Geometries of the projective matrix space. II (Q1080174)

Recently [cf. part I, ibid. 95, 263-307 (1985; Zbl 0589.51011)] the authors investigated the Euclidean, spherical and non-Euclidean metrics on suitable subspaces of the complex matrix projective line \(P_ 1(M_ n({\mathbb{C}}))\). Their interest is now focused at the Möbius transformations \(W(Z)=(ZC+D)^{-1}(ZA+B)\), where A, B, C, D, Z are \(n\times n\) complex matrices. The concepts ''complex plane'', ''Riemann sphere'' and ''unit disk'' are generalized accordingly. In chapter 1 the spherical, Euclidean and non-Euclidean circles are defined and studied in terms of (i) points at given distance from some point resp. (ii) zeroes of some (Hermitian) quadratic form. Chapter 2 deals with Möbius transformations that carry the unit disk into itself, as well as those that carry (or interchange) Hermitian and unitary matrices. Finally (chapter 3), the stereographic projection from the generalized Riemann sphere to the Euclidean plane is introduced; various properties of this map are dicussed.