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28-08-2017, 19:10
Ημερομηνία: Πέμπτη 31-8-2017
Ώρα: 13:00
Αίθουσα: B/M 342
Ομιλητής: Ιωάννης Χρυσικός, Dipartimento di Matematica, Universitá di Torino, Italy
Τίτλος: The Ricci endomorphism on the spinor bundle and applications


Consider a Riemannian spin manifold (M^n, g) (n> 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 talk we shall introduce a generalized $\frac{1}{2}$-Ricci type formula for the spinorial action of the Ricci endomorphism $\Ric^{s}(X)$, 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 shall describe a new alternative proof of the generalized Schr"odinger-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 K"ahler manifolds and nearly parallel $\G_2$-manifolds, in dimensions 5, 6 and 7, respectively.