While reading monograph on the Ricci flow, I came accross a fact (at least I think it is a fact), which is not proved explicitly in that book.
Assume a smooth 1-parameter family of Riemannian metrics $g_{t}$. The evolution of the Laplacian is as follows: $$\partial_{t} \nabla_{g(t)}=\partial_{t}g^{ij}\nabla_{i}\nabla_{j}=-\frac{\partial}{\partial t}g_{ij}\nabla_{i}\nabla_{j}+g_{ij}\partial_{t}(\nabla_{i}\nabla_{j}).$$
This seems to imply, that $\partial_{t}g^{ij}=-\partial_{t}g_{ij}$, but I cannot prove it. I have a hunch, that it is related to the following:
For Riemannian metric we have $g^{ij}g_{ij}=dim(manifold)$, thus by differentiating we ontain: $$g_{ij}\partial_{t}g^{ij}=-g^{ij}\partial_{t}g_{ij}.$$
But I cannot validate that $\partial_{t}g^{ij}=-\partial_{t}g_{ij}$. What am I missing?
You get, by differentiating $g^{ij}g_{jk}=\delta^i_k$ and subsequently multiplying from the right with $g^{kl}$ $$\partial_t g^{il}=-g^{ij}\,\partial_tg_{jk}\, g^{kl}$$ I doubt that you can do better in general.