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 $T\in {\Lambda }^3{T}^{\ast }M$, 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 ${\mathrm{G}}_2$-manifolds, in dimensions 5, 6 and 7, respectively.

References in corpus (3)

• Almost Hermitian Structures with Parallel Torsion
• Sasakian Geometry, Holonomy, and Supersymmetry
• Cocalibrated G_2-manifolds with Ricci flat characteristic connection

Cited by in corpus (2)

• Editorial note to: Erwin Schrödinger, Dirac electron in the gravitational field I
• A note on the volume of $\nabla$-Einstein manifolds with skew-torsion