Results in Mathematics

, Volume 64, Issue 3–4, pp 357–369 | Cite as

Homogeneous 4-Dimensional Kähler–Weyl Structures

  • M. Brozos-Vázquez
  • E. García-Río
  • P. Gilkey
  • R. Vázquez-Lorenzo
Article
  • 97 Downloads

Abstract

Any pseudo-Hermitian or para-Hermitian manifold of dimension 4 admits a unique Kähler–Weyl structure; this structure is locally conformally Kähler if and only if the alternating Ricci tensor ρa vanishes. The tensor ρa takes values in a certain representation space. In this paper, we show that any algebraic possibility Ξ in this representation space can in fact be geometrically realized by a left-invariant Kähler–Weyl structure on a 4-dimensional Lie group in either the Hermitian or the para-Hermitian setting.

Mathematics Subject Classification (2010)

53A15 53C15 15A72 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Belgun F., Moroianu A.: Weyl-parallel forms, conformal products and Einstein-Weyl manifolds. Asian J. Math. 15, 499–520 (2011)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Brozos-Vázquez M., Gilkey P., Nikčević S.: Geometric realizations of curvature. Imperial College Press, (2012)Google Scholar
  3. 3.
    Dunajski, M., Gutowski, J., Sabra, W., Tod, P.: Cosmological Einstein-Maxwell instantons and Euclidean supersymmetry: anti-self-dual solutions. Classical Quantum Gravity 28, no. 2, 025007, 16 pp (2011)Google Scholar
  4. 4.
    Eastwood M., Tod K.P.: Local constraints on Einstein–Weyl geometries: the 3-dimensional case. Ann. Global Anal. Geom. 18, 1–27 (2000)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Eastwood M., Tod K.P.: Local constraints on Einstein–Weyl geometries. J. Reine Angew. Math. 491, 183–198 (1997)MathSciNetGoogle Scholar
  6. 6.
    Ganchev G., Ivanov S.: Semi-symmetric W-metric connections and the W-conformal group. God. Sofij. Univ. Fak. Mat. Inform. 81, 181–193 (1994)MathSciNetGoogle Scholar
  7. 7.
    Gilkey P., Nikčević S.: Kähler and para-Kähler curvature Weyl manifolds. Math. Debrecen 80, 369–384 (2012)CrossRefMATHGoogle Scholar
  8. 8.
    Gilkey, P., Nikčević, S.: (para)-Kähler Weyl Structures. In: Sanchez, M., Ortega, M., Romero, A. (eds.) Recent Trends in Lorentzian Geometry, pp. 335–353. Springer (2013). ISBN 978-1-4614-4896-9Google Scholar
  9. 9.
    Gilkey, P., Nikčević, S.: Kähler–Weyl manifolds of dimension 4. Rendiconti del Seminario Matematico (to appear)Google Scholar
  10. 10.
    Gilkey P., Nikčević S., Simon U.: Curvature properties of Weyl geometries. Results Math. 59, 523–544 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Grumiller D., Jackiw R.: Einstein–Weyl from Kaluza-Klein. Phys. Lett. A 372, 2547–2551 (2008)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Reese Harvey, F., Blaine Lawson, H.: Split special Lagrangian geometry. In: Dai, X., Rong, X. (eds.) Metric and differential geometry. The Jeff Cheeger anniversary volume. Selected papers based on the presentations at the international conference on metric and differential geometry, Tianjin and Beijing, China, 11–15 May 2009. Progress in Mathematics, vol. 297, pp. 43–89. Springer, Berlin (2012)Google Scholar
  13. 13.
    Hayden H.: Sub-spaces of a space with torsion. Proc. Lond. Math. Soc. II 34, 27–50 (1932)CrossRefGoogle Scholar
  14. 14.
    Honda N., Nakata F.: Minitwistor spaces, Severi varieties, and Einstein-Weyl structure. Ann. Global Anal. Geom. 39, 293–323 (2011)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kalafat M.: Geometric invariant theory and Einstein–Weyl geometry. Expo. Math. 29, 220–230 (2011)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Kokarev G., Kotschick D.: Fibrations and fundamental groups of Kähler–Weyl manifolds. Proc. Amer. Math. Soc. 138, 997–1010 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Kerr M.: Homogeneous Einstein–Weyl structures on symmetric spaces. Ann. Global Anal. Geom. 15, 437–445 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Kundu S., Tarafdar M.: Almost Hermitian manifolds admitting Einstein–Weyl connection. Lobachevskii J. Math. 33, 5–9 (2012)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    LeBrun C., Mason L.J.: The Einstein–Weyl equations, scattering maps, and holomorphic disks. Math. Res. Lett. 16, 291–301 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Lunsford D.R.: Gravitation and electrodynamics over SO(3,3). Internat. J. Theoret. Phys. 43, 161–177 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Matzeu P.: Closed Einstein–Weyl structures on compact Sasakian and cosymplectic manifolds. Proc. Edinb. Math. Soc. 54, 149–160 (2011)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Meessena P., Ortnb T., Palomo-Lozano A.: On supersymmetric Lorentzian Einstein–Weyl spaces. J. Geom. Phys. 2, 301–311 (2012)CrossRefGoogle Scholar
  23. 23.
    Ornea L., Verbitsky M.: Einstein–Weyl structures on complex manifolds and conformal version of Monge-Ampère equation. Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 51(99), 339–353 (2008)MathSciNetGoogle Scholar
  24. 24.
    Ovando, G.: Complex, symplectic and Kähler structures on four dimensional Lie groups. Rev. Unión Mat. Argent. 45(2), 55–67 (2004)Google Scholar
  25. 25.
    Ovando G.: Indefinite pseudo-Kähler metrics in dimension four. J. Lie Theory 16, 371–391 (2006)MathSciNetMATHGoogle Scholar
  26. 26.
    Patera J., Sharp R.T., Winternitz P., Zassenhaus Z.: Invariants of real low dimensional Lie algebras. J. Math. Phys. 17, 986–994 (1976)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Pedersen H., Poon Y., Swann A.: The Einstein–Weyl equations in complex and quaternionic geometry. Differential Geom. Appl. 3, 309–321 (1993)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Pedersen H., Swann A.: Riemannian submersions, four manifolds, and Einstein–Weyl geometry. Proc. Lond. Math. Soc. 66, 381–399 (1991)MathSciNetGoogle Scholar
  29. 29.
    Pedersen H., Tod K.: Three-dimensional Einstein–Weyl geometry. Adv. Math. 97, 74–109 (1993)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Rezaei-Aghdam, A., Sephid, M.: Complex and bi-Hermitian structures on four dimensional real Lie algebras. J. Phys. A: Math. Theor. 43, 325210 (14pp) (2010)Google Scholar
  31. 31.
    Svensson, V.: Curvatures of Lie Groups. Bachelor’s thesis, Lund University 2009:K1Google Scholar
  32. 32.
    Vaisman I.: Generalized Hopf manifolds. Geom. Dedicata 13, 231–255 (1982)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Vaisman, I.: A survey of generalized Hopf manifolds. Differential Geometry on Homogeneous Spaces, Proc. Conf. Torino Italy (1983), Rend. Semin. Mat. Torino, Fasc. Spec. 205–221Google Scholar
  34. 34.
    Weyl, H.: Space–time–matter. Dover Publication, New York (1922)Google Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  • M. Brozos-Vázquez
    • 1
  • E. García-Río
    • 2
  • P. Gilkey
    • 3
  • R. Vázquez-Lorenzo
    • 2
  1. 1.Department of MathematicsUniversity of A CoruñaA CoruñaSpain
  2. 2.Faculty of MathematicsUniversity of Santiago de CompostelaSantiago de CompostelaSpain
  3. 3.Mathematics DepartmentUniversity of OregonEugeneUSA

Personalised recommendations