subgroup Γ of a connected real semisimple Lie group G with finite center there is a natural homomorphism $$j_\Gamma ^q:I_G^q \to {H^q}\left( {\Gamma ;c} \right)\quad \left( {q = 0,1, \ldots } \right),$$ (1) where I G q denotes the space of G-invariant harmonic q-forms on the symmetric space quotient X=G/K of G by a maximal compact subgroup K. If Γ is cocompact, this homomorphism is injective in all dimensions and the main objective of Matsushima in [19] is to give a range m(G), independent of Γ, in which j Γ q is also surjective. The main argument there is to show that if a certain quadratic form depending on q is positive non-degenerate, then any Γ-invariant harmonic q-form is automatically G-invariant. In [3], we proved similarly the existence of a range in which j Γ q is bijective when Γ is arithmetic, but not necessarily cocompact. There are three main steps to the proof: (i) The cohomology of Γ can be computed by using differential forms which satisfy a certain growth condition, “logarithmic growth,” at infinity; (ii) up to some range c(G), these forms are all square integrable; and (iii) use the fact, pointed out in [16] , that for q ≦ m(G), Matsushima’s arguments remain valid in the non-compact case for square integrable forms.

696 citations


Network Information
Related Topics (5)
Cohomology
21.5K papers, 389.8K citations
93% related
Lie group
18.3K papers, 381K citations
93% related
Manifold
18.7K papers, 362.8K citations
91% related
Moduli space
15.9K papers, 410.7K citations
91% related
Abelian group
30.1K papers, 409.4K citations
91% related
Performance
Metrics
No. of papers in the topic in previous years
YearPapers
202315
202242
202188
202079
201975
201882