Antipodal sets in oriented real Grassmann manifolds (Q2855481)

Let \(M\) be a Riemannian symmetric space of compact type and denote by \(s_x\) the geodesic symmetry of \(M\) at \(x \in M\). A subset \(S\) of \(M\) is called an antipodal set if \(s_x(y) = y\) for all \(x,y \in S\). The maximal possible cardinality \(\# S\) of an antipodal set \(S\) in \(M\) is called the \(2\)-number of \(M\) and denoted by \(\#_2M\). It is known that \(\#_2M\) is finite. A great antipodal set \(S\) of \(M\) is an antipodal set of \(M\) so that \(\# S = \#_2M\). These notions were introduced in [\textit{B.-Y. Chen} and \textit{T. Nagano}, Trans. Am. Math. Soc. 308, No. 1, 273--297 (1988; Zbl 0656.53049)]. NEWLINENEWLINENEWLINEIn the present paper the author investigates great antipodal sets in the real Grassmann manifolds \(\tilde{G}_k({\mathbb R}^n)\) of oriented \(k\)-planes in \({\mathbb R}^n\). He reduces the classification problem to that of finding maximal subsets satisfying certain conditions in the set consisting of subsets of cardinality \(k\) in \(\{1,\ldots,n\}\). Using this reduction approach he can determine all great antipodal sets in \(\tilde{G}_k({\mathbb R}^n)\) for \(k \leq 4\). For \(k > 4\) he constructs explicitly some great antipodal sets.