Abstract
We prove two Kazdan-Warner-type identities involving the renormalized volume coefficients nu((2k)) of a Riemannian manifold (M(n), g), the Gauss-Bonnet curvature G(2r), and a conformal Killing vector field on (M(n), g). In the case when the Riemannian manifold is locally conformally flat, we find nu((2k)) = (-2)(-k)sigma(k) and G(2r)(g) = 4(r)(n - r) !r! / (n - 2r)! sigma(r) and our results reduce to earlier ones established by Viaclovsky in 2000 and the second author in 2006.