Transverse Killing and twistor spinors associated to the basic Dirac operators

TL;DR

This paper explores the relationship between basic Dirac operators and special spinors in Riemannian foliations, extending classical results to more general geometric settings.

Contribution

It introduces a generalized framework for transverse Killing and twistor spinors linked to the basic Dirac operator in Riemannian foliations with bundle-like metrics.

Findings

Transverse Killing and twistor spinors coincide with classical definitions under certain conditions.

Extended classical results to Riemannian foliations with basic-harmonic mean curvature.

Provided new insights into the interplay between Dirac operators and spinors in foliated geometries.

Abstract

We study the interplay between basic Dirac operator and transverse Killing and twistor spinors. In order to obtain results for general Riemannian foliations with bundle-like metric we consider transverse Killing spinors that appear as natural extension of the harmonic spinors associated with the basic Dirac operator. In the case of foliations with basic-harmonic mean curvature it turns out that these Killing spinors and twistor spinors coincide with the standard definition. We obtain the corresponding version of classical results on closed Riemannian manifold with spin structure, extending some previous results.