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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

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
MathSciNet review: 869220
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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 $.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1987-0869220-4
PII: S 0002-9947(1987)0869220-4
Article copyright: © Copyright 1987 American Mathematical Society