Conformal geometry of symmetric spaces and generalized linear-fractional maps of Krein–Shmul'yan

The matrix balls $\mathrm B_{p,q}$ consisting of $p\times q$-matrices of norm $<1$ over $\mathbb C$ are considered. These balls are one possible realization of the symmetric spaces $\mathrm B_{p,q}=\operatorname U(p,q)/\operatorname U(p)\times\operatorname U(q)$. Generalized linear-fractional maps are maps $\mathrm B_{p,q}\to\mathrm B_{r,s}$ of the form $Z\mapsto K+LZ(1-NZ)^{-1}$ (they are in general neither injective nor surjective). Characterizations of generalized linear-fractional maps in the spirit of the “fundamental theorem of projective geometry” are obtained: for a certain family of submanifolds of $\mathrm B_{p,q}$ (“quasilines”) it is shown that maps taking quasilines to quasilines are generalized linear-fractional. In addition, for the standard field of cones on $\mathrm B_{p,q}$ (described by the inequality $\operatorname{rk}dZ\leqslant 1$) it is shown that maps taking cones to cones are generalized linear-fractional.