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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
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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

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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