Annals of Global Analysis and Geometry

, Volume 50, Issue 4, pp 381–394 | Cite as

Conformal Killing 2-forms on four-dimensional manifolds

  • Adrián Andrada
  • María Laura Barberis
  • Andrei Moroianu
Article
  • 107 Downloads

Abstract

We study four-dimensional simply connected Lie groups G with a left invariant Riemannian metric g admitting non-trivial conformal Killing 2-forms. We show that either the real line defined by such a form is invariant under the group action, or the metric is half-conformally flat. In the first case, the problem reduces to the study of invariant conformally Kähler structures, whereas in the second case, the Lie algebra of G belongs (up to homothety) to a finite list of families of metric Lie algebras.

Keywords

Conformal Killing forms Invariant conformally Kähler structures Half-conformally flat metrics 

Mathematics Subject Classification

53C25 53C15 53C30 

References

  1. 1.
    Alekseevsky, D.V., Cortes, V., Hasegawa, K., Kamishima, Y.: Homogeneous locally conformally Kähler and Sasaki manifolds. Inter. J. Math. 26(6), 1–29 (2015)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Andrada, A., Barberis, M.L., Dotti, I.G., Ovando, G.: Product structures on four dimensional solvable Lie algebras. Homol. Homot. Appl. 7(1), 9–37 (2005)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Andrada, A., Barberis, M.L., Dotti, I.: Invariant solutions to the conformal Killing-Yano equation on Lie groups. J. Geom. Phys. 94, 199–208 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Andrada, A., Origlia, M.: Locally conformally Kähler structures on unimodular Lie groups. Geom. Dedic. 179(1), 197–216 (2015)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Barberis, M.L., Dotti, I., Santillán, O.: The Killing-Yano equation on Lie groups. Class. Quantum Grav. 29, 1–10 (2012)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    De Smedt, V., Salamon, S.: Anti-self-dual metrics on Lie groups. Differential geometry and integrable systems (Tokyo, 2000). Contemporary Mathematics, vol. 308, pp. 63–75. American Mathematical Society, Providence, RI (2002)Google Scholar
  7. 7.
    Gauduchon, P., Moroianu, A.: Killing 2-forms in dimension 4, arXiv:1506.04292
  8. 8.
    Kasuya, H.: Vaisman metrics on solvmanifolds and Oeljeklaus-Toma manifolds. Bull. Lond. Math. Soc. 45(1), 15–26 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Kuiper, N.: On conformally flat spaces in the large. Ann. Math. 50, 916–924 (1950)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Maier, S.: Conformally flat Lie groups. Math. Z. 228, 155–175 (1998)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Milnor, J.: Curvature of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Pontecorvo, M.: On twistor spaces of anti-self-dual Hermitian surfaces. Trans. Am. Math. Soc. 331, 653–661 (1992)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Sawai, H.: Locally conformal Kähler structures on compact solvmanifolds. Osaka J. Math. 49, 1087–1102 (2012)MathSciNetMATHGoogle Scholar
  14. 14.
    Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 243, 503–527 (2003)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Adrián Andrada
    • 1
  • María Laura Barberis
    • 1
  • Andrei Moroianu
    • 2
  1. 1.FAMAF-CIEM, Universidad Nacional de CórdobaCiudad UniversitariaCordobaArgentina
  2. 2.Laboratoire de Mathématiques de VersaillesUVSQ, CNRS, Université Paris-SaclayVersaillesFrance

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