Abstract
We construct new homogeneous Einstein spaces with negativeRicci curvature in two ways: First, we give a method for classifying andconstructing a class of rank one Einstein solvmanifolds whose derivedalgebras are two-step nilpotent. As an application, we describe anexplicit continuous family of ten-dimensional Einstein manifolds with atwo-dimensional parameter space, including a continuous subfamily ofmanifolds with negative sectional curvature. Secondly, we obtain newexamples of non-symmetric Einstein solvmanifolds by modifying thealgebraic structure of non-compact irreducible symmetric spaces of rankgreater than one, preserving the (constant) Ricci curvature.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevskii, D. V.: Classification of quaternionic spaces with a transitive solvable group of motions, Math. USSR-Izv. 9 (1975), 297-339.
Azencott, R. and Wilson, E. N.: Homogeneous Manifolds with Negative Curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323-362.
Berger, M.: Quelques formules de variation pour une structure Riemannienne, Ann. Sci. Ecole Norm. Sup. (4) 3 (1970), 285-294.
Besse, A.: Einstein Manifolds, Springer-Verlag, New York, 1987.
D'Atri, J. E. and Miatello, I. D.: A characterization of bounded symmetric domains by curvature, Trans. Amer. Math. Soc. 276(2) (1983), 531-540.
Deloff, E.: Naturally reductive metrics and metrics with volume preserving geodesic symmetries on NC algebras, Dissertation, Rutgers, 1979.
Damek, E. and Ricci, F.: Harmonic analysis on solvable extensions of H-type groups, J. Geom. Anal. 2(3) (1992), 213-248.
Eberlein, P. and Heber, J.: A differential geometric characterization of symmetric spaces of higher rank, Inst. H.E. Sci. Publ. Math. 71 (1990), 33-44.
Eberlein, P. and Heber, J.: Quarter pinched homogeneous spaces of negative curvature, Internat. J. Math. 7(4) (1996), 441-500.
Gindikin, S. G., Piatetskii-Shapiro, I. I. and Vinberg, E. B.: On the classification and canonical realization of bounded homogeneous domains, Transl. Moscow Math. Soc. 12 (1963), 404-437.
Gordon, C. S. and Wilson, E. N.: Isometry groups of Riemannian solvmanifolds, Trans. Amer. Math. Soc. 307(1) (1988), 245-269.
Heber, J.: Homogeneous spaces of nonpositive curvature and their geodesic flow, Internat. J. Math. 6(2) (1995), 279-296.
Heber, J.: Noncompact homogeneous Einstein spaces, Invent. Math. 133 (1998), 279-352.
Heintze, E.: On homogeneous manifolds of negative curvature, Math. Ann. 211 (1974), 23-34.
Helgason, S.: Differential Geometry, Lie Groups, and Symmetric Spaces, Academic Press, New York, 1978.
Hengesch, D.: Exemples de variétés homogènes d'Einstein à courbure scalaire négative, PhD Thesis, Université Henri Poincaré-Nancy 1, 1996.
Hilbert, D.: Die Grundlagen der Physik, Nachr. Akad. Wiss. Gött. (1915), 395-407.
Kaplan, A.: On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983), 35-42.
Leukert, S.: Representations and nonpositively curved solvmanifolds, PhD Thesis, University of North Carolina at Chapel Hill, 1988.
Piatetskii-Shapiro, I. I.: On the classification of bounded homogeneous domains in n-dimensional complex space, Soviet Math. Dokl. 2 (1962), 1460-1463.
Wolf, J. A.: Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 14-18.
Wolter, T. H.: Homogeneous manifolds with nonpositive curvature operator, Geom. Dedicata 37 (1991), 361-370.
Wolter, T. H.: Einstein metrics on solvable groups, Math. Z. 206 (1991), 457-471.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gordon, C.S., Kerr, M.M. New Homogeneous Einstein Metrics of Negative Ricci Curvature. Annals of Global Analysis and Geometry 19, 75–101 (2001). https://doi.org/10.1023/A:1006767203771
Issue Date:
DOI: https://doi.org/10.1023/A:1006767203771