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Einstein homogeneous bisymmetric fibrations

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Abstract

We consider a homogeneous fibration G/L → G/K, with symmetric fiber and base, where G is a compact connected semisimple Lie group and L has maximal rank in G. We suppose the base space G/K is isotropy irreducible and the fiber K/L is simply connected. We investigate the existence of G-invariant Einstein metrics on G/L such that the natural projection onto G/K is a Riemannian submersion with totally geodesic fibers. These spaces are divided in two types: the fiber K/L is isotropy irreducible or is the product of two irreducible symmetric spaces. We classify all the G-invariant Einstein metrics with totally geodesic fibers for the first type. For the second type, we classify all these metrics when G is an exceptional Lie group. If G is a classical Lie group we classify all such metrics which are the orthogonal sum of the normal metrics on the fiber and on the base or such that the restriction to the fiber is also Einstein.

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References

  1. Alekseevsky D., Dotti I., Ferraris C.: Homogeneous Ricci positive 5-manifolds. Pacific J. Math. 175(1), 1–12 (1996)

    MathSciNet  Google Scholar 

  2. Araújo, F.: Einstein Homogeneous Riemannian fibrations. PhD thesis of The University of Edinburgh, arXiv:0905.3143v2 [math.DG] (2008)

  3. Araújo F.: Some Einstein homogeneous Riemannian fibrations. Differ. Geom. Appl. 28, 241–263 (2010)

    Article  MATH  Google Scholar 

  4. Bröcker T., Dieck T.: Representations of compact lie groups. Springer, Berlin (1985)

    MATH  Google Scholar 

  5. Besse A.: Einstein manifolds. Springer, Berlin (1987)

    MATH  Google Scholar 

  6. Boyer C.P., Galicki K.: On Sasakian-Einstein geometry. Internat. J. Math. 11, 873–909 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  7. Böhm C., Kerr M.M.: Low-dimensional homogeneous Einstein metrics. Trans. Am. Math. Soc. 358(4), 1455–1468 (2005)

    Article  Google Scholar 

  8. Castellani L., Romans L.J., Warner N.P.: A classification of compactifying solutions for d = 11 supergravity. Nuclear Phys. 241, 429–462 (1984)

    Article  MathSciNet  Google Scholar 

  9. Dickinson W., Kerr M.M.: The geometry of compact homogeneous spaces with two isotropy summands. Ann. Glob. Anal. Geom. 34, 329–350 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Helgason S.: Differential Geometry, Lie Groups and Symmetric Spaces, Mathematics 80. Academic Press, London (1978)

    Google Scholar 

  11. Humphreys J.E.: Introduction to Lie Algebras and Representation Theory. Springer, Berlin (1972)

    MATH  Google Scholar 

  12. Jensen G.R.: Homogeneous Einstein spaces of dimension 4. J. Diff. Geom. 3, 309–349 (1969)

    MATH  Google Scholar 

  13. Jensen G.R.: Einstein metrics on principal fiber bundles. J. Diff. Geom. 8, 599–614 (1973)

    MATH  Google Scholar 

  14. Jimenez J.A.: Riemannian 4-symmetric spaces. Trans. Am. Math. Soc. 306(2), 715–734 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  15. Nikonorov Y.G., Rodionov D.E.: Compact 6-dimensional homogeneous Einstein manifolds. Dokl. Math. RAR 336, 599–601 (1999)

    MathSciNet  Google Scholar 

  16. Onishchik L.A., Vinberg E.B.: Lie Groups and Algebraic Groups, Springer Series in Soviet Mathematics. Springer, Berlin (1990)

    Google Scholar 

  17. Panyushev D.: Isotropy representations, eingenvalues of a casimir element and commutative lie subalgebras. J. Lond. Math. Soc. 64(2), 61–80 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Wang, M.Y.: Einstein Metrics from Symmetry and Bundle Constructions, Surveys in Differential Geometry, VI: Essays on Einstein Manifolds, pp. 287–325. International Press (1999)

  19. Wolf J.A., Gray A.: Homogeneous spaces defined by lie group automorphisms I, II. J. Diff. Geom. 2, 77–114, 115–159 (1968)

    Google Scholar 

  20. Wang M.Y., Ziller W.: On normal homogeneous Einstein manifolds. Ann. Sci. l’E.N.S. 18(4), 563–633 (1985)

    MathSciNet  MATH  Google Scholar 

  21. Wang M.Y., Ziller W.: Existence and non-existence of homogeneous Einstein metrics. Invent. Math. 84, 177–194 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wang M.Y., Ziller W.: Einstein metrics on principal torus bundles. J. Diff. Geom. 31, 215–248 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Yau S.T.: On the Ricci curvature of a compact Kähler Manifold and the Complex Monge-Ampère Equation I. Comm. Pure Appl. Math. 31, 339–411 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ziller W.: Homogeneous Einstein metrics on spheres and projective spaces. Math. Ann. 259, 351–358 (1982)

    Article  MathSciNet  MATH  Google Scholar 

Download references

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  1. School of Mathematics, The University of Edinburgh, JCMB, The Kings Buildings, Edinburgh, EH9 3JZ, UK

    Fátima Araújo

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  1. Fátima Araújo
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Correspondence to Fátima Araújo.

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Araújo, F. Einstein homogeneous bisymmetric fibrations. Geom Dedicata 154, 133–160 (2011). https://doi.org/10.1007/s10711-010-9572-x

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