Maximal normal sets of \(n\)-planes in \({\mathbb{R}{}}^{2n}\)
The paper investigates the relation between the notions of maximal normal sets and maximal isoclinic sets of \(n\)-planes in \(R^{2n}\). Two \(n\)- planes \(A\) and \(B\) in \(R^{2n}\) are normally related if the intersection of \(A\) and \(B\) and also of \(A\) and the orthogonal complement of \(B\) both are 0. A normal set is a set of \(n\)-planes such that every two distinct \(n\)-planes are either orthogonal or normally related. Two \(n\)-planes are isoclinic if the angle between an arbitrary vector in one plane and its orthogonal projection on the other plane remains constant. An isoclinic set is a set of mutually isoclinic \(n\)-planes. It is shown that every maximal normal set that is parametrized by a subspace of nonsingular matrices is a maximal isoclinic set. Conversely it is proved by methods from homotopy theory: Every maximal isoclinic set is a maximal normal set.
scientific article; zbMATH DE number 167843