Hostname: page-component-586b7cd67f-dlnhk Total loading time: 0 Render date: 2024-11-29T19:37:01.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Classification of five-dimensional naturally reductive spaces

Published online by Cambridge University Press:  24 October 2008

Oldřich Kowalski  and
Lieven Vanhecke
Oldřich Kowalski
Affiliation:
Faculty of Mathematics and Physics, Charles University, 18600 Praha, Czechoslovakia
Lieven Vanhecke
Affiliation:
Department of Mathematics, Katholieke Universiteit Leuven, B-3030 Leuven, Belgium
Get access Rights & Permissions [Opens in a new window]

Extract

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 97 , Issue 3 , May 1985 , pp. 445 - 463
Copyright
Copyright © Cambridge Philosophical Society 1985

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]D'Atri, J. E. and Nickerson, H. K.. Geodesic symmetries in spaces with special curvature tensor. J. Differential Geometry 9 (1974), 251–262.CrossRefGoogle Scholar
[2]D'Atri, J. E.. Geodesic spheres and symmetries in naturally reductive homogeneous spaces. Michigan Math. J. 22 (1975), 71–76.Google Scholar
[3]D'Atri, J. E. and Ziller, W.. Naturally Reductive Metrics and Einstein Metrics on Compact Lie groups (Mem. Amer. Math. Soc. vol. 215, 1979).Google Scholar
[4]Helgason, S.. Differential Geometry and Symmetric Spaces (Academic Press, 1962).Google Scholar
[5]Kaplan, A.. On the geometry of groups of Heisenberg type. Bull. London Math. Soc. 15 (1983), 35–42.Google Scholar
[6]Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, vol. II (Interscience Publishers, 1969).Google Scholar
[7]Kowalski, O.. Classification of Generalized Symmetric Riemannian Spaces of Dimension n ≤ 5 (Rozpravy ČSAV, Řada MPV, no. 8, 85, Prague, 1975).Google Scholar
[8]Kowalski, O.. Generalized Symmetric Spaces. Lecture Notes in Mathematics, vol. 805 (Springer-Verlag, 1980).Google Scholar
[9]Kowalski, O.. Spaces with volume-preserving symmetries and related classes of Riemannian manifolds. Rend. Sem. Mat. Univ. e Politec. Torino (To appear).Google Scholar
[10]Kowalski, O. and Vanhecke, L.. Four-dimensional naturally reductive homogeneous spaces. Rend. Sem. Mat. Univ. e Politec. Torino (To appear).Google Scholar
[11]Kowalski, O. and Vanhecke, L.. Classification of four-dimensional commutative spaces. Quart. J. Math. Oxford (To appear).Google Scholar
[12]Lichnerowicz, A.. Opérateurs différentiels invariants sur un espace homogène. Ann. Sc. Ecole Norm. Sup. 81 (1964), 341–385.CrossRefGoogle Scholar
[13]Nomizu, K.. Invariant affine connections on homogeneous spaces. Amer. J. Math. 76 (1954), 33–65.Google Scholar
[14]Tricerri, F. and Vanhecke, L.. Homogeneous Structures on Riemannian Manifolds (London Math. Soc. Lecture Note Series, vol. 83, Cambridge Univ. Press, 1983).CrossRefGoogle Scholar