Su(3)-structures on Submanifolds of a Spin(7)-manifold

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 Spin(7)-structure. An application to Bryant [5] and Calabi [10] 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 [5] 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 [10]. In this case the holomorphic (3, 0)-form is parallel with respect to the unique Hermitian connection with totally skew-symmetric torsion.