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Suppose $M$ is a manifold and $f:M \rightarrow \bf{R}$ is a $\mathcal{C}^{\infty}$ function. Let $X$ and $Y$ denote vector fields on $M$, and let $\varphi_Y^t$ denote the flow of $Y$. Fix a point $p \in M$, and consider the real valued function $$u(t):=df_{\varphi_Y^t (p)}(X(\varphi_Y^t (p))).$$ When is $u$ identically zero? Is it always zero if the Lie bracket $\mathcal{L}_X Y$ is zero, or does the condition depend on the function $f$?

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    I know that it is kind of obvious to say, but we cannot take any function $f$. For instane, constant functions will produce $u(t)=0$, because $\textrm{d}f=0$, independently of the conditions on $X$ and $Y$. I think we may have to ask conditions on $f$ to take some interesting result.2012-04-22

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