Let $X,Y$ be smooth vector fields of a manifold $M$ (equipped with a metric) and $q\in M$.
We denote by $e^{tX}q$ the point of $M$ such that we integrate the ODE $$ \dot x = X(x) $$ during time $t$ and starting from $q$.
Is it true that $$ || e^{tX}e^{sY}q||\le ||e^{tX+sY}q|| ? $$
Applying BCH formula, we get:
$$ e^{tX}e^{sY}q = e^{tX+sY+1/2st[X,Y]+\ldots} $$
but I am trying to find a counterexample to reject the inequality. I mean if $X$,$Y$ are such that $X\cdot Y\ge 0$ and $[X,Y]\le 0$, the terms in $[X,Y]$ will bring the final point closer to the initial point.
I tried on $\mathbb{R}$, with $X(x)=1$ and $Y(x)=x$ so $[X,Y]=-1$ but in fact I get the equality ...
Source : http://sci-hub.tw/10.1137/0316047
