Generalized Killing spinors and Lagrangian graphs

We study generalized Killing spinors on the standard sphere S3S3, which turn out to be related to Lagrangian embeddings in the nearly Kähler manifold S3×S3S3×S3 and to great circle flows on S3S3. Using our methods we generalize a well known result of Gluck and Gu [5] concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of S3×S3S3×S3 has at least three connected components.