scholarly journals Invariant plurisubharmonic functions on non-compact Hermitian symmetric spaces

2021 ◽  
Author(s):  
Laura Geatti ◽  
Andrea Iannuzzi
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Hermitian Symmetric Space ◽  
Hermitian Symmetric Spaces ◽  
Plurisubharmonic Functions ◽  
Invariant Domain ◽  
Compact Hermitian Symmetric Space ◽  
Invariant Domains

AbstractLet $$\,G/K\,$$ G / K be an irreducible non-compact Hermitian symmetric space and let $$\,D\,$$ D be a $$\,K$$ K -invariant domain in $$\,G/K$$ G / K . In this paper we characterize several classes of $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D in terms of their restrictions to a slice intersecting all $$\,K$$ K -orbits. As applications we show that $$\,K$$ K -invariant plurisubharmonic functions on $$\,D\,$$ D are necessarily continuous and we reproduce the classification of Stein $$\,K$$ K -invariant domains in $$\,G/K\,$$ G / K obtained by Bedford and Dadok. (J Geom Anal 1:1–17, 1991).

scholarly journals Classification of totally umbilical submanifolds in symmetric spaces

1980 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Symmetric Space ◽  
Fundamental Form ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Totally Umbilical Submanifold ◽  
Totally Umbilical Submanifolds ◽  
First Fundamental Form

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

The Focal Locus of a Riemannian 4-Symmetric Space

1988 ◽  
Vol 31 (2) ◽  
pp. 175-181
Author(s):  
J. Alfredo Jimenez
Keyword(s):  
Symmetric Space ◽  
Normal Bundle ◽  
Symmetric Spaces ◽  
Fibre Bundle ◽  
Exponential Map ◽  
Focal Points ◽  
Cut Points ◽  
Covering Map ◽  
Simply Connected

Abstractcompact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

scholarly journals Pseudo-Riemannian Symmetries on Heisenberg Groups

2014 ◽  
Vol 0 (0) ◽  
Author(s):  
Michel Goze ◽  
Paola Piu ◽  
Elisabeth Remm
Keyword(s):  
Symmetric Space ◽  
Heisenberg Group ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Finite Abelian Group ◽  
Riemannian Metrics ◽  
Heisenberg Groups ◽  
Lorentzian Metrics ◽  
Symmetric Structures

Abstract The notion of Γ-symmetric space is a natural generalization of the classical notion of symmetric space based on Z2-grading on Lie algebras. We consider homogeneous spaces G/H such that the Lie algebra g of G admits a Γ-grading where Γ is a finite abelian group. In this work we study Riemannian metrics and Lorentzian metrics on the Heisenberg group H3 adapted to the symmetries of a Γ-symmetric structure on H3. We prove that the classification of Riemannian and Lorentzian Zl-symmetric metrics on H3 corresponds to the classification of its left-invariant Riemannian and Lorentzian metrics, up to isometry. We study also the Z§-symmetric structures on G/H when G is the (2p + 1)-dimensional Heisenberg group. This gives examples of non-Riemannian symmetric spaces. When k > 1, we show that there exists a family of flat and torsion free affine connections adapted to the Z§-symmetric structures.

Classification of homogeneous holomorphic hermitian principal bundles over G/K

2015 ◽  
Vol 27 (2) ◽  
Author(s):  
Indranil Biswas
Keyword(s):  
Symmetric Space ◽  
Half Plane ◽  
Complete Classification ◽  
Hermitian Symmetric Space ◽  
Principal Bundles ◽  
Noncompact Type ◽  
Isomorphism Classes ◽  
Upper Half Plane

AbstractWe give a complete classification of isomorphism classes of homogeneous holomorphic hermitian principal bundles over an irreducible hermitian symmetric space of noncompact type. This generalizes the results of our earlier work [Kyoto J. Math. 50 (2010), 325–363] where homogeneous holomorphic hermitian principal bundles over the upper half-plane were considered.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

scholarly journals On some recent progress in the classification of $$(P\hbox { and }Q)$$-polynomial association schemes

2019 ◽  
Author(s):  
Alexander L. Gavrilyuk ◽  
Jack H. Koolen
Keyword(s):  
Symmetric Spaces ◽  
Recent Progress ◽  
Association Schemes ◽  
Compact Symmetric Spaces ◽  
Expository Paper

AbstractThe problem of classification of $$(P\hbox { and }Q)$$(PandQ)-polynomial association schemes, as a finite analogue of E. Cartan’s classification of compact symmetric spaces, was posed in the monograph “Association schemes” by E. Bannai and T. Ito in the early 1980s. In this expository paper, we report on some recent results towards its solution.

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