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

Local SU(3)SU(3)-structures on an oriented submanifold of Spin(7)Spin(7)-manifold are determined and their types are characterized in terms of the shape operator and the type of Spin(7)Spin(7)-structure. An application to Bryant [R.L. Bryant, Submanifolds and special structures on the octonions, J. Diff. Geom. 17 (1982) 185–232] and Calabi [E. Calabi, Construction and properties of some 6-dimensional almost complex manifolds, Trans. Amer. Math. Soc. 87 (1958) 407–438] 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 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 this case the holomorphic (3,0)(3,0)-form is parallel with respect to the unique Hermitian connection with totally skew-symmetric torsion.