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- W4301359912 abstract "Consider a Riemannian spin manifold $(M^{n}, g)$ $(ngeq 3)$ endowed with a non-trivial 3-form $TinLambda^{3}T^{*}M$, 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 ${rm Ric}^{s}(X)$, 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 ${rm G}_2$-manifolds, in dimensions 5, 6 and 7, respectively." @default.
- W4301359912 created "2022-10-05" @default.
- W4301359912 creator A5048525568 @default.
- W4301359912 date "2017-03-12" @default.
- W4301359912 modified "2023-09-26" @default.
- W4301359912 title "A new $frac{1}{2}$-Ricci type formula on the spinor bundle and applications" @default.
- W4301359912 doi "https://doi.org/10.48550/arxiv.1703.04121" @default.
- W4301359912 hasPublicationYear "2017" @default.
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