The Frölicher–Nijenhuis bracket and the geometry of \(G_2\)-and Spin(7)-manifolds

  • Kotaro Kawai
  • Hông Vân Lê
  • Lorenz Schwachhöfer


We extend the characterization of the integrability of an almost complex structure J on differentiable manifolds via the vanishing of the Frölicher–Nijenhuis bracket \([J, J]^{FN}\) to an analogous characterization of torsion-free \(G_2\)-structures and torsion-free \(\text{Spin(7) }\)-structures. We also explain the Fernández–Gray classification of \(G_2\)-structures and the Fernández classification of \(\text{Spin(7) }\)-structures in terms of the Frölicher–Nijenhuis bracket.


Frölicher–Nijenhuis bracket \(G_2\)-manifold \(\text{Spin(7) }\)-manifold Fernández–Gray classification Fernández classification 

Mathematics Subject Classification

53C25 53C29 


  1. 1.
    Brown, R.B., Gray, A.: Vector cross product. Comment. Math. Helv. 42, 222–236 (1967)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Bryant, R.: Metrics with exceptional holonomy. Ann. Math. 126, 525–576 (1987)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bryant, R.: Some remarks on \(G_2\)-structures. In: Proceedings of Gökova Geometry-Topology Conference 2005, Gökova Geometry/Topology Conference (GGT), Gökova, pp. 75–109 (2006)Google Scholar
  4. 4.
    Fernández, M.: A classification of Riemannian manifolds with structure group Spin(7). Ann. Mat. Pura Appl. 143, 101–122 (1986)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Fernández, M., Gray, A.: Riemannian manifolds with structure group \(G_{2}\). Ann. Mat. Pura Appl. 32, 19–45 (1982)CrossRefMATHGoogle Scholar
  6. 6.
    Frölicher, A., Nijenhuis, A.: Theory of vector-valued differential forms. I. Derivations of the graded ring of differential forms. Indag. Math. 18, 338–359 (1956)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Frölicher, A., Nijenhuis, A.: Some new cohomology invariants for complex manifolds. I, II. Indag. Math. 18(540–552), 553–564 (1956)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Harvey, R., Lawson, H.B.: Calibrated geometry. Acta Math. 148, 47–157 (1982)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Humphreys, J.: Introduction to Lie Algebras and Representation Theory, Second Printing, Graduate Texts in Mathematics, vol. 9. Springer, New York (1978)Google Scholar
  10. 10.
    Karigiannis, S.: Deformations of \(G_2\) and Spin(7)-structures. Can. J. Math. 57, 1012–1055 (2005)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Kawai, K.: Deformations of homogeneous associative submanifolds in nearly parallel \(G_2\)-manifolds Asian J. Math. 21, 429–462 (2017). arXiv: 1407.8046
  12. 12.
    Kolar, I., Michor, P.W., Slovak, J.: Natural Operators in Differential Geometry. Springer, Berlin (1993)CrossRefMATHGoogle Scholar
  13. 13.
    Lê , H. V., Vanžura, J.: McLean’s second variation formula revisited. J. Geom. Phys. 113, 188–196. arXiv:1605.01267 (2017)
  14. 14.
    McLean, R.: Deformations of calibrated submanifolds. Commun. Anal. Geom. 6, 705–747 (1998)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Salmon, D., Walpuski, T.: Notes on the octonions. arXiv: 1005.2820

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  • Kotaro Kawai
    • 1
    • 2
  • Hông Vân Lê
    • 3
  • Lorenz Schwachhöfer
    • 4
  1. 1.Graduate School of Mathematical SciencesUniversity of TokyoMeguroJapan
  2. 2.Gakushuin UniversityToshimaJapan
  3. 3.Institute of Mathematics CASPrague 1Czech Republic
  4. 4.Fakultät für MathematikTechnische Universität DortmundDortmundGermany

Personalised recommendations