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Noncompact homogeneous Einstein manifolds attached to graded Lie algebras

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Abstract

In this paper, we study the nilradicals of parabolic subalgebras of semisimple Lie algebras and the natural one-dimensional solvable extensions of them. We investigate the structures, curvatures and Einstein conditions of the associated nilmanifolds and solvmanifolds. We show that our solvmanifold is Einstein if the nilradical is two-step. New examples of Einstein solvmanifolds with three-step and four-step nilradicals are also given.

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References

  1. Alekseevskii, D.V.: Homogeneous Riemannian spaces of negative curvature. Mat. Sb. 25, 87–109 (1975); English translation. Math. USSR-Sb. 96, 93–117 (1975)

    Google Scholar 

  2. Alekseevskii D.V. (1975). Classification of quaternionic spaces with transitive solvable group of motions. Math. USSR-Izv. 9: 297–339

    Article  Google Scholar 

  3. Berndt J., Tricerri F. and Vanhecke L. (1995). Generalized Heisenberg groups and Damek-Ricci harmonic spaces (Lecture Notes in Mathematics 1598). Springer, Berlin

    Google Scholar 

  4. Besse A.L. (1987). Einstein manifolds (Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 10). Springer, Berlin

    Google Scholar 

  5. Boggino J. (1985). Generalized Heisenberg groups and solvmanifolds naturally associated. Rend. Sem. Mat. Univ. Politec. Torino 43: 529–547

    MATH  MathSciNet  Google Scholar 

  6. Bourbaki N. (1968). Groupes et algèbres de Lie, Chapitre IV–VI (Éléments de mathématique). Hermann, Paris

    Google Scholar 

  7. Cortés V. (1996). Alekseevskian spaces. Diff. Geom. Appl. 6: 129–168

    Article  MATH  Google Scholar 

  8. Damek E. and Ricci F. (1992). Harmonic analysis on solvable extensions of H-type groups. J. Geom. Anal. 2: 213–248

    MATH  MathSciNet  Google Scholar 

  9. Eberlein P. and Heber J. (1996). Quarter pinched homogeneous spaces of negative curvature. Internat. J. Math. 7: 441–500

    Article  MATH  MathSciNet  Google Scholar 

  10. Gordon C.S. and Kerr M. (2001). New homogeneous Einstein metrics of negative Ricci curvature. Ann. Global Anal. Geom. 19: 75–101

    Article  MATH  MathSciNet  Google Scholar 

  11. Heber J. (1998). Noncompact homogeneous Einstein spaces. Invent. Math. 133: 279–352

    Article  MATH  MathSciNet  Google Scholar 

  12. Helgason, S.: Differential geometry, Lie groups, and symmetric spaces (Graduate Studies in Mathematics 34). American Mathematical Society, Providence, RI (2001)

  13. Kaneyuki S. (1993). On the subalgebras \(\mathfrak{g}_{0}\) and \(\mathfrak{g}_{\rm ev}\) of semisimple graded Lie algebras J. Math. Soc. Jpn. 45: 1–19

    Article  MATH  MathSciNet  Google Scholar 

  14. Kaneyuki S. and Asano H. (1988). Graded Lie algebras and generalized Jordan triple systems. Nagoya Math. J. 112: 81–115

    MATH  MathSciNet  Google Scholar 

  15. Lauret J. (2001). Ricci soliton homogeneous nilmanifolds. Math. Ann. 319: 715–733

    Article  MATH  MathSciNet  Google Scholar 

  16. Lauret J. (2002). Finding Einstein solvmanifolds by a variational method. Math. Z. 241: 83–99

    Article  MATH  MathSciNet  Google Scholar 

  17. Lauret, J.: Minimal metrics on nilmanifolds. In: Bureš, J., Kowalski, O., Krupka, D., Slovak, J. (eds.) Differential geometry and its applications Proceedings of the 9th International Conference (DGA 2004) held in Prague, August 30–September 3, 2004, 79–97. Matfyzpress, Prague (2005)

  18. Mori K. (2002). Einstein metrics on Boggino-Damek-Ricci-type solvable Lie groups. Osaka J. Math. 39: 345–362

    MATH  MathSciNet  Google Scholar 

  19. Nishikawa, S.: Harmonic maps and negatively curved homogeneous spaces. In: Chen, W.H., et. al. (eds) Geometry and topology of submanifolds, X, Beijing/Berlin, 1999, 200–215. World Sci. Publ., River Edge, NJ (2000)

  20. Pyateskii-Shapiro I.I. (1969). Automorphic functions and the geometry of classical domains. Gordon and Breach Science Publishers, New York

    Google Scholar 

  21. Tamaru H. (1999). The local orbit types of symmetric spaces under the actions of the isotropy subgroups. Diff. Geom. Appl. 11: 29–38

    Article  MATH  MathSciNet  Google Scholar 

  22. Tamaru H. (1999). On certain subalgebras of graded Lie algebras. Yokohama Math. J. 46: 127–138

    MATH  MathSciNet  Google Scholar 

  23. Tamaru, H.: A class of noncompact homogeneous Einstein manifolds. In: Bureš, J., Kowalski, O., Krupka, D., Slovak, J. (eds.) Differential geometry and its applications, Proceedings of the 9th International Conference (DGA 2004) held in Prague, August 30–September 3, 2004, 119–127. Matfyzpress, Prague (2005)

  24. Wolter T.H. (1991). Einstein metrics on solvable Lie groups. Math. Z. 206: 457–471

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

    Hiroshi Tamaru

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  1. Hiroshi Tamaru
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Correspondence to Hiroshi Tamaru.

Additional information

This work was partially supported by Grant-in-Aid for Young Scientists (B) 14740049 and 17740039, The Ministry of Education, Culture, Sports, Science and Technology, Japan.

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Tamaru, H. Noncompact homogeneous Einstein manifolds attached to graded Lie algebras. Math. Z. 259, 171–186 (2008). https://doi.org/10.1007/s00209-007-0217-1

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  • DOI: https://doi.org/10.1007/s00209-007-0217-1

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