Spinorially Twisted Spin Structures, III: CR Structures

Abstract

We develop a spinorial description of CR structures of arbitrary codimension. More precisely, we characterize almost CR structures of arbitrary codimension on (Riemannian) manifolds by the existence of a Spin\(^{c, r}\) structure carrying a partially pure spinor field. We study various integrability conditions of the almost CR structure in our spinorial setup, including the classical integrability of a CR structure as well as those implied by Killing-type conditions on the partially pure spinor field. In the codimension one case, we develop a spinorial description of strictly pseudoconvex CR manifolds, metric contact manifolds, and Sasakian manifolds. Finally, we study hypersurfaces of Kähler manifolds via partially pure Spin\(^c\) spinors.

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

References

  1. 1.

    Blair, D.E.: Contact Manifolds in Riemannian Geometry, vol. 509. Springer, Berlin (1976)

    Google Scholar 

  2. 2.

    Borisov, L., Salamon, S., Viaclovsky, J.: Twistor geometry and warped product orthogonal complex structures. Duke Math. J. 156(1), 125–166 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Cartan, E.: Lecons sur la théorie des spineurs. Hermann, Paris (1937)

    Google Scholar 

  4. 4.

    Cartan, E.: The Theory of Spinors. Hermann, Paris (1966)

    Google Scholar 

  5. 5.

    Chevalley, C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)

    Google Scholar 

  6. 6.

    Dadok, J., Harvey, R.: Calibrations on \({\mathbb{R}}^6\). Duke Math. J. 4, 1231–1243 (1983)

    Article  MATH  Google Scholar 

  7. 7.

    Dearricott, O.: Lectures on n-Sasakian Manifolds. Geometry of Manifolds with Non-negative Sectional Curvature. Lecture Notes in Mathematics, vol. 2110, pp. 57–109. Springer, Cham (2014)

    Google Scholar 

  8. 8.

    Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds. Progress in Mathematics, vol. 246. Springer, Berlin (2007)

    Google Scholar 

  9. 9.

    Espinosa, M., Herrera, R.: Spinorially twisted Spin structures, I: curvature identities and eigenvalue estimates. Differ. Geom. Appl. 46, 79–107 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Friedrich, T.: Dirac Operator’s in Riemannian Geometry. Graduate Studies in Mathematics, vol. 25. American Mathematical Society, Providence (2000)

    Google Scholar 

  11. 11.

    Harvey, R., Lawson, H.B.: Calibrated geometries. Acta Math. 148, 47–157 (1982)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Herrera, R., Tellez, I.: Twisted partially pure spinors. J. Geom. Phys. 106, 6–25 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Morel, B.: Tenseur d’impulsion-énergie et géométrie spinorielle extrinsèque, Ph.D. thesis, Institut Elie Cartan (2002)

  15. 15.

    Morgan, J.W.: The Seiberg-Witten equations and applications to the topology of smooth four-manifolds. Mathematical Notes, vol. 44. Princeton University Press, Princeton, NJ. ISBN: 0-691-02597-5 , viii+128 pp (1996)

  16. 16.

    Nakad, R.: The energy-momentum tensor on \(Spin^{c}\) manifolds. IJGMMP 8(2), 345–365 (2011)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Nicolaescu, L.: Geometric connections and geometric Dirac operators on contact manifolds. Differ. Geom. Appl. 22, 355–378 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Ornea, L., Verbitsky, M.: Sasakian structures on CR-manifolds. Geom Dedicata 125, 159–173 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Penrose, R.: Twistor theory, its aims and achievements. In: Isham, C.J., Penrose, R., Sciama, D. (eds.) Quantum Gravity: An Oxford Symposium, pp. 268–407. Oxford University Press, Oxford (1975)

    Google Scholar 

  20. 20.

    Penrose, P., Rindler, W.: Spinors and Space-Time, vol. 1. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  21. 21.

    Penrose, R., Rindler, W.: Spinors and Space-Time, vol. 2. Cambridge University Press, Cambridge (1986)

    Google Scholar 

  22. 22.

    Petit, R.: Spin\(^c\) structures and Dirac operators on contact manifolds. Differ. Geom. Appl. 22, 229–252 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

The authors are grateful to Oussama Hijazi for his encouragement and valuable comments. The authors thank Helga Baum and the Institute of Mathematics of the University of Humboldt-Berlin for their hospitality and support. The first author would also like to thank the hospitality and support of the International Centre for Theoretical Physics and the Institut des Hautes Études Scientifiques. The second author gratefully acknowledges the support and hospitality of the Centro de Investigación en Matemáticas A.C. (CIMAT). Rafael Herrera was partially supported by grants of CONACyT, LAISLA (CONACyT-CNRS), and the IMU Berlin Einstein Foundation Program. Iván Téllez was supported by a CONACyT scholarship.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Rafael Herrera.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Herrera, R., Nakad, R. & Téllez, I. Spinorially Twisted Spin Structures, III: CR Structures. J Geom Anal 28, 3223–3277 (2018). https://doi.org/10.1007/s12220-017-9958-1

Download citation

Keywords

  • Twisted spin structure
  • Partially pure spinor
  • Twisted Dirac operator
  • CR structure

Mathematics Subject Classification

  • 53C10
  • 53C25
  • 53C27
  • 58J50
  • 58J60
  • 32V05