Almost complex submanifolds of the six sphere

Abstract

The proof is given in [1]. Almost complex manifolds having this property are called nearly Kahlerian. In [2] it is shown that S6 has some almost complex structures, defined by means of a 3-fold vector cross product on R8, which are different from the usual almost complex structure, but nonetheless nearly Kahlerian. Our theorem will apply to these almost complex structures, also. We next give an account of our machinery for describing the geometry of submanifolds. Let M and M be CIO Riemannian manifolds with M isometrically embedded in M. Let X(M) and X(M) denote respectively the Lie algebras of vector fields on M and the restrictions to M of vector fields on M. Then we may write X(M) =X(M) @3(M)'. The configuration tensor is the function T: X(M) X(M) -4(M) defined by the formulas