A New $\frac{1}{2}$-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 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.