Spheres in Hermitian symmetric spaces and flag manifolds (Q679357)

A new property of compact irreducible Hermitian symmetric spaces \(M\) is given here, by proving the following result. If \(M=G/K\), where \(G\) is a compact simply connected simple Lie group, \(T\) a maximal torus of \(G\) and \(F(T,M) =\{E_1, \dots, E_m\}\) is the fixed point set of \(T\) on \(M\), then for each pair \(E_i\), \(E_j\) there is a 2-dimensional sphere \(N_{ij} \subset M\) such that \(E_i\) and \(E_j\) are antipodal points of \(N_{ij}\). \textit{B.-Y. Chen} and \textit{T. Nagano} introduced in [Trans. Am. Math. Soc. 308, 273-297 (1988; Zbl 0656.53049)] the 2-number \(\#_2\). The holomorphic version \(\#_2^H\) of \(\#_2\) is defined here and for the above space \(M\) it is shown that \(\#_2^H\) is equal to \(\#_2\) and the Euler-Poincaré characteristic.