The variation of the de Rham zeta function

Abstract:

Special values of the zeta function $\zeta (s)$ for the Laplacian on forms $\Delta$ 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 $\zeta ’(0)$, or equivalently of det $\Delta$. Finally, flat manifolds are characterized by flatness at a point and a condition on the amplitudes of the eigenforms of $\Delta$.