Octonionic Calabi-Yau theorem

Abstract

A new class of Riemannian metrics, called octonionic K\"ahler, is introduced and studied on a certain class of 16-dimensional manifolds. It is an octonionic analogue of K\"ahler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge-Amp\`ere equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi-Yau theorem from K\"ahler geometry.