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

• Autores: Reggiani, Silvio Nicolás
• Año de publicación: 2012
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: Metric Connection; Flat Connection; Skew-Symmetric Torsion; Holonomy
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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.
Fil: Reggiani, Silvio Nicolás. Universidad Nacional de Córdoba. Facultad de Matemática, Astronomía y Física; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina