I am fiddling around with some computations in (semi-)Riemannian geometry and have come across the following term:
$$ \int_{M} \phi L_{X}(\Delta_{g}\phi)dvol_{g}, $$
where $\Delta_{g}$ is the Laplace-Beltrami operator and $L_{X}$ is the Lie derivative. Ideally, I would like this term to be $0$ or have it be a "surface" term, either in the way that the scalar part in front of the volume form is a covariant derivative or that the entire integrand is an exact form. Of course, if $\phi$ is harmonic, this whole thing is zero, which is good for other reasons, but I'm curious to see how this could be transfigured anyway.