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I have the following question: Let $M$ be a smooth manifold and let $p \in M$. Furthermore let $X$ and $Y$ be two vector fields in a neighbourhood $U$ of $p$ and consider their flows $\varphi^{X}(t,x)$ and $\varphi^{Y}(t,x)$. Define now $s(t):= \varphi^{X}_{\sqrt{t}} \circ \varphi^{Y}_{\sqrt{t}} \circ \varphi^{X}_{-\sqrt{t}} \circ \varphi^{Y}_{-\sqrt{t}}$. How can one show that $\dot{s}(0) = [X,Y]|_{0}$ ? Or is this statement true? If yes, why? Thanks in advance.

Eric

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    applying this I get $\frac{d}{dt}|_{t=t_{0}}s(t) = \dot{\varphi^{X}}_{t_{0}} \frac{1}{2\sqrt{t_{0}}} + D\varphi^{X}_{t_{0}} \frac{d}{dt}|_{t=t_{0}}\varphi^{Y}_{\sqrt{t}} \circ \varphi^{X}_{-\sqrt{t}} \circ \varphi^{Y}_{-\sqrt{t}}$. How do I go further on ?2012-06-26

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