Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-23T23:29:27.839Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetric homogeneous convex domains

Published online by Cambridge University Press:  22 January 2016

Tadashi Tsuji
Tadashi Tsuji*
Affiliation:
Department of Mathematics, Mie University, Tsu, Mie 514, Japan
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let D be a convex domain in the n-dimensional real number space Rn, not containing any affine line and A(D) the group of all affine transformations of Rn leaving D invariant. If the group A(D) acts transitively on D, then the domain D is said to be homogeneous. From a homogeneous convex domain D in Rn, a homogeneous convex cone V = V(D) in Rn+1 = Rn × R is constructed as follows (cf. Vinberg [11]):

which is called the cone fitted on the convex domain D.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 93 , March 1984 , pp. 1 - 17
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1984

References

[ 1 ] Asano, H., On the irreducibility of homogeneous convex cones, J. Fac. Sci. Univ. Tokyo, 15 (1968), 201–208.Google Scholar
[ 2 ] Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry, vol. 2, J. Wiley-Interscience, New York, 1969.Google Scholar
[ 3 ] Meschiari, M., Isometrie dei coni convessi regolari omogenei, Atti Sem. Mat. Fis. Univ. Modena, 27 (1978), 297–314.Google Scholar
[ 4 ] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. J. Math., 76 (1954), 33–65.CrossRefGoogle Scholar
[ 5 ] Rothaus, O. S., Domains of positivity, Abh. Math. Sem. Univ. Hamburg, 24 (1960), 189–235.Google Scholar
[ 6 ] Shima, H., Homogeneous convex domains of negative sectional curvature, J. Differential Geom., 12 (1977), 327–332.Google Scholar
[ 7 ] Shima, H., A differential geometric characterization of homogeneous self-dual cones, Tsukuba J. Math., 6 (1982), 79–88.Google Scholar
[ 8 ] Tsuji, T., A characterization of homogeneous self-dual cones, Tokyo J. Math., 5 (1982), 1–12.Google Scholar
[ 9 ] Tsuji, T., On the homogeneous convex cones of non-positive curvature, ibid., 405–417.Google Scholar
[10] Tsuji, T., On infinitesimal isometries of homogeneous convex cones, Japan. J. Math., 8 (1982), 383–406.Google Scholar
[11] Vinberg, E. B., The theory of convex homogeneous cones, Trans. Moscow Math. Soc, 12 (1963), 340–403.Google Scholar
[12] Vinberg, E. B., The structure of the group of automorphisms of a homogeneous convex cone, Trans. Moscow Math. Soc, 13 (1965), 63–93.Google Scholar