A new $\frac{1}{2}$-Ricci type formula on the spinor bundle and applications

Consider a Riemannian spin manifold $(M^{n}, g)$ $(n\geq 3)$ endowed with a non-trivial 3-form $T\in\Lambda^{3}T^{*}M$, such that $\nabla^{c}T=0$, where $\nabla^{c}:=\nabla^{g}+\frac{1}{2}T$ is the metric connection with skew-torsion $T$. In this note we introduce a generalized $\frac{1}{2}$-Ricci type formula for the spinorial action of the Ricci endomorphism ${\rm Ric}^{s}(X)$, induced by the one-parameter family of metric connections $\nabla^{s}:=\nabla^{g}+2sT$. This new identity extends a result described by Th. Friedrich and E. C. Kim, about the action of the Riemannian Ricci endomorphism on spinor fields, and allows us to present a series of applications. For example, we describe a new alternative proof of the generalized Schr\"odinger-Lichnerowicz formula related to the square of the Dirac operator $D^{s}$, induced by $\nabla^{s}$, under the condition $\nabla^{c}T=0$. In the same case, we provide integrability conditions for $\nabla^{s}$-parallel spinors, $\nabla^{c}$-parallel spinors and twistor spinors with torsion. We illustrate our conclusions for some non-integrable structures satisfying our assumptions, e.g. Sasakian manifolds, nearly K\"ahler manifolds and nearly parallel ${\rm G}_2$-manifolds, in dimensions 5, 6 and 7, respectively.