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

JOURNAL OF GEOMETRY AND PHYSICS

卷:65

Article

Reggiani, Silvio

关键词: Metric connection;    Flat connection;    Skew-symmetric torsion;    Holonomy;   

DOI  :  10.1016/j.geomphys.2012.11.012

来源: Elsevier

【 摘 要 】

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. In particular, we get a refinement of a well-known theorem of Cartan and Schouten. Finally, we show that the holonomy group of a metric connection with skew-symmetric torsion on these spaces generically coincides with the Riemannian holonomy. (c) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

Free