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

  • Kotaro Kawai
  • Hông Vân Lê
  • Lorenz Schwachhöfer
Article

Abstract

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.

Keywords

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

Mathematics Subject Classification

53C25 53C29 

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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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