I suppose the two Levi-Civita parallelogramoids on a riemannian manifold equipped with Levi-Civita connection (https://en.wikipedia.org/wiki/Levi-Civita_parallelogramoid).
Given $p \in M$ and $\xi_1, \xi_2 \in T_p M$, we can construct two parallelogramoids:
- Parallel transport of $\xi_2$ along the geodesic of initial direction $\xi_1$ first
- Parallel transport of $\xi_1$ along the geodesic of initial direction $\xi_2$ first
In [http://www.yann-ollivier.org/rech/publs/visualcurvature.pdf], it is said that when $\xi_1$ and $\xi_2$ are of small amplitude, the two endpoints coincide at first order, up to $o(|\xi_1||\xi_2|)$.
I was wondering under which conditions (on the metric certainly) the two endpoints would coincide for any $p, \xi_1, \xi_2$. It is clear that it does in an Euclidean setting, but is it the only case?