Octonion-Valued Forms and the Canonical 8-Form on Riemannian Manifolds with a Spin (9)-Structure
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Octonion-Valued Forms and the Canonical 8-Form on Riemannian Manifolds with a Spin(9)-Structure Jan Kotrbatý1 Received: 13 September 2018 © Mathematica Josephina, Inc. 2019
Abstract 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. Keywords Spin(9) · Octonions · Kähler form · Kraines form Mathematics Subject Classification 53A55 · 53C10 · 53C27 · 14L24 · 15A21
1 Introduction One of the most common features of the reals as well as of the complex numbers is the compatibility of their product with the norm. Famously, besides R and C, this is an exclusive property of only two other spaces: the four-dimensional algebra H of quaternions and the eight-dimensional algebra O of octonions (sometimes also referred to as Cayley numbers or octaves). Since this striking result of Hurwitz [51] was published, the fact that precisely four normed division algebras exist has turned out to be extremely generic as it has been observed to underlie a wide variety of other classification theorems. Remember,
Supported by DFG Grant WA3510/1-1.
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Jan Kotrbatý [email protected] Friedrich-Schiller-Universität Jena, Fakultät für Mathematik und Informatik, Jena, Germany
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for instance, simple Lie algebras: the three classical series correspond to R, C and H (see e.g. [53], §I.8) while the five exceptions are closely tied to O (see §4 of the excellent survey [16]). Formally real Jordan algebras are categorised similarly, see [52]. Or, as shown by Adams [3], there exist precisely four Hopf fibrations; see §2.2 for construction of the ‘octonionic’ one, the others are obtained simply replacing O by the other normed division algebras (see [16,43]). And the list continues. In the forties, Borel [30] and Montgomery with Samelson [55] classified compact connected Lie groups acting transitively and effectively on a unit sphere. These are S O(n) = S O(Rn ); U (n), SU (n) ⊂ S O(Cn ); Sp(n), Sp(n)U (1), Sp(n)Sp(1) ⊂ S O(Hn );
(1)
G 2 ⊂ S O(Im O); Spin(7) ⊂ S O(O); Spin(9) ⊂ S O(O2 ); in each case the action on the unit sphere in V is inherited from S O(V ). The reason to write Hn instead of R4n , Im O (octonions with zero real part, see §2.1) rather than R7 , etc., is that all the listed groups are naturally realised in terms of the respective normed-division-algebra structures (see §2.2 for Spin(9)). A few years later, Berger’s classification [
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