Maximal antipodal sets of compact classical symmetric spaces and their cardinalities. I
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 \(G\).
scientific article; zbMATH DE number 7315175