14 nov 2016 -- 16:30
Aula 5, Dipartimento di Matematica, Università di Torino
Abstract.
Given a compact Riemannian spin manifold endowed with a non-integrable $G$-structure, we shall speak about spinor fields which are parallel with respect to the characteristic connection $\nabla^{c}=\nabla^{g}+\frac{1}{2}T$. First, and under the condition $\nabla^{c}T=0$, we will show that the twistor equation with torsion with respect to the family $\nabla^{s}=\nabla^{g}+2sT$ can be viewed as a parallelism condition under a suitable connection on the bundle $\Sigma \oplus\Sigma$, where $\Sigma$ is the associated spinor bundle. Consequently, we prove that a twistor spinor with torsion has isolated zero points, generalizing a well-known property of Riemannian twistor spinors in the case of (parallel) skew-torsion. Next we shall focus on $\nabla^{c}$-parallel spinors which satisfy the Killing spinor equation with torsion for some $s\neq 0, 1/4$ with respect to the family $\nabla^{s}$. We show that their existence implies that our spin Riemannian manifold is both Einstein and $\nabla^{c}$-Einstein. Finally, we describe 1-parameter families of non-trivial Killing spinors with torsion (or twistor spinors with torsion) on nearly K\"ahler manifolds and nearly parallel $G_2$-manifolds, in dimensions 6 and 7 respectively.