A note on invariant description of $SU(2)$-structures in dimension 5

TL;DR

This paper develops an invariant spinor-based method to analyze $SU(2)$-structures on 5-dimensional spin manifolds, characterizing subspaces and inducing quaternionic structures, with implications for understanding intrinsic torsion.

Contribution

It introduces a spinor approach to characterize $SU(2)$-structures and their intrinsic torsion invariants on 5-manifolds, providing a new perspective.

Findings

Characterization of subspaces inducing $SU(2)$-structures

Induction of quaternionic structures on contact distributions

Invariance of certain covariant derivative components of spinor fields

Abstract

We develop an invariant approach to $SU(2)$--structures on spin $5$--manifolds. We characterize (via spinor approach) the subspaces in the spinor bundle which induce the same group isomorphic to $SU(2)$. Moreover, we show how to induce quaternionic structure on the contact distribution of considered $SU(2)$--structure. We conclude with the invariance of certain components of the covariant derivative $\nabla\varphi$, where $\varphi$ is any spinor field defining considered $SU(2)$--structure. This shows, what expected, that (at least some of) the intrinsic torsion modules, can be derived invariantly with the spinorial approach.