Most natural connection on Lie group: comparison of different pictures

Question

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?

Answer 1

From the formula for the Levi-Civita connection you can check that the LC connection for a left invariant metric will satisfy 1, but will not satisfy 2 unless the metric is bi-invariant.

For 3, you have to specify what parallel translation is taken along. If it is along geodesics then both the geodesics and parallel translation are given by the group action on a symmetric space.

Answer 2

Let $\nabla$ be a left invariant connection defined by a differentiable metric on a Lie group $G$. As you mentioned, $\nabla$ induces on the Lie algebra ${\cal G}$ of $G$ a bilinearr product defined by $\nabla_xy=b(x,y)$.

Consider a geodesic $c(t)$, write $x(t)=dL_{c(t)}^{-1}{d\over{dt}}c(t)\in {\cal G}$, it satisfies the equation:

${d\over{dt}}x(t)+b(x(t),x(t))=0$. We deduce that $c(t)$ coincide with the one paremeter if $b(x(t),x(t))=0$.

Geodesics equation on Lie groups with left invariant metrics