Maximal antipodal sets of compact classical symmetric spaces and their cardinalities. I
type: Item
instance of: scholarly article
title: Maximal antipodal sets of compact classical symmetric spaces and their cardinalities. I
zbMATH Open document ID: 1468.53048
author: Makiko S. Tanaka
author: Hiroyuki Tasaki
published in: Differential Geometry and its Applications
publication date: 2021-02-24T00:00:00Z
review text: In a compact symmetric space \(M\), a set \(A\subset M\) is said to be antipodal if, for every \(x\) and \(y\) in \(A\), \(s_x(y)=y\), where \(s_x\) is the geodesic symmetry at \(x\). An antipodal set is finite. The goal of the paper is to determine the antipodal sets and their cardinalities. An antipodal set with maximal cardinality is said to be a great antipodal set and its cardinality the 2-number of the Riemannian symmetric space \(M\). The authors study first the case of a compact Lie group. Then they consider the Grassmannian manifolds \[G_k(\mathbb{K}^n)\simeq \operatorname{O}(n,\mathbb{K})/\operatorname{O}(k,\mathbb{K})\times \operatorname{O}(n-k,\mathbb{K}),\] where \(\mathbb{K}=\mathbb{R}\), \(\mathbb{C}\), or \(\mathbb{H}\), the set of real, complex, or quaternonic numbers. In the case of \(G_m(\mathbb{K}^{2m})\), the authors consider the quotient \[G_m(\mathbb{K}^{2m})^*=G_m(\mathbb{K}^{2m})/\{Id,\gamma \},\] where \(\gamma \) is the isometry \(x\mapsto x^{\perp}\). Further the following compact symmetric spaces are considered: \begin{align*} CI(n)&\simeq \operatorname{Sp}_n/\operatorname{U}(n),\quad & CI(n)^*&=CI(n)/\{\pm I_n\},\\ DIII(n)&\simeq \operatorname{SO}(2n)/\operatorname{U}(n),\quad & DIII(n)^*&=DIII(n)/\{\pm I_{2n}\}. \end{align*} In each case the authors determine the antipodal sets, their cardinalities and the 2-number. The proofs use an embedding of the compact symmetric space \(M=G/K\) into the compact Lie group \(G\).
Mathematics Subject Classification ID: 53C35
Mathematics Subject Classification ID: 22F30
zbMATH DE Number: 7315175
zbMATH Keywords: symmetric space
zbMATH Keywords: antipodal set
reviewed by: Jacques Faraut
MaRDI profile type: Publication
full work available at URL: j.difgeo.2020.101682
OpenAlex ID: W3086230914
Wikidata QID: Q115354710
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DOI: 10.1016/J.DIFGEO.2020.101682
description: scientific article; zbMATH DE number 7315175