I have some question about detail on Chapter 9 Riemannian submersion in the book [Be]. 


[Be] A. L. Besse, Einstein Manifolds, Springer-Verlag 


In the above book, in the proof of 9.24 Proposition (See p. 240) it is described that 
 $[X,U]$ is vertical. 


Let me explain details : On $(M = B\times\_f F, g=\hat{g} + f\check{g})$ with a  

 submersion $\pi : M \rightarrow B$, let $U$ be a vertical vector field and $X$
 be a horizontal vector field. 


The proof used the argument that $[X,U]$ is vertical, which is not proved. 
 I can not understand why $[X,U]$ is vertical. 


In pp. 239-240, 9.22 also says that $[X,U]$ is vertical, but no explanation. And I know 
 that $d\pi[X,Y] = [d\pi X, d\pi Y]$. From this can we derive that $[X,U]$ is vertical ?


Please give me an explanation



Is [U,X] vertical on a warped product ?
Fix arbitrary $\phi:B\to \mathbb R$ and let $f=\phi\circ\pi$.
It is sufficient to show that $[X,U]f=0$.


Since $U$ is vertical,
$Uf=0$.
SInce $X$ is a horizontal lift,
$Xf$ is constant on each fiber and therefore $U(Xf)=0$.
Hence
$$[X,U]f=X(Uf)-U(Xf)=0.$$


(In case if $X$ is only horizontal, the function $Xf$ is arbitrary,
in this case $[X,U]f=-U(Xf)\ne 0$.)


