The Ray-Singer paper "R-torsion and the Laplacian on Riemannian manifolds" claimed that one may prove the metric invariance of analytical torsion by forming a homotopy between metric $\rho_{0},\rho_{1}$ using $\rho_{\mu}=\mu\rho_{0}+(1-\mu)\rho_{1}$. Then one differentiate $$ \frac{\partial }{\partial u}f(u,s)=\frac{1}{2}\sum^{N}_{q=0}(-1)^{q}q\int^{\infty}_{0}t^{s-1}Tr(e^{t\Delta_{q}(u)}\dot{\Delta_{q}})dt $$ here the $\frac{1}{\Gamma(s)}$ is omitted as it is entire, and if we know the differential of original function with $u$ is $0$, then with respect to $s$ should be $0$ as well. Now the differential of the fundamental solution of the initial value problem ("heat kernel") is given by $$ \frac{d}{d\mu}Tr e^{t\Delta_{q}(\mu)}=-tTr((\delta \alpha d-d\alpha\delta-\alpha \delta d+\alpha d\delta))e^{t\Delta_{q}(\mu)} $$ hence in the first equation we actually have $$ \dot{\Delta}_{q}=\alpha \delta d-\delta \alpha d+d\alpha \delta-d\delta\alpha,\alpha=*^{-1}\dot{*}=*^{-1}(\frac{\partial}{\partial \mu}*) $$ The proof then proceed by substituting the later formula into the above identity, using the fact that the trace of trace-class operator and bounded operator commutes to reduce the sum to a telescoping sum, which then reduces to the following: $$ \frac{1}{2}\sum^{N}_{q=0}(-1)^{q}\int^{\infty}_{0}t^{s}\frac{d}{dt}Tr(e^{t\Delta_{q}(\mu)}\alpha)dt $$ using integration by parts, we gain an additional $s$ on the front, and the desired result $\frac{\partial}{\partial \mu}f(\mu,s)=0$ now follows by regularity of the heat kernel for $\operatorname{Re}(s)$ large enough.
I noticed a seemingly discrepancy in the above proof,namely the setting of the theorem is when $W$ is a closed manifold, and the formula (Proposition 6.1 of the paper) is derived in the context of a manifold with two boundary components $M_1,M_2$, on which the Laplacian satisfies certain boundary conditions. In the case of the closed manifold, the formula does not easily reduce from the manifold with boundary case (see the remark in page 195), and the same strategy seems no longer works (possibly) due to the failure of using integration by parts in the second to last step. It seems the proof works mainly because in the detailed computation in page 196 we actually do not have any integral over the boundary components, hence the same strategy would work trivially for a closed manifold.
My question is: Is there a simpler proof for this in the case of a closed manifold? I am sure Ray and Singer must have tried and thought about it, but somehow still decided to use the strategy presented in paper to finish the proof. After all the proof presented in page 195-199 is really involved. So if there is a "clear cut" solution, it most likely would not be easy. I checked the literature but I am not sure if someone else has done any work related to it. It really isn't clear to me how to analyze the regularity of the differential of the Laplacian $\frac{\partial}{\partial \mu}\Delta_{q}(\mu)$ when $\operatorname{Re}{s}\rightarrow \infty$. I am not sure if the strategy in page 154 still works in this context.
