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

TL;DR

This paper establishes two Kazdan-Warner type identities linking renormalized volume coefficients and Gauss-Bonnet curvatures with conformal Killing fields, extending previous results in special cases of locally conformally flat manifolds.

Contribution

It introduces two new identities connecting geometric invariants and conformal symmetries, generalizing earlier specific cases to broader classes of Riemannian manifolds.

Findings

Proves identities involving $v^{(2k)}$ and $G_{2r}$ with conformal Killing vector fields.

Reduces to known results in the locally conformally flat case.

Extends Kazdan-Warner type identities to new geometric invariants.

Abstract

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.