I am dealing with matrix Lie groups. I want to know the relation of Riemannian operators under left invariant and right invariant metrics. Riemannian operators I am interested include geodesic, exponential map, logarithmic map, parallel transportation.
For example, I see the following equation on matrix Lie groups [1], \begin{aligned} \operatorname{Exp}_I^{\text {right }}(U) & =\left(\operatorname{Exp}_I^{\text {left }}(-U)\right)^{-1} \\ \text { distance }^{\text {right }}(X, Y) & =\text { distance }^{\text {left }}\left(X^{-1}, Y^{-1}\right) \end{aligned}
So, can I obtain geodesic, exponential map or parallel transportation of the Right invariant metric, if I have already known these operators under the left invariant metric.(Group structure is the same one.)
[1] Zacur E, Bossa M, Olmos S. Left-invariant Riemannian geodesics on spatial transformation groups[J]. SIAM Journal on Imaging Sciences, 2014, 7(3): 1503-1557.