The Variation of the De Rham Zeta Function

Special values of the zeta function for the Laplacian on forms on a compact Riemannian manifold are known to have geometric significance. We compute the variation of these special values with respect to the variation of the metric and write down the Euler-Lagrange equation for conformal variations. The invariant metric on a locally symmetric space is shown to be critical for every local Lagrangian. We also compute the variation of <!-- MATH $\zeta '(0)$ --> , or equivalently of det . Finally, flat manifolds are characterized by flatness at a point and a condition on the amplitudes of the eigenforms of .