Grassmann geometry and \(J\)-holomorphic curves of a 6-dimensional sphere (Q2752837)

This is a short survey about submanifolds of the 6-dimensional sphere \(S^6\), which admits, as is well known, a homogeneous almost Hermitian structure. The author notes that there does not exist a unified theory of submanifolds of \(S^6\) yet. He gives a summary (together with references) of the now existing results about 1) totally real submanifolds, 2) \(J\)-holomorphic curves, and 3) CR-submanifolds in \(S^6\). The Grassmann geometry for \(S^6\) means the geometry of submanifolds whose tangent bundles belong to some subbundle of the Grassmann bundle \(G^k(TS^6)\to S^6\). By means of the known result (Fukami and Ishihara, 1955), that the automorphism group of the almost Hermitian manifold \(S^6\) coincides with the Lie group \(G_2\) of automorphisms of the octionions, there is given now a classification of Grassmann geometries for \(S^6\). Also some problems are stated. Considering \(J\)-holomorphic curves of \(S^6\) new results are announced about their rigidity and curvature properties, which will be, together with the proofs, published elsewhere.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00035].