Octonionic Calabi–Yau Theorem

On a certain class of 16-dimensional manifolds a new class of Riemannian metrics, called octonionic Kähler, is introduced and studied. It is an octonionic analogue of Kähler metrics on complex manifolds and of HKT-metrics of hypercomplex manifolds. Then for this class of metrics an octonionic version of the Monge–Ampère equation is introduced and solved under appropriate assumptions. The latter result is an octonionic version of the Calabi–Yau theorem from Kähler geometry.