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

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

2003-05-16

arxiv.org/abs/math/0305233v2

Mathematics

Differential Geometry

Latex2e, 15 pages

Scientific paper

10.1016/j.geomphys.2003.11.001

For any triple $(M^n, g, \nabla)$ 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 $\nabla$-parallel spinors.

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