2005/10/31 by Stefan Ivanov, Francisco Martı́n Cabrera, Francisco Martín Cabrera
Mathematics · Physics and Astronomy · #Combinatorics #Complex plane #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Geometry #Geometry and complex manifolds #Holomorphic function #Mathematical analysis #Mathematics #Plane curve #Pure mathematics #Submanifold #hep-th #math.DG #msc:53C15 #msc:53C26
paper · pdf · doi:10.1016/j.difgeo.2007.11.006
published as Diff.Geom.Appl.26:113-132,2008 · 25 pages, no figures, some improvements and clarifications are made, final version to appear in Diff. Geom. Appl
arxiv created 2007/06/28 · openalex publication_date 2008/03/06 · arxiv updated 2009/12/01 · openalex created_date 2022/08/13 · openalex updated_date 2026/07/31
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 \citeMR89b:53084 and Calabi \citeMR24 #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 \citeMR89b: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 \citeMR24 #A558. In this case the holomorphic (3,0)-form is parallel with respect to the unique Hermitian connection with totally skew-symmetric torsion.