Let [itex]M[/itex] be a diff. manifold, [itex]X[/itex] a complete vectorfield on [itex]M[/itex] generating the 1-parameter group of diffeomorphisms [itex]\phi_t[/itex]. If I now define the Lie Derivative of a real-valued function [itex]f[/itex] on [itex]M[/itex] by(adsbygoogle = window.adsbygoogle || []).push({});

[tex]\mathscr{L}_Xf=\lim_{t\rightarrow 0}\left(\frac{\phi_t^*f-f}{t}\right)=\frac{d}{dt}\phi_{t}^{*}f |_{t=0}[/tex]

(where [itex]...^{*}[/itex] denotes the pull-back by ...) that's mere notation, right? I.e. the limit is not a functional limit, right? This simply defines how I should evaluate the Lie Derivative, true?

Now, if I know that

(1) [tex]\phi_t^*\theta\cdot\phi_{-t*}Y\s=\s\phi_t^*(\theta\cdot Y)[/tex]

where [itex]Y[/itex] is another vectorfield on [itex]M[/itex], how could I prove that

(2) [tex](\mathscr{L}_X\Theta)\cdot Y + \Theta\cdot (\mathscr{L}_XY)=\mathscr{L}_X(\Theta\cdot Y)[/tex]?

(Here, the Lie Derivative is defined correspondingly.) If I take the time derivative at [itex]t=0[/itex] of both sides in (1), I can't apply the standard (i.e. banach space) product rule, because the constituents are not real functions! What can I do?

Thanks in advance. Best regards...Cliowa

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# Lie Derivatives

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