Homogeneous surfaces admitting invariant connections (Q2107688)

A homogeneous space \(M\) is a smooth manifold endowed with a transitive smooth action of a Lie group \(G\). An infinitesimal homogeneous space is a manifold \(M\) endowed with a transitive Lie algebra representation of a Lie algebra \(g\) into the Lie algebra of smooth vector fields of \(M\). The authors ask the following questions. Which homogeneous spaces of a fixed dimension do admit exactly one, or strictly more than one invariant connections? Is it possible to give a complete list? In the paper it is proved that there is a finite list of simply connected homogeneous surfaces admitting invariant connections. In particular, there are six non-equivalent simply connected homogeneous surfaces admitting more than one invariant connections and four classes of simply connected homogeneous surfaces admitting exactly one invariant connection.