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

Abstract

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 Schrodinger–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 Kahler manifolds and nearly parallel $$\hbox {G}_2$$ -manifolds, in dimensions 5, 6 and 7, respectively.