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Communications in Mathematical Physics

, Volume 369, Issue 2, pp 637–673 | Cite as

Einstein Warped G2 and Spin(7) Manifolds

  • Víctor Manero
  • Luis UgarteEmail author
Article

Abstract

In this paper most of the classes of G2-structures with Einstein induced metric of negative, null, or positive scalar curvature are realized. This is carried out by means of warped G2-structures with fiber an Einstein SU(3) manifold. The torsion forms of any warped G2-structure are explicitly described in terms of the torsion forms of the SU(3)-structure and the warping function, which allows to give characterizations of the principal classes of Einstein warped G2 manifolds. Similar results are obtained for Einstein warped Spin(7) manifolds with fiber a G2 manifold.

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Notes

Acknowledgements

We are grateful to Stefan Ivanov for useful comments about Question 5.3. This work has been partially supported by the project MTM2017-85649-P (AEI/Feder, UE), and E22-17R “Algebra y Geometría” (Gobierno de Aragón/FEDER). We would like to thank the referees for their valuable suggestions and remarks that have improved the paper.

References

  1. 1.
    Agricola I., Chiossi S.G., Friedrich T., Höll J.: Spinorial description of SU(3)- and G2-manifolds. J. Geom. Phys. 98, 535–555 (2015)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Apostolov V., Drăghici T., Moroianu A.: A splitting theorem for Kähler manifolds whose Ricci tensor have constant eigenvalues. Int. J. Math. 12, 769–789 (2001)CrossRefzbMATHGoogle Scholar
  3. 3.
    Bär C.: Real killing spinors and holonomy. Commun. Math. Phys. 154, 509–521 (1993)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bedulli L., Vezzoni L.: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys. 57, 1125–1146 (2007)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Besse, A.: Einstein Manifolds. Springer, Berlin (1987)Google Scholar
  6. 6.
    Bilal A., Metzger S.: Compact weak G2-manifolds with conical singularities. Nuclear Phys. B 663, 343–364 (2003)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Boyer, C., Galicki, K.: Sasakian Geometry, Oxford Mathematical Monographs. Oxford University Press, Oxford (2008)Google Scholar
  8. 8.
    Bryant R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Bryant, R.L.: Some remarks on G2 structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, pp. 75–109 (2006)Google Scholar
  10. 10.
    Bryant R.L., Salamon S.: On the construction of some complete metrics with exceptional holonomy. Duke Math. J. 58, 829–850 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Butruille J.-B.: Classification des variétés approximativement kähleriennes homogènes. Ann. Glob. Anal. Geom. 27, 201–225 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Cabrera F.M., Monar M.D., Swann A.F.: Classification of G2-structures. J. Lond. Math. Soc. 53, 407–416 (1996)CrossRefzbMATHGoogle Scholar
  13. 13.
    Cleyton R., Ivanov S.: On the geometry of closed G2-structures. Commun. Math. Phys. 270, 53–67 (2007)ADSCrossRefzbMATHGoogle Scholar
  14. 14.
    Cleyton R., Ivanov S.: Conformal equivalence between certain geometries in dimension 6 and 7. Math. Res. Lett. 15, 631–640 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cleyton R., Ivanov S.: Curvature decomposition of G2-manifolds. J. Geom. Phys. 58, 1429–1449 (2008)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chiossi, S., Salamon, S.: Intrinsic Torsion of SU(3) and G2-Structures. Differential Geometry (Valencia, 2001). World Scientific Publishing, River Edge, NJ, pp. 115–133 (2002)Google Scholar
  17. 17.
    Eells J., Salamon S.: Twistorial construction of harmonic maps of surfaces into four-manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 12, 589–640 (1985)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Fernández M.: A classification of Riemannian manifolds with structure group Spin(7). Ann. Mat. Pura Appl. 143, 101–122 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Fernández M., Fino A., Manero V.: G2-structures on Einstein solvmanifolds. Asian J. Math. 19, 321–342 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Fernández M., Gray A.: Riemannian manifolds with structure group G2. Ann. Mat. Pura Appl. 132, 19–45 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fernández M., Ivanov S., Muñoz V., Ugarte L.: Nearly hypo structures and compact nearly Kähler 6-manifolds with conical singularities. J. Lond. Math. Soc. 78, 580–604 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fino A., Raffero A.: Coupled SU(3)-structures and supersymmetry. Symmetry 7, 625–650 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Fino A., Raffero A.: Einstein locally conformal calibrated G2-structures. Math. Z. 280, 1093–1106 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Foscolo L., Haskins M.: New G2-holonomy cones and exotic nearly Kähler structures on S 6 and \({S^3 \times S^3}\). Ann. Math. 185, 59–130 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Friedrich T., Kath I., Moroianu A., Semmelmann U.: On nearly parallel G2-structures. J. Geom. Phys. 23, 259–286 (1997)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Goldberg S.I.: Integrability of almost Kähler manifolds. Proc. Am. Math. Soc. 21, 96–100 (1969)CrossRefzbMATHGoogle Scholar
  27. 27.
    Gibbons G.W., Page D.N., Pope C.N.: Einstein metrics on S 3, \({{\mathbb{R}}^3}\) and \({{\mathbb{R}}^4}\) bundles. Commun. Math. Phys. 127, 529–553 (1990)ADSCrossRefGoogle Scholar
  28. 28.
    Grigorian S.: Deformations of G2-structures with torsion. Asian J. Math. 20, 123–155 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Hitchin N.: The geometry of three-forms in six dimensions. J. Differ. Geom. 55, 547–576 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Hitchin, N.: Stable Forms and Special Metrics. Global Differential Geometry: The mathematical Legacy of Alfred Gray, Bilbao, 2000, Contemporary Mathematics 288, American Mathematical Society, Providence, RI, 2001, pp. 70–89 (2001)Google Scholar
  31. 31.
    Ivanov S.: Connetions with torsion, parallel spinors and geometry of Spin(7)-manifolds. Math. Res. Lett. 11, 171–186 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Joyce, D.: Compact Riemannian 7-manifolds with holonomy G2. I, II, J. Differ. Geom. 43 291–328, 329–375 (1996).Google Scholar
  33. 33.
    Joyce D.: Compact 8-manifolds with holonomy Spin(7). Invent. Math. 123, 507–552 (1996)ADSMathSciNetzbMATHGoogle Scholar
  34. 34.
    Karigiannis S.: Deformations of G2 and Spin(7) structures. Can. J. Math. 57, 1012–1055 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lauret J.: Einstein solvmanifolds are standard. Ann. Math. 172, 1859–1877 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Lin, C.: Torsion-free G2-structures with identical Riemannian metric. J. Topol. Anal. 10, 915–932 (2018)Google Scholar
  37. 37.
    Manero, V.: Closed G2 Forms and Special Metrics. Ph.D. Thesis, University of the Basque Country (2015)Google Scholar
  38. 38.
    O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Pure and Applied Mathematics, vol. 103. Academic Press, New York (1983)Google Scholar
  39. 39.
    Puhle C.: Spin(7) manifolds with parallel torsion form. Commun. Math. Phys. 291, 303–320 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Salamon S.: A tour of exceptional geometry. Milan J. Math. 71, 59–94 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Schulte-Hengesbach, F.: Half-flat Structures on Lie Groups. Ph.D. Thesis, Hamburg (2010)Google Scholar
  42. 42.
    Sekigawa K.: On some compact Einstein almost Kähler Einstein manifolds. J. Math. Soc. Jpn. 39, 677–684 (1987)CrossRefzbMATHGoogle Scholar
  43. 43.
    Tomasiello, A.: New string vacua from twistor spaces. Phys. Rev. D78(4), 046007 (2008)Google Scholar

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© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Departamento de Matemáticas - I.U.M.A.Universidad de ZaragozaZaragozaSpain

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