This follows from another post: What is exponential map in differential geometry. Actually it's implied there, but not explicitly stated.
In exponential maps of Lie group, we have BCH formula $\exp(X)\exp_{q}(Y)=\exp((X+Y)+[X, Y]+ \dots)$
Is there a similar formula $\exp_{q}(v_1)\exp_{q}(v_2)=\exp_{q}((v_1+v_2)+[v_1, v_2]+ \dots)$?
Considering there is an analog between operation of tangent vectors $X_e,Y_e$ (or $v_1, v_2$ in the above formula) and operations of vector spaces (tangent spaces of Lie groups) $\widetilde{X},\widetilde{Y}$, namely $[X_e,Y_e]=[\widetilde{X},\widetilde{Y}]_e$, we may define $[v_1, v_2]$ in general for Riemannian manifolds (or we can't do so for those manifolds we can't impose Lie group structure on?), therefore the above formula seems to exist (or only exist for manifolds w Lie group structure?); unless in all Riemannian manifolds $[v_1, v_2]=0$, which implies we always have $\exp_{q}(v_1)\exp_{q}(v_2)=\exp_{q}(v_1+v_2)$.
