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Sbornik: Mathematics, 1999, Volume 190, Issue 2, Pages 255–283
DOI: https://doi.org/10.1070/sm1999v190n02ABEH000385
(Mi sm385)
 

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
References:
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
Russian version:
Matematicheskii Sbornik, 1999, Volume 190, Number 2, Pages 93–122
DOI: https://doi.org/10.4213/sm385
Bibliographic databases:
UDC: 514.76
MSC: Primary 32M15, 53C35; Secondary 53C10
Language: English
Original paper language: Russian
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
Citation in format AMSBIB
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  • This publication is cited in the following 8 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник - 1992–2005 Sbornik: Mathematics
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    Abstract page:653
    Russian version PDF:248
    English version PDF:12
    References:46
    First page:3
     
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