Octonion-valued forms and the canonical 8-form on Riemannian manifolds with a -structure
arXiv:1808.02452 · doi:10.1007/s12220-019-00209-z
Abstract
It is well known that there is a unique -invariant 8-form on the octonionic plane that naturally yields a canonical differential 8-form on any Riemannian manifold with a weak -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 -invariant 8-form is given. The approach we use generalizes 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.
18 pages, minor language corrections