# Result of Parallel transport along geodesic on $U(n)$

I am reading this paper Conjugate Gradient Algorithm for Optimization Under Unitary Matrix Constraint. It basically describes a variant of the conjugate gradient method when optimizing for unitary matrices which takes advantage of the Lie group structure of $$U(n)$$.

The question however is basic differential geometry and maybe Lie theory and this is what I am struggling with. Specifically, the following is stated:

Now, the parallel transport of a tangent vector $$\tilde{X} = XW \in T_W U(n) , X \in \mathfrak{u}(n)$$, w.r.t. the Riemannian connection along the geodesic $$\mathcal{G}_W (t) = \exp(tS) W,\quad S\in \mathfrak{u}(n), \quad t\in \mathbb{R} \tag{1}$$ is given by $$\tau \tilde{X}(t) = \exp(tS/2) \; X \; \exp(−tS/2) \;G_W(t) \tag{2}$$ where $$\tau$$ denotes the parallel transport.

Here, $$\mathfrak{u}(n)$$ denotes the Lie algebra to $$U(n)$$ and the geodesic $$\mathcal{G}_W(t)$$ is such that it emanates from $$W$$ in the direction of the vector $$(dR_W)_e(S)=SW \in T_WU(n)$$. I should further note that $$U(n)$$ was equipped with a bi-invariant metric!

My questions:

1. From the formula of the geodesic $$\mathcal{G}_W(t)$$, it seems to be that one can transport every vector on the Lie group just by left/right translation to every point. This seems much more natural than parallel transport. In the case of a bi-invariant metric, does it differ from parallel transport (which I understand depends on the geometry crucially)

2. Could someone walk me trough how one obtains the formula for the parallel transport, namely (2)?

In particular, this actually looks a like a right translation of the vector $$\exp(tS/2) \; X \; \exp(−tS/2)$$, too, i.e. $$\tau \tilde{X}(t) = dR_{G_W(t)}\left(\exp(tS/2) \; X \; \exp(−tS/2)\right) \tag{3}$$ and the thing in the bracket is somehow the adjoint map but I still cannot puzzle these things together.