Advertisement

Annals of Global Analysis and Geometry

, Volume 53, Issue 4, pp 467–501 | Cite as

The Ricci tensor of almost parahermitian manifolds

  • Diego Conti
  • Federico A. Rossi
Article

Abstract

We study the pseudoriemannian geometry of almost parahermitian manifolds, obtaining a formula for the Ricci tensor of the Levi–Civita connection. The formula uses the intrinsic torsion of an underlying \(\mathrm {SL}(n,\mathbb {R})\)-structure; we express it in terms of exterior derivatives of some appropriately defined differential forms. As an application, we construct Einstein and Ricci-flat examples on Lie groups. We disprove the parakähler version of the Goldberg conjecture and obtain the first compact examples of a non-flat, Ricci-flat nearly parakähler structure. We study the paracomplex analogue of the first Chern class in complex geometry, which obstructs the existence of Ricci-flat parakähler metrics.

Keywords

Einstein pseudoriemannian metrics Almost parahermitian structures Intrinsic torsion Ricci tensor 

Mathematics Subject Classification

53C15 53C10 53C29 53C50 

References

  1. 1.
    Alekseevsky, D.V., Kimel’fel’d, B.N.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funkcional. Anal. i PriloŽen. 9(2), 5–11 (1975)MathSciNetGoogle Scholar
  2. 2.
    Alekseevsky, D.V., Medori, C., Tomassini, A.: Homogeneous para-Kählerian Einstein manifolds. Uspekhi Mat. Nauk 64(1(385)), 3–50 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Angella, D., Rossi, F.A.: Cohomology of \({\bf D}\)-complex manifolds. Differ. Geom. Appl 30(5), 530–547 (2012).  https://doi.org/10.1016/j.difgeo.2012.07.003 CrossRefMATHGoogle Scholar
  4. 4.
    Aubin, T.: Équations du type Monge-Ampère sur les variétés kählériennes compactes. Bull. Sci. Math. 102(1), 63–95 (1978)MathSciNetMATHGoogle Scholar
  5. 5.
    Bedulli, L., Vezzoni, L.: The Ricci tensor of SU(3)-manifolds. J. Geom. Phys 57(4), 1125–1146 (2007)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bryant, R.L.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Bryant, R.L.: Some remarks on \(G_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005. Gökova Geometry/Topology Conference (GGT), pp. 75–109. Gökova (2006)Google Scholar
  8. 8.
    Chiossi, S., Salamon, S.: The intrinsic torsion of \(SU(3)\) and \(G_2\) structures. In: Differential Geometry, Valencia 2001, pp. 115–133. World Scientific (2002)Google Scholar
  9. 9.
    Chursin, M., Schäfer, L., Smoczyk, K.: Mean curvature flow of space-like Lagrangian submanifolds in almost para-Kähler manifolds. Calc. Var. Partial Differ. Equ. 41(1–2), 111–125 (2011).  https://doi.org/10.1007/s00526-010-0355-x CrossRefMATHGoogle Scholar
  10. 10.
    Conti, D.: Intrinsic torsion in quaternionic contact geometry. Ann. Sc. Norm. Super. Pisa Cl. Sci. 16(2), 625–674 (2016)MathSciNetMATHGoogle Scholar
  11. 11.
    Cortés, V., Leistner, T., Schäfer, L., Schulte-Hengesbach, F.: Half-flat structures and special holonomy. Proc. Lond. Math. Soc. 102(1), 113–158 (2011).  https://doi.org/10.1112/plms/pdq012 MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Cortés, V., Mayer, C., Mohaupt, T., Saueressig, F.: Special geometry of Euclidean supersymmetry. I. Vector multiplets. J. High Energy Phys. 3, 028. (electronic) (2004).  https://doi.org/10.1088/1126-6708/2004/03/028
  13. 13.
    Cortés, V., Schäfer, L.: Geometric structures on Lie groups with flat bi-invariant metric. J. Lie Theory 19(2), 423–437 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26(1), 83–115 (1996).  https://doi.org/10.1216/rmjm/1181072105 MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Etayo, F., Santamaría, R., Trías, U.R.: The geometry of a bi-Lagrangian manifold. Differ. Geom. Appl. 24(1), 33–59 (2006).  https://doi.org/10.1016/j.difgeo.2005.07.002 MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Fulton, W., Harris, J.: Representation Theory, Graduate Texts in Mathematics, vol. 129. Springer, New York (1991).  https://doi.org/10.1007/978-1-4612-0979-9 Google Scholar
  17. 17.
    Gadea, P.M., Masque, J.M.: Classification of almost para-Hermitian manifolds. Rend. Mat. Appl. 11(2), 377–396 (1991)MathSciNetMATHGoogle Scholar
  18. 18.
    Gong, M.P.: Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R). ProQuest LLC, Ann Arbor, MI (1998). http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqdiss&rft_dat=xri:pqdiss:NQ30613. Thesis (Ph.D.)–University of Waterloo (Canada)
  19. 19.
    Gray, A., Hervella, L.: The sixteen classes of almost Hermitian manifolds. Ann. Mat. Pura e Appl. 282, 1–21 (1980)MATHGoogle Scholar
  20. 20.
    Harvey, F.R., Lawson, J.H.B.: Split special Lagrangian geometry. In: Metric and differential geometry, Progr. Math., vol. 297, pp. 43–89. Birkhäuser/Springer, Basel (2012).  https://doi.org/10.1007/978-3-0348-0257-4_3
  21. 21.
    Ivanov, S., Zamkovoy, S.: Parahermitian and paraquaternionic manifolds. Differ. Geom. Appl. 23(2), 205–234 (2005).  https://doi.org/10.1016/j.difgeo.2005.06.002 MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Kim, Y.H., McCann, R.J., Warren, M.: Pseudo-Riemannian geometry calibrates optimal transportation. Math. Res. Lett. 17(6), 1183–1197 (2010).  https://doi.org/10.4310/MRL.2010.v17.n6.a16 MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Libermann, P.: Sur les variétés presque paracomplexes. In: Colloque de topologie et géométrie différentielle, Strasbourg, 1952, no. 5, p. 10. La Bibliothèque Nationale et Universitaire de Strasbourg (1953)Google Scholar
  24. 24.
    Martín Cabrera, F., Swann, A.: Curvature of almost quaternion-Hermitian manifolds. Forum Math. 22(1), 21–52 (2010).  https://doi.org/10.1515/FORUM.2010.002 MathSciNetMATHGoogle Scholar
  25. 25.
    Matsushita, Y.: Counterexamples of compact type to the Goldberg conjecture and various version of the conjecture. In: Topics in contemporary differential geometry, complex analysis and mathematical physics, pp. 222–233. World Sci. Publ., Hackensack (2007).  https://doi.org/10.1142/9789812709806_0024
  26. 26.
    Rossi, F.A.: D-complex structures: cohomological properties and deformations. Ph.D. thesis, Università degli Studi di Milano–Bicocca (2013). http://hdl.handle.net/10281/41976
  27. 27.
    Schäfer, L.: Conical Ricci-flat nearly para-Kähler manifolds. Ann. Global Anal. Geom. 45(1), 11–24 (2014).  https://doi.org/10.1007/s10455-013-9385-x MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Schäfer, L., Schulte-Hengesbach, F.: Nearly pseudo-Kähler and nearly para-Kähler six-manifolds. In: Handbook of pseudo-Riemannian geometry and supersymmetry, IRMA Lect. Math. Theor. Phys., vol. 16, pp. 425–453. Eur. Math. Soc., Zürich (2010).  https://doi.org/10.4171/079-1/12
  29. 29.
    Sekigawa, K.: On some compact Einstein almost Kähler manifolds. J. Math. Soc. Jpn 39(4), 677–684 (1987).  https://doi.org/10.2969/jmsj/03940677 CrossRefMATHGoogle Scholar
  30. 30.
    Steenrod, N.: The Topology of Fibre Bundles. Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton, NJ (1951). Reprinted in Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ (1999)Google Scholar
  31. 31.
    Yau, S.T.: On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation, I. Comm. Pure Appl. Math. 31(3), 339–411 (1978).  https://doi.org/10.1002/cpa.3160310304 MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2017

Authors and Affiliations

  1. 1.Dipartimento di Matematica e ApplicazioniUniversità di Milano BicoccaMilanItaly

Personalised recommendations