We study twistor spinors (with torsion) on Riemannian spin manifolds $(M^{ n}, g, T)$ carrying metric connections with totally skew-symmetric torsion. We consider the characteristic connection $\nabla^{ c}=\nabla^{ g}+\frac{ 1}{ 2}T$ and under the condition $\nabla^{ c}T=0$, we show that the twistor equation with torsion w.r.t. 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. Next we study a special class of twistor spinors with torsion, namely these which are $T$-eigenspinors and parallel under the characteristic connection; we show that the existence of such a spinor for some $s\neq 1/4$ implies that $(M^{ n}, g, T)$ is both Einstein and $\nabla^{ c}$-Einstein, in particular the equation ${ \rm Ric}^{ s}=\frac{ { \rm Scal}^{ s}}{ n}g$ holds for any $s\in\mathbb{ R}$. In fact, for $\nabla^{ c}$-parallel spinors we provide a correspondence between the Killing spinor equation with torsion and the Riemannian Killing spinor equation. This allows us to describe 1-parameter families of non-trivial Killing spinors with torsion on nearly K\"ahler manifolds and nearly parallel ${ \rm G}_2$-manifolds, in dimensions 6 and 7, respectively, but also on the 3-dimensional sphere ${ \rm S}^{ 3}$. We finally present applications related to the universal and twistorial eigenvalue estimate of the square of the cubic Dirac operator.
Categories: math.DG