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Naturally Reductive Homogeneous Riemannian Manifolds

Published online by Cambridge University Press:  20 November 2018

Carolyn S. Gordon
Carolyn S. Gordon*
Affiliation:
Washington University, St. Louis, Missouri
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The simple algebraic and geometric properties of naturally reductive metrics make them useful as examples in the study of homogeneous Riemannian manifolds. (See for example [2], [3], [15]). The existence and abundance of naturally reductive left-invariant metrics on a Lie group G or homogeneous space G/L reflect the structure of G itself. Such metrics abound on compact groups, exist but are more restricted on noncompact semisimple groups, and are relatively rare on solvable groups. The goals of this paper are

  • (i) to study all naturally reductive homogeneous spaces of G when G is either semisimple of noncompact type or nilpotent and

  • (ii) to give necessary conditions on a Riemannian homogeneous space of an arbitrary Lie group G in order that the metric be naturally reductive with respect to some transitive subgroup of G.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 3 , 01 June 1985 , pp. 467 - 487
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Azencott, R. and Wilson, E. N., Homogeneous manifolds with negative curvature 1, Trans. Amer. Math. Soc. 215 (1976), 323–362.Google Scholar
2. D'Atri, J. E., Geodesic spheres and symmetries in naturally reductive spaces, Mich. 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. 18, No. 215 (1979).Google Scholar
4. Deloff, E. D., Naturally reductive metrics and metrics with volume preserving geodesicsymmetries on NC algebras, thesis, Rutgers (1979).Google Scholar
5. Gordon, C. S., Riemannian isometry groups containing transitive reductive subgroups, Math. Ann. 248 (1980), 185–192.Google Scholar
6. Gordon, C. S., Transitive Riemannian isometry groups with nilpotent radicals, Ann. Inst. Four., Grenoble 31 (1981), 193–204.Google Scholar
7. Gordon, C. S., Semisimple normal subgroups of transitive Riemannian isometry groups, Osaka J. Math. 70 (1982), 283–286.Google Scholar
8. Gordon, C. S. and Wilson, E. N., The fine structure of transitive Riemannian isometry groups, Part I, to appear in Trans. Amer. Math. Soc.CrossRefGoogle Scholar
9. Gordon, C. S. and Ziller, W., Naturally reductive metrics of non-positive Ricci curvature, to appear in Proc. Amer. Math. Soc.Google Scholar
10. Helgason, S., Differential geometry, Lie groups, and symmetric spaces (Academic Press, New York, 1978).Google Scholar
11. Kaplan, A., On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35–42.Google Scholar
12. Kobayashi, S. and Nomizu, K., Foundations of differential geometry II (Interscience, New York, 1969).Google Scholar
13. Milnor, J., Curvatures of left-invariant metrics on Lie groups, Adv. in Math. 21 (1976), 293–329.Google Scholar
14. O'Neill, B., The fundamental equations of a submersion, Mich. Math. J. 13 (1966), 459–469.Google Scholar
15. Wang, M. and Ziller, W., On normal homogeneous Einstein metrics, Preprint.Google Scholar
16. Wilson, E. N., Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), 337–346.Google Scholar