Holomorphic sectional curvature of quasisymmetric domains

Author:
R. Zelow

Journal:
Proc. Amer. Math. Soc. **76** (1979), 299-301

MSC:
Primary 53C55; Secondary 32M10

MathSciNet review:
537092

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Abstract: It is well known that the holomorphic sectional curvature of a bounded symmetric domain is bounded above by a negative constant. In this paper we show that this is true more generally for a quasi-symmetric Siegel domain, and the proof is based on a formula for the curvature from the author's thesis. The bounded homogeneous domains are, as is well known, biholomorphic to homogeneous Siegel domains and the bounded symmetric domains are biholomorphic to those quasi-symmetric (homogeneous) Siegel domains that satisfy a certain algebraic identity (which we do not need here).

**[1]**Soji Kaneyuki,*On the automorphism groups of homogeneuous bounded domains*, J. Fac. Sci. Univ. Tokyo Sect. I**14**(1967), 89–130 (1967). MR**0227472****[2]**Shingo Murakami,*On automorphisms of Siegel domains*, Lecture Notes in Mathematics, Vol. 286, Springer-Verlag, Berlin-New York, 1972. MR**0364690****[3]**I. I. Pyateskii-Shapiro,*Automorphic functions and the geometry of classical domains*, Translated from the Russian. Mathematics and Its Applications, Vol. 8, Gordon and Breach Science Publishers, New York-London-Paris, 1969. MR**0252690****[4]**I. Satake,*On classification of quasi-symmetric domains*, Nagoya Math. J.**62**(1976), 1–12. MR**0412491****[5]**-, Forthcoming book about algebraic structures on symmetric domains (to appear).**[6]**Masaru Takeuchi,*Homogeneous Siegel domains*, Study Group of Geometry, Institute of Mathematics, Yoshida College, Kyoto University, Kyoto, 1973. Publications of the Study Group of Geometry, Vol. 7. MR**0407332****[7]**R. Zelow Lundquist,*Curvature of quasisymmetric Siegel domains*, J. Differential Geom.**14**(1979), no. 4, 629–655 (1981). MR**600619****[8]**Rune Zelow Lundquist,*On the geometry of some Siegel domains*, Nagoya Math. J.**73**(1979), 175–195. MR**524015**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1979-0537092-4

Keywords:
Bounded homogeneous domain,
bounded symmetric domain,
Bergman metric,
quasi-symmetric Siegel domain,
cone,
tube domain

Article copyright:
© Copyright 1979
American Mathematical Society