The intersection of two real forms in Hermitian symmetric spaces of compact type (Q1928342)

A subset \(S\) of a Riemannian symmetric space \(M\) is an antipodal set if for every point \(x\in S\) the geodesic symmetry \(s_x\) fixes the whole set. The supremum of the cardinality of the antipodal sets in \(M\) is by definition the \(2\)-number \(\#_2 M\) of \(M\). For \(R\)-symmetric spaces, and in particular for compact Hermitian symmetric spaces, the 2-number coincides with \(\dim H_*(M,\text{Z}_2)\). This paper is concerned with intersections of real forms (i.e. fixed point sets of anti-holomorphic isometries, and therefore totally geodesic Lagrangian submanifolds) in Hermitian compact symmetric spaces \(M\). The following results are obtained: {(a)} If two real forms \(L_1\) and \(L_2\) intersect transversally, then \(L_1\cap L_2\) is an antipodal set in \(M\). {(b)}Replacing \(L_1\) and \(L_2\) with congruent real forms, transversality is preserved and so is the cardinality of the intersection. {(c)} If \(L_1\) and \(L_2\) intersect transversally and are congruent, then \(\# L_1\cap L_2=\#_2 L_1=\#_2 L_2\). As a consequence, any real form of a compact Hermitian symmetric space is a globally tight Lagrangian submanifold. In the irreducible case, if \(M=G_{2m}^{\mathbb C}({\mathbb C}^{4m})\), for \(m\geq 2\), \(L_1\) is congruent to \(G_{m}^{\mathbb H}({\mathbb H}^{2m} )\), and \(L_2\) is congruent to \(U(2m)\), then \[ \#L_1\cap L_2=2^m < {2m\choose m} =\#_2L_1<2^m=\#_2 L_2 . \] Otherwise, \(L_1\cap L_2\) is a great antipodal set (i.e. attains \(\#_2 M\)) of one of the \(L_i\)'s and \[ \#L_1\cap L_2=\text{min}\{\#_2 L_1,\#_2 L_2\}. \]