Octonion-Valued Forms and the Canonical 8-Form on Riemannian Manifolds with a Spin(9)-Structure

  • Jan KotrbatýEmail author


It is well known that there is a unique Spin(9)-invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak Spin(9)-structure. Over the decades, this invariant has been studied extensively and described in several equivalent ways. In the present article, a new explicit algebraic formula for the Spin(9)-invariant 8-form is given. The approach we use generalises the standard expression of the Kähler 2-form. Namely, the invariant 8-form is constructed only from the two octonion-valued coordinate 1-forms on the octonionic plane. For completeness, analogous expressions for the Kraines form, the Cayley calibration and the associative calibration are also presented.


Spin(9) Octonions Kähler form Kraines form 

Mathematics Subject Classification

53A55 53C10 53C27 14L24 15A21 



I would like to thank my advisor Prof. Thomas Wannerer, who gave me the initial impulse to deal with this problem and supported my work by plenty of useful ideas and comments.


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Authors and Affiliations

  1. 1.Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und InformatikJenaGermany

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