The uniqueness of the canonical connection

SILVIO REGGIANI

Sao Paulo

Congreso; 16th School of Differential Geometry; 2010

University of Sao Paulo

In a previous work we proved that the canonical connection on a compact naturally reductive space is unique (provided the space does not split off locally, is not a sphere nor a compact Lie group with a bi-invariant metric). Recently, we have extended such a result to the noncompact case: the canonical connection is unique, in the irreducible case, except for the symmetric dual of compact Lie groups. Moreover, the hyperbolic space (the dual of the sphere) admits only one naturally reductive decomposition. This result is a consequence of a decomposition theorem for naturally reductive spaces and a Berger-type theorem: The Skew-torsion Holonomy Theorem. We prove such results in a geometric way.