A Berger-type theorem for metric connections with skew-symmetric torsion

Details A Berger-type theorem for metric connections with skew-symmetric torsion A Berger-type theorem for metric connections with skew-symmetric torsion

2011-11-21

arxiv.org/abs/1111.5044v1

Mathematics

Differential Geometry

10 pages

Scientific paper

We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled-back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy.

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