×

Isospectral deformations of closed Riemannian manifolds with different scalar curvature. (English) Zbl 0922.58083

The main aim of this paper is to look more closely to the basic question: to what extent the eigenvalue spectrum of a compact Riemannian manifold determines the geometry of the manifold? The authors show that for \(n\geq 4\), there exist continuous \(d\)-parameter families \( {g_t}\) of isospectral, non-isometric Riemannian metrics on the manifold \(S^n\times T\), where \(T\) is the \(2\)-dimensional torus and \(S^n\) is the \(n\)-dimensional sphere. Here \(d\) is of order at least \(O(n^2)\). These metrics are non-homogeneous. For some of the deformations, the maximum scalar curvature of \(g_t\) depends non-trivially on \(t\). They give the first examples of continuous isospectral deformations of closed manifolds for which the metrics are not locally isometric, as well as they give the first examples of isospectral manifolds with different scalar curvature.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
22E25 Nilpotent and solvable Lie groups
53C20 Global Riemannian geometry, including pinching
PDFBibTeX XMLCite
Full Text: DOI arXiv Numdam EuDML

References:

[1] [Be] , Variétés riemanniennes isospectrales non isométriques, Séminaire Bourbaki 705, no 177-178 (1988-1989), 127-154. · Zbl 0703.53035
[2] [Br] , Constructing isospectral manifolds, Amer. Math. Monthly, 95 (1988), 823-839. · Zbl 0673.58046
[3] [BT] and , Isospectral surfaces of small genus, Nagoya Math. J., 107 (1987), 13-24. · Zbl 0605.58041
[4] [Bu] , Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), 36-2 (1986), 167-192. · Zbl 0579.53036
[5] [D] , Audible and inaudible geometric properties, Rend. Sem. Fac. Sci. Univ. Cagliari, 58 (supplement 1988), 1-26.
[6] [DG1] and , Isospectral deformations I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387. · Zbl 0649.53025
[7] [DG2] and , Isospectral deformations II: Trace formulas, metrics, and potentials, Comm. Pure Appl. Math., 42 (1989), 1067-1095. · Zbl 0709.53030
[8] [E] , Geometry of two-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup., (4) 27 (1994), 611-660. · Zbl 0830.53039
[9] [G1] , 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. · Zbl 0772.58062
[10] [G2] , Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom., 37 (1993), 639-649. · Zbl 0792.53037
[11] [G3] , 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. · Zbl 0811.58063
[12] [GGt] and , Spectral geometry of nilmanifolds, Proceedings of the Summer University of Southern Stockholm: Advances in Inverse Spectral Geometry, Birkhäuser, 1997, 23-49. · Zbl 0892.58076
[13] [GWW] , , and , Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22. · Zbl 0778.58068
[14] [GW1] and , Isospectral deformations of compact solvmanifolds, J. Differential Geom., 19 (1984), 241-256. · Zbl 0523.58043
[15] [GW2] and , The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. · Zbl 0599.53038
[16] [GW3] and , Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differential Geom., to appear. · Zbl 0915.58104
[17] [Gt1] , A new construction of isospectral Riemannian manifolds with examples, Michigan Math. J., 43 (1996), 159-188. · Zbl 0851.53024
[18] [Gt2] , Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal., to appear. · Zbl 1009.58023
[19] [I] , On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup., (4) 13 (1980), 303-315. · Zbl 0451.58037
[20] [M] , Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542. · Zbl 0124.31202
[21] [Sch] , Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds, Comment. Math. Helv., 70 (1995), 434-454. · Zbl 0847.58078
[22] [Su] , Riemannian coverings and isospectral manifolds, Ann. of Math., (2) 121 (1985), 169-186. · Zbl 0585.58047
[23] [Sz] , Locally nonisometric yet super isospectral spaces, preprint. · Zbl 0964.53026
[24] [V] , Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., (2) 112 (1980)? 21-32. · Zbl 0445.53026
[25] [W] , Isometry groups on homogeneous nilmanifolds, Geom. Dedicata, 12 (1982), 337-346. · Zbl 0489.53045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.