The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3223–3277 | Cite as

Spinorially Twisted Spin Structures, III: CR Structures

  • Rafael HerreraEmail author
  • Roger Nakad
  • Iván Téllez


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.


Twisted spin structure Partially pure spinor Twisted Dirac operator CR structure 

Mathematics Subject Classification

53C10 53C25 53C27 58J50 58J60 32V05 



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.


  1. 1.
    Blair, D.E.: Contact Manifolds in Riemannian Geometry, vol. 509. Springer, Berlin (1976)CrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cartan, E.: Lecons sur la théorie des spineurs. Hermann, Paris (1937)zbMATHGoogle Scholar
  4. 4.
    Cartan, E.: The Theory of Spinors. Hermann, Paris (1966)zbMATHGoogle Scholar
  5. 5.
    Chevalley, C.: The Algebraic Theory of Spinors. Columbia University Press, New York (1954)CrossRefzbMATHGoogle Scholar
  6. 6.
    Dadok, J., Harvey, R.: Calibrations on \({\mathbb{R}}^6\). Duke Math. J. 4, 1231–1243 (1983)CrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  8. 8.
    Dragomir, S., Tomassini, G.: Differential Geometry and Analysis on CR Manifolds. Progress in Mathematics, vol. 246. Springer, Berlin (2007)zbMATHGoogle Scholar
  9. 9.
    Espinosa, M., Herrera, R.: Spinorially twisted Spin structures, I: curvature identities and eigenvalue estimates. Differ. Geom. Appl. 46, 79–107 (2016)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Herrera, R., Tellez, I.: Twisted partially pure spinors. J. Geom. Phys. 106, 6–25 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hitchin, N.: Harmonic spinors. Adv. Math. 14, 1–55 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Morel, B.: Tenseur d’impulsion-énergie et géométrie spinorielle extrinsèque, Ph.D. thesis, Institut Elie Cartan (2002)Google Scholar
  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)Google Scholar
  16. 16.
    Nakad, R.: The energy-momentum tensor on \(Spin^{c}\) manifolds. IJGMMP 8(2), 345–365 (2011)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Nicolaescu, L.: Geometric connections and geometric Dirac operators on contact manifolds. Differ. Geom. Appl. 22, 355–378 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ornea, L., Verbitsky, M.: Sasakian structures on CR-manifolds. Geom Dedicata 125, 159–173 (2007)MathSciNetCrossRefzbMATHGoogle 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)CrossRefzbMATHGoogle Scholar
  21. 21.
    Penrose, R., Rindler, W.: Spinors and Space-Time, vol. 2. Cambridge University Press, Cambridge (1986)CrossRefzbMATHGoogle Scholar
  22. 22.
    Petit, R.: Spin\(^c\) structures and Dirac operators on contact manifolds. Differ. Geom. Appl. 22, 229–252 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Witten, E.: Monopoles and four-manifolds. Math. Res. Lett. 1(6), 769–796 (1994)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Centro de Investigación en MatemáticasGuanajuatoMexico
  2. 2.Department of Mathematics and Statistics, Faculty of Natural and Applied SciencesNotre Dame University-LouaizéZouk MikaelLebanon

Personalised recommendations