Isospectral deformations of closed riemannian manifolds with different scalar curvature
Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 593-607.

On construit les premiers exemples de familles continues de métriques riemanniennes isospectrales, mais pas localement isométriques sur des variétés fermées, plus précisément sur S n ×T m , où T m est un tore de dimension m2 et S n est une sphère de dimension n4. Ces métriques ne sont pas localement homogènes ; en particulier, la courbure scalaire d’une telle métrique n’est pas constante. Dans certaines des déformations que l’on considère, la courbure scalaire maximale change pendant la déformation.

We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on S n ×T m , where T m is a torus of dimension m2 and S n is a sphere of dimension n4. These metrics are not locally homogeneous; in particular, the scalar curvature of each metric is nonconstant. For some of the deformations, the maximum scalar curvature changes during the deformation.

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     author = {Gordon, Carolyn S. and Gornet, Ruth and Schueth, Dorothee and Webb, David L. and Wilson, Edward N.},
     title = {Isospectral deformations of closed riemannian manifolds with different scalar curvature},
     journal = {Annales de l'Institut Fourier},
     pages = {593--607},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {48},
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     year = {1998},
     doi = {10.5802/aif.1630},
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     url = {http://www.numdam.org/articles/10.5802/aif.1630/}
}
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Gordon, Carolyn S.; Gornet, Ruth; Schueth, Dorothee; Webb, David L.; Wilson, Edward N. Isospectral deformations of closed riemannian manifolds with different scalar curvature. Annales de l'Institut Fourier, Tome 48 (1998) no. 2, pp. 593-607. doi : 10.5802/aif.1630. http://www.numdam.org/articles/10.5802/aif.1630/

[Be] P. Bérard, Variétés riemanniennes isospectrales non isométriques, Séminaire Bourbaki 705, no 177-178 (1988-1989), 127-154. | Numdam | Zbl

[Br] R. Brooks, Constructing isospectral manifolds, Amer. Math. Monthly, 95 (1988), 823-839. | MR | Zbl

[BT] R. Brooks and R. Tse, Isospectral surfaces of small genus, Nagoya Math. J., 107 (1987), 13-24. | MR | Zbl

[Bu] P. Buser, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), 36-2 (1986), 167-192. | Numdam | MR | Zbl

[D] D. Deturck, Audible and inaudible geometric properties, Rend. Sem. Fac. Sci. Univ. Cagliari, 58 (supplement 1988), 1-26.

[DG1] D. Deturck and C. Gordon, Isospectral deformations I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387. | MR | Zbl

[DG2] D. Deturck and C. Gordon, Isospectral deformations II: Trace formulas, metrics, and potentials, Comm. Pure Appl. Math., 42 (1989), 1067-1095. | MR | Zbl

[E] P. Eberlein, Geometry of two-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup., (4) 27 (1994), 611-660. | Numdam | MR | Zbl

[G1] C.S. Gordon, You can't hear the shape of a manifold, New Developments in Lie Theory and Their Applications (J. Tirao and N. Wallach, eds.), Birkhäuser, 1992. | MR | Zbl

[G2] C.S. Gordon, Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom., 37 (1993), 639-649. | MR | Zbl

[G3] C.S. Gordon, Isospectral closed Riemannian manifolds which are not locally isometric, Part II, Contemporary Mathematics: Geometry of the Spectrum (R. Brooks, C. Gordon, P. Perry, eds.), vol. 173, Amer. Math. Soc., 1994, 121-131. | MR | Zbl

[GGt] C.S. Gordon and R. Gornet, Spectral geometry of nilmanifolds, Proceedings of the Summer University of Southern Stockholm: Advances in Inverse Spectral Geometry, Birkhäuser, 1997, 23-49. | MR | Zbl

[GWW] C.S. Gordon, D. Webb, and S. Wolpert, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22. | MR | Zbl

[GW1] C.S. Gordon and E.N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom., 19 (1984), 241-256. | MR | Zbl

[GW2] C.S. Gordon and E.N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. | MR | Zbl

[GW3] C.S. Gordon and E.N. Wilson, Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differential Geom., to appear. | Zbl

[Gt1] R. Gornet, A new construction of isospectral Riemannian manifolds with examples, Michigan Math. J., 43 (1996), 159-188. | MR | Zbl

[Gt2] R. Gornet, Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal., to appear. | Zbl

[I] A. Ikeda, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup., (4) 13 (1980), 303-315. | Numdam | MR | Zbl

[M] J. Milnor, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542. | MR | Zbl

[Sch] D. Schueth, Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds, Comment. Math. Helv., 70 (1995), 434-454. | MR | Zbl

[Su] T. Sunada, Riemannian coverings and isospectral manifolds, Ann. of Math., (2) 121 (1985), 169-186. | MR | Zbl

[Sz] Z. Szabo, Locally nonisometric yet super isospectral spaces, preprint. | Zbl

[V] M.F. Vignéras, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., (2) 112 (1980)? 21-32. | MR | Zbl

[W] E.N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata, 12 (1982), 337-346. | MR | Zbl

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