Generalized Killing spinors and Lagrangian graphs

TL;DR

This paper explores generalized Killing spinors on the 3-sphere, linking them to Lagrangian embeddings in a nearly Kähler manifold and analyzing their geometric properties, including divergence-free flows and the topology of Lagrangian submanifolds.

Contribution

It introduces a novel connection between generalized Killing spinors and Lagrangian embeddings, extending known results and revealing multiple connected components of Lagrangian submanifolds.

Findings

Relation between generalized Killing spinors and Lagrangian embeddings.

Extension of Gluck and Gu's divergence-free geodesic vector fields result.

Identification of at least three connected components of Lagrangian submanifolds.

Abstract

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.