# Principal Symbol for Ricci-DeTurck Flow

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 $$G(T)$$:

$$\bigg(\frac{\partial}{\partial t}\delta G(T) \bigg)Z = -T \bigg((\delta G(h))^{\#},Z\bigg) + \bigg,$$

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.