Let $G$ be a compact Lie group with Haar measure $\mu$. Let $X\in\mathfrak{X} (G)$ be such that, if $T(x)=\exp_x(X_x)$, $$T_*\mu=\mu,$$ then $\operatorname{div}(X)=0$?
This is true when $G=S^1$, because the diffeomorfisms that let the Lebesgue measure on $S^1$ invariant are the rigid rotations and the rigid reflections, thus, such an $X$ should be constant, hence without divergence.
I'm trying this line of reasoning: if we let $T_t(x)=\exp_x(tX_x)$, and consider de Jacobian $J_t(x)$ given by $$(T_t)_*\mu=J_t(x)\mu$$ Be derivating at $t=0$ we get $$\dot{J_0}(x)=-\operatorname{div}(X)(x).$$ Since $T_0(x)=x$, we have that, by integrating, $$J_t(x)=e^{-\int_0^t\operatorname{div}(X)(T_s(x)) ds}.$$ Since $(T_1)_*\mu=\mu$, $J_1\equiv 1$ and thus, \begin{equation}\label{integraljacobiano} \int_0^1\operatorname{div}(X)(T_s(x)) ds=0, \end{equation} for every $x\in G$. But I cannot go any further.
