Generalized Killing spinors and Lagrangian graphs

We study generalized Killing spinors on the standard sphere ${\mathbb{S}}^3$, which turn out to be related to Lagrangian embeddings in the nearly K"ahler manifold $S^3\times S^3$ and to great circle flows on ${\mathbb{S}}^3$. Using our methods we generalize a well known result of Gluck and Gu concerning divergence-free geodesic vector fields on the sphere and we show that the space of Lagrangian submanifolds of $S^3\times S^3$ has at least three connected components.