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Sfb288 logo Sfb 288 Differential Geometry and Quantum Physics

Abstract for Sfb Preprint No. 588


The Casimir operator of a metric connection with skew-symmetric torsion

I. Agricola, Th. Friedrich

For any triple $(M^n, g, abla)$ consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator $Omega$ acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry group. Several non-homogeneous geometries (Sasakian, nearly K"ahler, cocalibrated $mathrm{G}_2$-structures) admit unique connections with skew-symmetric torsion. We study the corresponding Casimir operator and compare its kernel with the space of $ abla$-parallel spinors.


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