scholarly journals Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram

2008 ◽  
Vol 138 (1) ◽  
pp. 25-50 ◽  
Author(s):  
Sebastian Klein
Keyword(s):  
Symmetric Space ◽  
Geometric Structure ◽  
Riemannian Symmetric Space

scholarly journals Minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces

2019 ◽  
Vol 6 (1) ◽  
pp. 303-319
Author(s):  
Yoshihiro Ohnita
Keyword(s):  
Symmetric Space ◽  
Floer Homology ◽  
Lagrangian Submanifolds ◽  
Symplectic Manifolds ◽  
Lagrangian Submanifold ◽  
Riemannian Symmetric Space ◽  
Canonical Embedding ◽  
Real Form ◽  
Isotropy Representation ◽  
Totally Geodesic

AbstractAn R-space is a compact homogeneous space obtained as an orbit of the isotropy representation of a Riemannian symmetric space. It is known that each R-space has the canonical embedding into a Kähler C-space as a real form, and thus a compact embedded totally geodesic Lagrangian submanifold. The minimal Maslov number of Lagrangian submanifolds in symplectic manifolds is one of invariants under Hamiltonian isotopies and very fundamental to study the Floer homology for intersections of Lagrangian submanifolds. In this paper we show a Lie theoretic formula for the minimal Maslov number of R-spaces canonically embedded in Einstein-Kähler C-spaces, and provide some examples of the calculation by the formula.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

Distribution Calibration in Riemannian Symmetric Space

2011 ◽  
Vol 41 (4) ◽  
pp. 921-930 ◽  
Author(s):  
Si Si ◽  
Wei Liu ◽  
Dacheng Tao ◽  
Kwok-Ping Chan
Keyword(s):  
Symmetric Space ◽  
Riemannian Symmetric Space

scholarly journals Reduction Theorem for Connections and its Application to the Problem of Isotropy and Holonomy Groups of a Riemannian Manifold

1955 ◽  
Vol 9 ◽  
pp. 57-66 ◽  
Author(s):  
Katsumi Nomizu
Keyword(s):  
Differential Geometry ◽  
Riemannian Manifold ◽  
Symmetric Space ◽  
Homogeneous Spaces ◽  
Holonomy Group ◽  
Complete Riemannian Manifold ◽  
Reduction Theorem ◽  
Riemannian Symmetric Space ◽  
Isotropy Group ◽  
Holonomy Groups

The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

scholarly journals A Note on Triple Systems and Totally Geodesic Submanifolds in a Homogeneous Space

1968 ◽  
Vol 32 ◽  
pp. 5-20 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Symmetric Space ◽  
Jordan Algebras ◽  
Riemannian Symmetric Space ◽  
Lie Triple System ◽  
Totally Geodesic ◽  
Triple Systems ◽  
Geodesic Submanifolds ◽  
Totally Geodesic Submanifolds ◽  
Nonassociative Algebras ◽  
Torsion Free

In the study of nonassociative algebras various “triple systems” frequently arise from the associator function and other multilinear objects. In particular Lie triple systems arise in the study of Jordan algebras and a generalization of a Lie triple system arises in Malcev algebras. Lie triple systems also are used to study totally geodesic submanifolds of a Riemannian symmetric space. We shall show how a generalization of Lie triple systems also arises from the study of curvature and geodesies of a torsion free connexion on a manifold and bring out the relation of this to various nonassociative algebras.

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