The variation of the de Rham zeta function

Author:
Steven Rosenberg

Journal:
Trans. Amer. Math. Soc. **299** (1987), 535-557

MSC:
Primary 58G10; Secondary 58G25

DOI:
https://doi.org/10.1090/S0002-9947-1987-0869220-4

MathSciNet review:
869220

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Abstract: 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 , or equivalently of det . Finally, flat manifolds are characterized by flatness at a point and a condition on the amplitudes of the eigenforms of .

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DOI:
https://doi.org/10.1090/S0002-9947-1987-0869220-4

Article copyright:
© Copyright 1987
American Mathematical Society