Recently, I have read the paper
Munteanu, Ovidiu; Sesum, Natasa, On gradient Ricci solitons, J. Geom. Anal. 23, No. 2, 539-561 (2013). ZBL1275.53061. In that paper at the page 543, there is a calculation, saying that
$$-\int_MR_{ij}f_{ij}e^{-\lambda f}\phi^2=\int_Mf_i\nabla_j(R_{ij}e^{-\lambda f}\phi^2),$$
where $R_{ij}$ is the Ricci tensor, $f\in C^\infty(M)$ and $\phi$ is a cut-off function. Also the author used the fact $\nabla_i(R_{ij}e^{-f})=0$.
Please tell me the intermediate calculation. I have found that
$$R_{ij}f_{ij}e^{-\lambda f}\phi^2=\nabla_j(R_{ij}f_ie^{-\lambda f}\phi^2)-f_i\nabla_j(R_{ij}e^{-\lambda f}\phi^2).$$ And by integrating we get an extra term $\int_M\nabla_j(R_{ij}f_ie^{-\lambda f}\phi^2)$. How to prove that this term is zero?
Please help me.
By definition, $\phi$ is compactly supported. Let $K$ be a sufficiently smooth compact set containing the support of $\phi$. Then $$ \int_M \nabla_j (R_{ij}f_i e^{-\lambda f}\phi^2) = \int_K \nabla_j (R_{ij}f_i e^{-\lambda f}\phi^2) = \int_{\partial K} n_j R_{ij}f_i e^{-\lambda f}\phi^2 = 0 . $$ The first equality is because the integrand is zero outside $K$, the second equality follows from the Divergence Theorem, and the third because $\phi$ vanishes on $\partial K$.