SU(3)-structures on submanifolds of a Spin(7)-manifold
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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.
25 pages, no figures, some improvements and clarifications are made, final version to appear in Diff. Geom. Appl
25 pages, no figures, some improvements and clarifications are made, final version to appear in Diff. Geom. Appl