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I am following some lecture notes on Ricci flow and linearizing an operator to obtain its principal symbol. We have $T \in \: \Gamma(Sym^2 T^{*}M)$ smooth, fixed and positive definite and then compute the time derivative for the divergence of $G(T)$:

$\bigg(\frac{\partial}{\partial t}\delta G(T) \bigg)Z = -T \bigg((\delta G(h))^{\#},Z\bigg) + \bigg<h,\nabla T(.,.,Z) - \frac{1}{2}\nabla_{Z}T \bigg>, $

where $\frac{\partial g}{\partial t}=h$, $Z$ an arbitrary vector field and $G(T)=T-\frac{1}{2}(tr \: T)h$.

The notes then say that this implies

$\frac{\partial}{\partial t}T^{-1}\delta G(T) = -\delta G(h)\: + \:\: ...$

where the dots indicate terms which don't have derivatives of $h$. I see that you can apply $T^{-1}$ as it doesn't depend on $t$ and then lose the $Z$ as it is arbitrary, but not sure exactly what happens to the right-hand side such that we end up with $-\delta G(h)$ or where the sharp goes.

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