I am a physicist, and thus my question might be not precise and/or even not well posed. The problem I encounter is to characterize all `natural' connections on the group manifold AN(3). AN(3) is a 4-dimensional Lie group with the Lie algebra generated by $x_i$, $i=1,2,3$, $t$ satisfying
$[t, x_i]=x_i$, $[x_i, x_j]=0$
The AN(3) group is a 4-dimensional manifold, so I can clearly construct a Levi-Civita connection on it (with non-zero curvature and zero torsion). I can also take Maurer-Cartan forms (zero curvature and non-zero torsion). Are there any other natural ones?
More generally. As a manifold the AN(3) group is a submanifold of de Sitter space $SO(4,1))/SO(3,1)$. The Levi-Civita connection on de Sitter is well known in physics in the context of cosmology, for example, but are there any other natural/interesting ones?