Integration by parts on manifolds with Hessian

Question

I'm working on the following exercise in my Riemannian manifolds book: prove an "integration by parts" formula for the expression

$\int_{\Omega} f_{1} \text{Hessian}(f_{2})(X,Y) d \mu_{g} $

where $\Omega$ is a subset of a Riemannian manifold $(M,g)$, $X,Y$ are two tangent vectors and $f_{1},f_{2}$ are smooth functions on $(M,g)$. Here, $d \mu_{g}$ is the standard measure on $(M,g)$.

Also: $\text{Hessian}(f)(X,Y)=g(\nabla_{X}\text{grad}_{g}(f),Y)$.

Thanks in advance!

Answer 1

For those you encounter the same problem, here is the solution:

Proposition 6.2 (Integration by parts formula) page 27 https://arxiv.org/abs/1505.04817v1