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This article is cited in 8 scientific papers (total in 8 papers)
Conformal geometry of symmetric spaces and generalized linear-fractional maps of Krein–Shmul'yan
Yu. A. Neretin Moscow State Institute of Electronics and Mathematics
Abstract:
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.
Received: 12.05.1998
Citation:
Yu. A. Neretin, “Conformal geometry of symmetric spaces and generalized linear-fractional maps of Krein–Shmul'yan”, Mat. Sb., 190:2 (1999), 93–122; Sb. Math., 190:2 (1999), 255–283
Linking options:
https://www.mathnet.ru/eng/sm385https://doi.org/10.1070/sm1999v190n02ABEH000385 https://www.mathnet.ru/eng/sm/v190/i2/p93
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Abstract page: | 653 | Russian version PDF: | 248 | English version PDF: | 12 | References: | 46 | First page: | 3 |
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