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 .

**[ABP]**M. Atiyah, R. Bott, and V. K. Patodi,*On the heat equation and the index theorem*, Invent. Math.**19**(1973), 279–330. MR**0650828**, https://doi.org/10.1007/BF01425417**[B1]**David D. Bleecker,*Critical Riemannian manifolds*, J. Differential Geom.**14**(1979), no. 4, 599–608 (1981). MR**600616****[B2]**David Bleecker,*Determination of a Riemannian metric from the first variation of its spectrum*, Amer. J. Math.**107**(1985), no. 4, 815–831. MR**796904**, https://doi.org/10.2307/2374358**[BE]**M. Berger and D. Ebin,*Some decompositions of the space of symmetric tensors on a Riemannian manifold*, J. Differential Geometry**3**(1969), 379–392. MR**0266084****[C]**Jeff Cheeger,*Analytic torsion and the heat equation*, Ann. of Math. (2)**109**(1979), no. 2, 259–322. MR**528965**, https://doi.org/10.2307/1971113

Werner Müller,*Analytic torsion and 𝑅-torsion of Riemannian manifolds*, Adv. in Math.**28**(1978), no. 3, 233–305. MR**498252**, https://doi.org/10.1016/0001-8708(78)90116-0**[CGW]**Robert S. Cahn, Peter B. Gilkey, and Joseph A. Wolf,*Heat equation, proportionality principle, and volume of fundamental domains*, Differential geometry and relativity, Reidel, Dordrecht, 1976, pp. 43–54. Mathematical Phys. and Appl. Math., Vol. 3. MR**0436222****[G1]**Peter B. Gilkey,*The spectral geometry of a Riemannian manifold*, J. Differential Geometry**10**(1975), no. 4, 601–618. MR**0400315****[G2]**Peter B. Gilkey,*Spectral geometry of symmetric spaces*, Trans. Amer. Math. Soc.**225**(1977), 341–353. MR**0423258**, https://doi.org/10.1090/S0002-9947-1977-0423258-3**[G3]**Peter B. Gilkey,*Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian*, Compositio Math.**38**(1979), no. 2, 201–240. MR**528840****[MS]**H. P. McKean Jr. and I. M. Singer,*Curvature and the eigenvalues of the Laplacian*, J. Differential Geometry**1**(1967), no. 1, 43–69. MR**0217739****[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 Geometry**5**(1971), 233–249. MR**0292114****[R]**Hans Rademacher,*Topics in analytic number theory*, Springer-Verlag, New York-Heidelberg, 1973. Edited by E. Grosswald, J. Lehner and M. Newman; Die Grundlehren der mathematischen Wissenschaften, Band 169. MR**0364103****[de R]**G. de Rham,*Variétés différentiables*, Hermann, Paris, 1960.**[RS]**D. B. Ray and I. M. Singer,*𝑅-torsion and the Laplacian on Riemannian manifolds*, Advances in Math.**7**(1971), 145–210. MR**0295381**, https://doi.org/10.1016/0001-8708(71)90045-4**[T]**C. L. Terng, unpublished.

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

Article copyright:
© Copyright 1987
American Mathematical Society