This question is in reference to the following Mathoverflow question and the accepted answer to it. It seems to me that it is taken for granted that if the metric $g_t$ perturbs real analytically in time, so does the Laplacian $\Delta_t$ associated to $g_t$. I am trying to prove this step by step by trying to demonstrate a Taylor series expansion for $\Delta_{a + h}$ in terms of powers of $h$ and $t$ derivatives of $\Delta$ evaluated at $a$. In doing this, I have started from the local coordinate expression of $\Delta_{a + h}$ in terms of $g_{a + h}$, which is given by $\Delta_{a + h}f = \frac{1}{\sqrt {|g_{a + h}|}} \partial_i \left(\sqrt{|g_{a + h}|} g_{a + h}^{ij} \partial_j f \right)$. But trying to use the power series expansion for $g_{a+ h}$ (which is given by the fact that $g_t$ is varying real analytically) in the above expression for the Laplacian is becoming real messy real quick.

So, my question is, is this the proper way to see that the Laplacian is real analytic? Or there is some slick coordinate-invariant, or other methods of seeing this? Sorry if the question is too basic, I just wish to make sure that I understand all the details.