Advertisement

Killing Forms on 2-Step Nilmanifolds

  • Viviana del BarcoEmail author
  • Andrei Moroianu
Article
  • 5 Downloads

Abstract

We study left-invariant Killing k-forms on simply connected 2-step nilpotent Lie groups endowed with a left-invariant Riemannian metric. For \(k=2,3\), we show that every left-invariant Killing k-form is a sum of Killing forms on the factors of the de Rham decomposition. Moreover, on each irreducible factor, non-zero Killing 2-forms define (after some modification) a bi-invariant orthogonal complex structure and non-zero Killing 3-forms arise only if the Riemannian Lie group is naturally reductive when viewed as a homogeneous space under the action of its isometry group. In both cases, \(k=2\) or \(k=3\), we show that the space of left-invariant Killing k-forms of an irreducible Riemannian 2-step nilpotent Lie group is at most one-dimensional.

Keywords

Killing forms 2-step nilpotent Lie groups Naturally reductive homogeneous spaces 

Mathematics Subject Classification

53D25 22E25 53C30 

Notes

Acknowledgements

We would like to thank the anonymous referee for several pertinent remarks and suggestions, which led to the low-dimensional classification results in Theorems 4.14 and 5.14, and to the global improvement of the presentation.

References

  1. 1.
    Andrada, A., Dotti, I.: Killing-Yano 2-forms on 2-step nilpotent Lie groups. arXiv: 1907.03662
  2. 2.
    Barberis, M.L., Dotti, I., Santillán, O.: The Killing–Yano equation on Lie groups. Class. Quantum Gravity 29(6), 1–10 (2012)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Barberis, M.L., Moroianu, A., Semmelmann, U.: Generalized vector cross products and Killing forms on negatively curved manifolds. Geom. Dedicata (2019).  https://doi.org/10.1007/s10711-019-00467-9
  4. 4.
    Belgun, F., Moroianu, A., Semmelmann, U.: Killing forms on symmetric spaces. Differ. Geom. Appl. 24, 215–222 (2006)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Cleyton, R., Moroianu, A., Semmelmann, U.: Metric connections with parallel skew-symmetric torsion. arXiv:1807.00191
  6. 6.
    del Barco, V., Moroianu, A.: Symmetric Killing tensors on nilmanifolds. arXiv:1811.09187
  7. 7.
    Eberlein, P.: Geometry of \(2\)-step nilpotent groups with a left invariant metric. Ann. Sci. École Norm. Sup. (4) 27(5), 611–660 (1994)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gauduchon, P., Moroianu, A.: Killing \(2\)-forms in dimension 4. In: S. Chiossi, A. Fino, E. Musso, F. Podestà, L. Vezzoni (eds.) Special metrics and group actions in geometry, volume 23 of Springer INdAM series (2017)Google Scholar
  9. 9.
    Gordon, C.: Naturally reductive homogeneous Riemannian manifolds. Can. J. Math. 37, 467–487 (1985)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Lauret, J.: Homogeneous nilmanifolds attached to representations of compact Lie groups. Manuscr. Math. 99, 287–309 (1999)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Magnin, L.: Sur les algèbres de Lie nilpotentes de dimension \(\le 7\). J. Geom. Phys. 3(1), 119–144 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Moroianu, A.: Conformally related Riemannian metrics with non-generic holonomy. J. Reine Angew. Math. 755, 279–292 (2019)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Moroianu, A., Semmelmann, U.: Killing forms on Quaternion-Kähler manifolds. Ann. Glob. Anal. Geom. 28, 319–335 (2005)CrossRefGoogle Scholar
  14. 14.
    Penrose, R., Walker, M.: On quadratic first integrals of the geodesic equations for type \(\{22\}\) spacetimes. Commun. Math. Phys. 18, 265–274 (1970)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245, 503–527 (2003)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Semmelmann, U.: Killing forms on \({\rm G}_2\)- and \({{\rm Spin}}_7\)-manifolds. J. Geom. Phys. 56, 1752–1766 (2006)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Wilson, E.: Isometry groups on homogeneous nilmanifolds. Geom. Dedicata 12, 337–346 (1982)MathSciNetCrossRefGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques d’OrsayUniversidad Nacional de Rosario, CONICETRosarioArgentina
  2. 2.Univ. Paris-SudOrsayFrance
  3. 3.Laboratoire de Mathématiques d’OrsayUniv. Paris-Sud, CNRS, Université Paris-SaclayOrsayFrance

Personalised recommendations