Skip to main content
SpringerLink
Log in

Lorentzian connections with parallel twistor-free torsion

  • Published:
Collectanea Mathematica Aims and scope Submit manuscript

Abstract

We describe Lorentzian manifolds that admit metric connections with parallel torsion having zero twistorial component and non-zero vectorial component. We also describe Lorentzian manifolds admitting metric connections with closed parallel skew-symmetric torsion.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Agricola, I., Friedrich, Th.: On the holonomy of connections with skew-symmetric torsion. Math. Annal. 328(4), 711–748 (2004)

    Article  MathSciNet  Google Scholar 

  2. Agricola, I.: Non-integrable geometries, torsion, and holonomy. In: Handbook of pseudo–Riemannian geometry and supersymmetry, IRMA, EMS, 277–346 (2010)

  3. Agricola, I., Ferreira, C., Friedrich, Th.: The classification of naturally reductive homogeneous spaces in dimensions \(n\le 6\). Diff. Geom. Appl. 39, 59–92 (2015)

    Article  Google Scholar 

  4. Alekseevsky, D., Cortes, V., Galaev, A., Leistner, Th.: Cones over pseudo-Riemannian manifolds and their holonomy. J. Reine Angew. Math. 635, 23–69 (2009)

    MathSciNet  Google Scholar 

  5. Blau, M., O’Loughlin, M.: Homogeneous plane waves. Nuclear Phys. B 654(1–2), 135–176 (2003)

    Article  MathSciNet  Google Scholar 

  6. Bohle, C.: Killing spinors on Lorentzian manifolds. J. Geom. Phys. 45, 285–308 (2003)

    Article  MathSciNet  Google Scholar 

  7. Brozos-Vázquez, M., García-Río, E., Gilkey, P., Nikčević, S., Vázquez-Lorenzo, R.: The geometry of Walker manifolds, Synth. Lect. Math. Stat., 5, Morgan & Claypool Publishers, Williston, VT (2009)

  8. Calvaruso, G., López, M.C.: Pseudo-Riemannian Homogeneous Structures. Springer, Berlin (2019)

    Book  Google Scholar 

  9. Cleyton, R., Moroianu, A., Semmelmann, U.: Metric connections with parallel skew-symmetric torsion. Adv. Math. 378, 107519 (2021)

    Article  MathSciNet  Google Scholar 

  10. Coley, A., Hervik, S., Papadopoulos, G., Pelavas, N.: Kundt spacetimes. Class. Quantum Grav. 26 no. 10, arc. num. 105016 (2009)

  11. Ernst, I., Galaev, A.S.: On Lorentzian connections with parallel skew torsion. Doc. Math. 27, 2333–2383 (2022)

    Article  MathSciNet  Google Scholar 

  12. Figueroa-O’Farrill, J., Philip, S., Meessen, P.: Homogeneity and plane-wave limits. J. High Energy Phys. 05(05) (2005)

  13. Friedrich, Th., Ivanov, S.: Parallel spinors and connections with skew-symmetric torsion in string theory. Asian J. Math. 6(2), 303–335 (2002)

    Article  MathSciNet  Google Scholar 

  14. Gadea, P.M., Oubiña, J.A.: Reductive homogeneous pseudo-Riemannian manifolds. Monatsh. Math. 124, 17–34 (1997)

    Article  MathSciNet  Google Scholar 

  15. Leistner, Th.: Screen bundles of Lorentzian manifolds and some generalisations of pp-waves. J. Geom. Phys. 56(10), 2117–2134 (2006)

    Article  MathSciNet  Google Scholar 

  16. Leistner, T., Schliebner, D.: Completeness of compact Lorentzian manifolds with abelian holonomy. Math. Annalen 364, 1469–1503 (2016)

    Article  MathSciNet  Google Scholar 

  17. Meessen, P.: Homogeneous Lorentzian spaces admitting a homogeneous structure of type \(T_1 \oplus T_3\). J. Geom. Phys. 56, 754–761 (2006)

    Article  MathSciNet  Google Scholar 

  18. Murcia, Á., Shahbazi, C.S.: Contact metric three manifolds and Lorentzian geometry with torsion in six-dimensional supergravity. J. Geom. Phys. 158, 103868 (2020)

    Article  MathSciNet  Google Scholar 

  19. Montesinos Amilibia, A.: Degenerate homogeneous structures of type \(S_1\) on pseudo-Riemannian manifolds. Rocky Mt. J. Math. 31, 561–579 (2001)

    MathSciNet  Google Scholar 

  20. Moroianu, A., Pilca, M.: Metric connections with parallel twistor-free torsion. Int. J. Math. 32, arc. num. 2140011 (2021)

  21. Strominger, A.: Superstrings with torsion. Nucl. Phys. B 274, 253–284 (1986)

    Article  MathSciNet  Google Scholar 

  22. Tricerri, F., Vanhecke, L.: Homogeneous Structures on Riemannian Manifolds. London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge (1983)

    Book  Google Scholar 

Download references

Acknowledgements

The authors are thankful to Andrei Moroianu, Thomas Leistner and Eivind Schneider for useful discussions. The authors are grateful to the Reviewer for the important comments. I.E. was supported by grant MUNI/A/1099/2022 of Masaryk University. A.G. acknowledges institutional support of University of Hradec Králové.

Author information

Authors and Affiliations

  1. Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37, Brno, Czech Republic

    Igor Ernst

  2. Faculty of Science, University of Hradec Králové, Rokitanského 62, 500 03, Hradec Králové, Czech Republic

    Anton S. Galaev

Authors
  1. Igor Ernst
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anton S. Galaev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anton S. Galaev.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ernst, I., Galaev, A.S. Lorentzian connections with parallel twistor-free torsion. Collect. Math. (2023). https://doi.org/10.1007/s13348-023-00430-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s13348-023-00430-8

Keywords

Mathematics Subject Classification

Search

Navigation