How well-known is the following result: 
Let $(M,g)$ be a closed Riemannian manifold and define $D:C^\infty(M,Sym^mT^*M)\to C^\infty(M,Sym^{m+1}T^*M)$, $D\alpha:=\mathbb S(\nabla\alpha)$ where $Sym^m$ denotes the symmetric power of the cotangent bundle, $\mathbb S$ denotes symmetrization and $\nabla$ denotes the covariant derivative on forms induced by the Levi-Civita connection. In the case $m=1$, I can prove $L_{X}g = 2D(X^b)$ for all vector fields $X\in C^\infty(M,TM)$ and $b:TM\to T^*M$ denotes the musical isomorphism. For the general case $m\geq0$, we introduce $\pi_m^*:C^\infty(M,Sym^mT^*M)\to C^\infty(SM)$, $(\pi_m^*f)(v):=f_{\pi(v)}(v,\ldots,v)$, where $SM$ denotes the unit tangent bundle and $\pi$ the footpoint projection. Now let $X\in C^\infty(SM,T(SM))$ be the generator of the geodesic flow. 

- Somehow one should be able to show that $X\pi_m^* = \pi_{m+1}^*D$. 
- With this, one can somehow deduce that if the geodesic flow is dense (meaning transitive) in $SM$, then there are no non-trivial Killing fields. 

Any complete proof or suggestions from the literature on both points would be highly appreciated. - Thanks