I asked this question on math.stackexchange too: it's not a homework problem, but something that came to my mind while thinking of commutation:
Let $(M,g)$ be a Riemannian manifold, and let $\gamma_{p,v}, \gamma_{p,w}$ be two geodesics starting from $p$ with directions/initial vectors $v,w$ respectively. Consider the two operations (to be formally stated in a moment):
I) Take the parallel transport $P^v$ of $w$ from time $0$ to $t$ along $\gamma_{p,v}$, then take exponential map at this point and travel for time $s$.
II)Take the parallel transport $P^w$ of $v$ from time $0$ to $s$ along $\gamma_{p,w}$, then take the exponential maps at this point and travel for time t.
My question is: how different are these two points for varying $s,t$?
To formalize things, consider $F(t,s):=exp_{exp_p(tv)}{(sP^v_{0,t}(w))}$ and $G(t,s)=exp_{exp_p(sw)}{(tP^w_{0,s}(v))}$.
Clearly, $F(t,s)$ and $G(t,s)$ are not necessarily equal, because if we fix $s$, $G$ is a geodesic but $F$ is not, unless the curature vanishes. But I'd like to know if we have results that estimate $||F(t,s)-G(t,s)||$.
