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 



A part of this project has been discussed during HVL’s visit to the Osaka City University in December 2015. She thanks Professor Ohnita for his invitation to Osaka and his hospitality. HVL and LS also thank the Max Planck Institute for Mathematics in the Sciences in Leipzig for its hospitality during extended visits. We also thank the referee for many helpful comments which enabled us to significantly improve the manuscript.


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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

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