Twistor sections of Dirac bundles

TL;DR

This paper explores twistor equations on Dirac bundles over Riemannian manifolds, providing vanishing theorems, characterizations of solutions, and demonstrating existence of solutions on spheres.

Contribution

It introduces the twistor equation in the context of Dirac bundles and characterizes solutions via the Dirac operator and curvature, with new existence results on spheres.

Findings

Vanishing theorems for solutions

Characterization of solutions using Dirac operator and curvature

Existence of nontrivial solutions on spheres

Abstract

A Dirac bundle is a euclidean bundle over a riemannian manifold $M$ which is a compatible left $C\ell(M)$-module, together with a metric connection also compatible with the Clifford action in a natural way. We prove some vanishing theorems and introduce the twistor equation within this framework. In particular, we exhibit a characterization of solutions for this equation in terms of the Dirac operator $D$ and a suitable Weitzenb\"ock-type curvature operator $\mathcal{R}$. Finally, we analyze the especial case of the Clifford bundle to prove existence of nontrivial solutions of the twistor equation on spheres.