SU(3)-structures on submanifolds of a Spin(7)-manifold

TL;DR

This paper characterizes local SU(3)-structures on submanifolds of Spin(7)-manifolds using shape operators and structure types, with applications to specific geometric examples involving Cayley planes and minimal surfaces.

Contribution

It provides a detailed analysis of SU(3)-structures on submanifolds of Spin(7)-manifolds, linking geometric properties to shape operators and applying results to classical examples by Bryant and Calabi.

Findings

Characterization of SU(3)-structures via shape operator and Spin(7)-structure type

Identification of conditions for holomorphic volume forms in specific submanifold configurations

Connection between geometric structures and classical Cayley and Calabi examples

Abstract

Local SU(3)-structures on an oriented submanifold of Spin(7)-manifold are determined and their types are characterized in terms of the shape operator and the type of the Spin(7)-structure. An application to Bryant \cite{MR89b:53084} and Calabi \cite{MR24 #A558} examples is given. It is shown that the product of a Cayley plane and a minimal surface lying in a four-dimensional orthogonal Cayley plane with the induced complex structure from the octonions described by Bryant in \cite{MR89b:53084} admits a holomorphic local complex volume form exactly when it lies in a three-plane, i.e. it coincides with the example constructed by Calabi in \cite{MR24 #A558}. In this case the holomorphic (3,0)-form is parallel with respect to the unique Hermitian connection with totally skew-symmetric torsion.