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Consider a Riemannian spin manifold \((M^{n}, g)\) \((n\ge 3)\) 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 \({{\mathrm{Ric}}}^{s}(X)\), 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 Schrödinger–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 Kähler manifolds and nearly parallel \(\hbox {G}_2\)-manifolds, in dimensions 5, 6 and 7, respectively.

A New $$\frac{1}{2}$$ 1 2 -Ricci Type Formula on the Spinor Bundle and Applications