Two Kazdan-Warner type identities for the renormalized volume coefficients and the Gauss-Bonnet curvatures of a Riemannian metric

In this note, we prove two Kazdan-Warner type identities involving $v^{(2k)}$, the renormalized volume coefficients of a Riemannian manifold $(M^n,g)$, and $G_{2r}$, the so-called Gauss-Bonnet curvature, and a conformal Killing vector field on $(M^n,g)$. In the case when the Riemannian manifold is locally conformally flat, $v^{(2k)}=(-2{)}^{-k}{\sigma }_k$, $G_{2r}(g)=\frac{4^r(n-r)!r!}{(n-2r)!}{\sigma }_r$, and our results reduce to earlier ones established by Viaclovsky and by the second author.