SU(3)-structures on submanifolds of a Spin(7)-manifold
arXiv:math/0510406 · doi:10.1016/j.difgeo.2007.11.006
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.
25 pages, no figures, some improvements and clarifications are made, final version to appear in Diff. Geom. Appl
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