On the geometry of a pair of oriented planes (Q2778001)

For some \(m\)-dimensional vector space \(V\) with an inner product \(g\) let \(p,q\) be two oriented planes (i.e. 2-dimensional subspaces of \(V\)). Two oriented orthonormal bases \(x,y\) and \(u,v\) are called related bases, if \(g(x,u)=0=g(y,v)\) holds. It is easy to see that such related bases always exist for any pair of oriented planes (Proposition 1). If, in addition, \(g(x,v)^2=g(y,u)^2\) holds, then any line in \(p\) (in \(q\)) forms the same angle \(\alpha\) (\(\beta\)) with the plane \(q\) (\(p\)) and \(\alpha=\beta\) holds (Proposition 3). NEWLINENEWLINENEWLINEThe author defines equivalence classes on the set of related bases and shows that there is a bijection between these equivalence classes and the set of non-ordered pairs \(r,s\) of non-oriented planes being transversal to both \(p\) and \(q\) and being strictly orthogonal (Proposition 4). NEWLINENEWLINENEWLINEThe last section of the paper deals with the Hermitian case.