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A Remark on Homogeneous Convex Domains

Published online by Cambridge University Press:  22 January 2016

Satoru Shimizu
Satoru Shimizu*
Affiliation:
Mathematical Institute Tohoku University, Sendai, 980, Japan
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In this note, by a homogeneous convex domain in Rn we mean a convex domain Ω in Rn containing no complete straight lines on which the group G(Ω) of all affine transformations of Rn leaving Ω invariant acts transitively. Let Ω be a homogeneous convex domain. Then Ω admits a G(©)-invariant Riemannian metric which is called the canonical metric (see [11]). The domain Ω endowed with the canonical metric is a homogeneous Riemannian manifold and we denote by I(Ω) the group of all isometries of it. A homogeneous convex domain Ω is called reducible if there is a direct sum decomposition of thé ambient space Rn = Rn1 × Rn2, ni > 0, such that Ω = Ω1 × 02 with Ωi a homogeneous convex domain in Rni; and if there is no such decomposition, then Ω is called irreducible.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 105 , March 1987 , pp. 1 - 7
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

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