Annals of Global Analysis and Geometry

, Volume 54, Issue 2, pp 301–313 | Cite as

Local geometry of even clifford structures on conformal manifolds

  • Charles Hadfield
  • Andrei Moroianu


We introduce the concept of a Clifford–Weyl structure on a conformal manifold, which consists of an even Clifford structure parallel with respect to the tensor product of a metric connection on the Clifford bundle and a Weyl structure on the manifold. We show that the Weyl structure is necessarily closed except for some “generic” low-dimensional instances, where explicit examples of non-closed Clifford–Weyl structures can be constructed.


Even Clifford structures Conformal manifolds Weyl structures 

Mathematics Subject Classification

Primary 53C26 53A30 


  1. 1.
    Arizmendi, G., Herrera, R.: Centralizers of spin subalgebras. J. Geom. Phys. 97, 77–92 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Arizmendi, G., Hadfield, C.: Twistor Spaces of Riemannian manifolds with even clifford structures. Ann. Glob. Anal. Geom. (2016).
  3. 3.
    Arizmendi, G., Herrera, R., and Santana, N.: Almost even-Clifford Hermitian manifolds with large automorphism group (2015). arXiv:1506.03713
  4. 4.
    Arizmendi, G., Garcia-Pulido, A., Herrera, R.: A note on the geometry and topology of almost even-Clifford Hermitian manifolds (2016). arXiv:1606.00774
  5. 5.
    Belgun, F., Moroianu, A.: Weyl-parallel forms, conformal products and Einstein–Weyl manifolds. Asian J. Math. 15, 499–520 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bourguignon, J.-P., Hijazi, O., Milhorat, J.-L., Moroianu, A., Moroianu, S.: A Spinorial Approach to Riemannian and Conformal Geometry. EMS Monographs in Mathematics. European Mathematical Society (EMS), Zürich (2015)CrossRefzbMATHGoogle Scholar
  7. 7.
    Garcia-Pulido, A., Herera, R.: Rigidity and vanishing theorems for almost even-Clifford Hermitian manifolds (2016). arXiv:1609.01509
  8. 8.
    Moroianu, A., Pilca, M.: Higher rank homogeneous Clifford structures. J. Lond. Math. Soc. 87(2), 384–400 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Moroianu, A., Semmelmann, U.: Clifford structure on Riemannian manifolds. Adv. Math. 228, 940–967 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Ornea, L.: Weyl structures on quaternionic manifolds. A state of the art (2001). arXiv:math/0105041
  11. 11.
    Parton, M., Piccini, P.: The even Clifford structure of the fourth Severi variety. Complex Manifolds 2, 89–104 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Parton, M., Piccini, P., Vuletescu, V.: Clifford systems in octonionic geometry (to appear in Rend. Sem. Mat. Torino, volume in memory of Sergio Console). arXiv:1511.06239

Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.École Normale SupérieureParis Cedex 05France
  2. 2.Laboratoire de Mathématiques de Versailles, UVSQ, CNRSUniversité, Paris-SaclayVersaillesFrance

Personalised recommendations