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.
Categories: math.DG