The variation of the de Rham zeta function
Author:
Steven Rosenberg
Journal:
Trans. Amer. Math. Soc. 299 (1987), 535557
MSC:
Primary 58G10; Secondary 58G25
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 EulerLagrange 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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 [B1]
 D. Bleecker, Critical Riemannian manifolds, J. Differential Geom. 14 (1979), 599608. MR 600616 (82g:58024)
 [B2]
 , Determination of a Riemannian metric from the first variation of its spectrum, Amer. J. Math. (to appear). MR 796904 (86k:58124)
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 M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Differential Geom. 3 (1969), 379392. MR 0266084 (42:993)
 [C]
 J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259322. MR 528965 (80j:58065a)
 [CGW]
 R. Cahn, P. B. Gilkey, and J. A. Wolf, Heat equation, proportionality principle, and volume of fundamental domains, Differential Geometry and Relativity (D. Cohen and M. Flato, eds.), Reidel, Holland, 1976, pp. 4354. MR 0436222 (55:9170)
 [G1]
 P. B. Gilkey, The spectral geometry of a Riemannian manifold, J. Differential Geom. 10 (1975), 601618. MR 0400315 (53:4150)
 [G2]
 , The spectral geometry of symmetric spaces, Trans. Amer. Math. Soc. 225 (1977), 341353. MR 0423258 (54:11238)
 [G3]
 , Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio Math. 38 (1979), 201240. MR 528840 (80i:53020)
 [MS]
 H. P. McKean and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1 (1967), 4369. MR 0217739 (36:828)
 [P1]
 R. Palais, Applications of the symmetric critically principle to mathematical physics and differential geometry, Sonderforschungsbereich 40, Universität Bonn.
 [P]
 V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geom. 5 (1971), 233249. MR 0292114 (45:1201)
 [R]
 H. Rademacher, Topics in analytic number theory, Grundlehren Math. Wiss., Band 169, SpringerVerlag, Berlin, and New York, 1973. MR 0364103 (51:358)
 [de R]
 G. de Rham, Variétés différentiables, Hermann, Paris, 1960.
 [RS]
 D. B. Ray and I. M. Singer, torsion and the Laplacian on Riemannian manifolds, Adv. in Math. 7 (1971), 145210. MR 0295381 (45:4447)
 [T]
 C. L. Terng, unpublished.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708692204
PII:
S 00029947(1987)08692204
Article copyright:
© Copyright 1987
American Mathematical Society
