Let $G$ be a Lie group (not necessarily compact). One can equip $G$ with the left invariant metric (or
right invariant but in general there is no biinvariant metric in the noncompact case). Once the metric on
$G$ is chosen we can consider the Levi Civita connection for this metric i.e. the unique metric connection
which has no torsion. If our metric was chosen to be (left)invariant we arrive at the connection which
is in some sense compatible with the group structure. However there are several notions of compatibility
with the group structure:

1. If $X,Y$ are left invariant vector fields then we can require $\nabla_X Y$ to be left invariant as well.

2. We may require that the geodesic defined by our connection coincide with the one parameter subgroups.

3. Finally we may require that the parallel transport defined by our connection coincide with the
(differential of) group translation.

Of course we have also the previous mentioned

4. Condition of being metric connection, with respect to some chosen (say) left invariant metric

I would like to understand what are the relations between conditions 1,2,3 and 4?

Invariant Affine Connections on Homogeneous Spaces, and the paper of Kostant,A Characterization of Invariant Affine Connections(projecteuclid.org/euclid.nmj/1118800357), and the many papers citing them. $\endgroup$ – Dan Fox Nov 22 '17 at 7:12