Skip to main content
SpringerLink
Log in

Killing spinor-valued forms and their integrability conditions

  • Published:
Annals of Global Analysis and Geometry Aims and scope Submit manuscript

A Correction to this article was published on 10 March 2021

This article has been updated

Abstract

We study invariant systems of PDEs defining Killing vector-valued forms, and then we specialize to Killing spinor-valued forms. We give a detailed treatment of their prolongation and integrability conditions by relating the pointwise values of solutions to the curvature of the underlying manifold. As an example, we completely solve the equations on model spaces of constant curvature producing brand-new solutions which do not come from the tensor product of Killing spinors and Killing–Yano forms.

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

Change history

References

  1. Armstrong, S.: Projective holonomy. II. Cones and complete classifications. Ann. Global Anal. Geom. 33(2), 137–160 (2008)

    Article  MathSciNet  Google Scholar 

  2. Bär, C.: Real Killing spinors and holonomy. Comm. Math. Phys. 154(3), 509–521 (1993)

    Article  MathSciNet  Google Scholar 

  3. Bailey, T.N., Eastwood, M.G., Gover, A.R.: Thomas’s structure bundle for conformal, projective and related structures. Rocky Mountain J. Math. 24(4), 1191–1217 (1994)

    Article  MathSciNet  Google Scholar 

  4. Baum, H., Friedrich, T., Grunewald, R., Kath, I.: Twistors and Killing spinors on Riemannian manifolds. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics], 124. B. G. Teubner Verlagsgesellschaft mbH, Stuttgart (1991)

  5. Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and Spin geometry. Math. Z. 249(3), 545–580 (2005)

    Article  MathSciNet  Google Scholar 

  6. Čap, A., Slovák J.: Parabolic Geometries I: Background and General Theory. Mathematical Surveys and Monographs, 154. American Mathematical Society, Providence (2009)

  7. Duff, M.J., Nilsson, B.E.W., Pope, C.N.: Kaluza-Klein supergravity. Phys. Rep. 130(1–2), 1–142 (1986)

    Article  MathSciNet  Google Scholar 

  8. Duff, M.J., Pope, C.N.: Kaluza-Klein supergravity and the seven sphere. In: Ferrara, S., Taylor J.G., van Nieuwenhuizen, P. (eds.) Supersymmetry and Supergravity ’82, Proceedings of the Trieste September 1982 School, pp. 183–228. World Scientific Press, Singapore (1983)

  9. Eastwood, M.: Notes on projective differential geometry. In: Eastwood, M., Miller, W. (eds.) Symmetries and Overdetermined Systems of Partial Differential Equations. The IMA Volumes in Mathematics and its Applications, 144, pp. 41–60. Springer Science & Business Media, New York (2008)

  10. Friedrich, T.: Der erste Eigenwert des Dirac-Operators einer kompakten, Riemannschen Mannigfaltigkeit nichtnegativer Skalarkrümmung. Math. Nachr. 97(1), 117–146 (1980)

    Article  MathSciNet  Google Scholar 

  11. Gover, A.R., Neusser, K., Willse, T.: Projective geometry of Sasaki-Einstein structures and their compactification. Dissertationes Math. 546, 1–64 (2019)

    Article  MathSciNet  Google Scholar 

  12. Hinterleitner, I., Berezovski, V., Chepurna, E., Mikeš, J.: On the concircular vector fields of spaces with affine connection. Acta Math. Acad. Paedagog. Nyházi. (N.S.) 33(1), 53–60 (2017)

    MathSciNet  MATH  Google Scholar 

  13. Hammerl, M., Somberg, P., Souček, V., Šilhan, J.: On a new normalization for tractor covariant derivatives. J. Eur. Math. Soc. (JEMS) 14(6), 1859–1883 (2012)

    Article  MathSciNet  Google Scholar 

  14. Hammerl, M., Somberg, P., Souček, V., Šilhan, J.: Invariant prolongation of overdetermined PDEs in projective, conformal, and Grassmannian geometry. Ann. Global Anal. Geom. 42(1), 121–145 (2012)

    Article  MathSciNet  Google Scholar 

  15. Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245(3), 503–527 (2003)

    Article  MathSciNet  Google Scholar 

  16. Stepanov, S.E.: The Killing–Yano tensor. Theor. Math. Phys. 134(3), 333–338 (2003)

    Article  MathSciNet  Google Scholar 

  17. Somberg, P., Zima, P.: Killing spinor-valued forms and the cone construction. Arch. Math. (Brno) 52(5), 341–355 (2016)

    Article  MathSciNet  Google Scholar 

  18. Yano, K.: Concircular geometry I. Concircular transformations. Proc. Imp. Acad. Tokyo 16(6), 195–200 (1940)

    MathSciNet  MATH  Google Scholar 

  19. Yano, K.: Some remarks on tensor fields and curvature. Ann. of Math. 2(55), 328–347 (1952)

    Article  MathSciNet  Google Scholar 

  20. Zima, P.: Software package for solving Killing-type equations on homogeneous spaces (2017). https://github.com/petr-zima/mac-homog

Download references

Funding

The authors gratefully acknowledge the support of the Grants GA19-06357S, GAUK 700217 and SVV-2017-260456.

Author information

Authors and Affiliations

  1. Mathematical Institute of Charles University, Sokolovská 83, Praha 8, Karlín, Czech Republic

    Petr Somberg & Petr Zima

Authors
  1. Petr Somberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr Zima
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petr Zima.

Additional information

Publisher's Note

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

The authors gratefully acknowledge the support of the Grants GA19-06357S, GAUK 700217 and SVV-2017-260456.

The original online version of this article was revised: The funding needs to read as follows: The authors gratefully acknowledge the support of the Grants GA19-06357S, GAUK 700217 and SVV-2017-260456.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Somberg, P., Zima, P. Killing spinor-valued forms and their integrability conditions. Ann Glob Anal Geom 58, 351–384 (2020). https://doi.org/10.1007/s10455-020-09730-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10455-020-09730-9

Keywords

Mathematics Subject Classification

Search

Navigation