17 mar 2017 -- 11:30
Aula Tricerri, DiMaI, Firenze
Abstract.
Let us fix a Riemannian spin manifold $(M, g)$ endowed with a non-trivial $3$-form $T$ and the $1$-parameter family of metric connections with skew torsion, $\nabla^{s}=\nabla^{g}+2sT$. In this talk we describe a new formula for the action of the Ricci operator associated to $\nabla^{s}$ on the corresponding spinor bundle. This is given in terms of the Dirac operator associated to $\nabla^{s}$ and generalises a result of Th. Friedrich and E.C. Kim from the Riemannian setting (i.e. $T=0$). When $T$ is $\nabla^{c}$-parallel, our new formula allows us to present an alternative proof of the generalized Schröndinger-Lichnerowicz formula associated to the Dirac operator $D^{s}$. We also discuss applications related to $\nabla^{s}$-parallel spinor fields, or $\nabla^{c}$-parallel spinor fields, where $\nabla^{c}=\nabla^{g}+(1/2)T$ is the so-called characteristic connection. We illustrate these results on some types of non-integrable geometric structures.