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During the study of a paper I see that its author defines $$\frac{dh^{ab}}{d\tau}:=h^{am}h^{bn}\frac{dh_{mn}}{d\tau}$$ and from this concludes that $$\frac{d}{d\tau}(h^{ab})=\frac{d}{d\tau}(h^{am}h^{bn}h_{mn})=-\frac{dh^{ab}}{d\tau}$$

where $h_{ab}$ is a Riemannian metric that depends on the parameter $\tau$. I Can not derive last equation. Can someone point me in the right direction? Sorry, if my question is very obvious!

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    $\begingroup$ Might be relevant to cite the paper? $\endgroup$
    – Kyle Kanos
    Dec 7, 2013 at 14:49

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I think you should have a lowered index on the RHS:

$$\begin{align} \frac{dh_{ab}}{dt} &= \frac{d}{dt}\left(h_{ma}h_{nb}h^{mn}\right)\\ &=\frac{dh_{ma}}{dt}\delta_{b}{}^{m} + \frac{dh_{nb}}{dt}\delta_{a}{}^{n} + h_{ma}h_{nb}\frac{dh^{mn}}{dt}\\ \frac{dh_{ab}}{dt}&= 2 \frac{dh_{ab}}{dt} + h_{ma}h_{nb}\frac{dh^{mn}}{dt}\\ \frac{dh_{ab}}{dt} &= -h_{ma}h_{nb}\frac{dh^{mn}}{dt} \end{align}$$

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