The triplets of the Grassmann manifold \(\text{G}_ 2(\mathbb{R}^ 6)\)
The author investigates triplets of two-dimensional linear subspaces of \({\mathbb{R}}^6\). The key notion in this study are the so-called critical angles. The critical angles between two (two-dimensional) planes \(V_1\), \(V_2\) in \({\mathbb{R}}^n\) are defined as the (two) critical values of the function which assigns to each pair of vectors \(v_i \in V_i\) their angle. It turns out that a \textit{couple} of two-planes is determined up to isometry by their critical angles. Having this result in mind one might ask whether or not a triplet \(T = \{V_1,V_2,V_2\}\) of 2-planes is determined up to isometry by knowing all critical values between any pair of 2-planes of \(T\). The answer to this conjecture is ``no''. It is necessary to add more invariants to the set of 6 critical values. These are three more angles (which the author calls inner angles) and four signs. An inner angle is defined as follows. Take the unit circles in \(V_1\) and \(V_2\), e.g., and project them orthogonally onto \(V_3\). The images of these two unit circles are ellipses in \(V_3\) and the angle between their bases is called an inner angle. The signs are derived from pairs of vectors which enclose critical angles. The author shows that \(T\) is uniquely determined by these invariants. He also gives partial answers to the question of the existence and the number of triplets \(T\) for a given set of invariants. This leads in a natural way to the notion of regular and semi-regular triplets. The interesting paper closes with a detailed study of some special triplets.
scientific article; zbMATH DE number 1098479