Isospectral deformations of closed riemannian manifolds with different scalar curvature

Carolyn S. Gordon; Ruth Gornet; Dorothee Schueth; David L. Webb; Edward N. Wilson

Annales de l'institut Fourier (1998)

  • Volume: 48, Issue: 2, page 593-607
  • ISSN: 0373-0956

Abstract

top
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 m ≥ 2 and S n is a sphere of dimension n ≥ 4 . 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.

How to cite

top

Gordon, Carolyn S., et al. "Isospectral deformations of closed riemannian manifolds with different scalar curvature." Annales de l'institut Fourier 48.2 (1998): 593-607. <http://eudml.org/doc/75294>.

@article{Gordon1998,
abstract = {We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. 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.},
author = {Gordon, Carolyn S., Gornet, Ruth, Schueth, Dorothee, Webb, David L., Wilson, Edward N.},
journal = {Annales de l'institut Fourier},
keywords = {spectral geometry; isospectral deformations; scalar curvature},
language = {eng},
number = {2},
pages = {593-607},
publisher = {Association des Annales de l'Institut Fourier},
title = {Isospectral deformations of closed riemannian manifolds with different scalar curvature},
url = {http://eudml.org/doc/75294},
volume = {48},
year = {1998},
}

TY - JOUR
AU - Gordon, Carolyn S.
AU - Gornet, Ruth
AU - Schueth, Dorothee
AU - Webb, David L.
AU - Wilson, Edward N.
TI - Isospectral deformations of closed riemannian manifolds with different scalar curvature
JO - Annales de l'institut Fourier
PY - 1998
PB - Association des Annales de l'Institut Fourier
VL - 48
IS - 2
SP - 593
EP - 607
AB - We construct the first examples of continuous families of isospectral Riemannian metrics that are not locally isometric on closed manifolds , more precisely, on $S^n\times T^m$, where $T^m$ is a torus of dimension $m\ge 2$ and $S^n$ is a sphere of dimension $n\ge 4$. 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.
LA - eng
KW - spectral geometry; isospectral deformations; scalar curvature
UR - http://eudml.org/doc/75294
ER -

References

top
  1. [Be] P. BÉRARD, Variétés riemanniennes isospectrales non isométriques, Séminaire Bourbaki 705, no 177-178 (1988-1989), 127-154. Zbl0703.53035
  2. [Br] R. BROOKS, Constructing isospectral manifolds, Amer. Math. Monthly, 95 (1988), 823-839. Zbl0673.58046MR89k:58285
  3. [BT] R. BROOKS and R. TSE, Isospectral surfaces of small genus, Nagoya Math. J., 107 (1987), 13-24. Zbl0605.58041MR88m:58182
  4. [Bu] P. BUSER, Isospectral Riemann surfaces, Ann. Inst. Fourier (Grenoble), 36-2 (1986), 167-192. Zbl0579.53036MR88d:58123
  5. [D] D. DETURCK, Audible and inaudible geometric properties, Rend. Sem. Fac. Sci. Univ. Cagliari, 58 (supplement 1988), 1-26. 
  6. [DG1] D. DETURCK and C. GORDON, Isospectral deformations I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math., 40 (1987), 367-387. Zbl0649.53025MR88m:58186
  7. [DG2] D. DETURCK and C. GORDON, Isospectral deformations II: Trace formulas, metrics, and potentials, Comm. Pure Appl. Math., 42 (1989), 1067-1095. Zbl0709.53030MR91e:58197
  8. [E] P. EBERLEIN, Geometry of two-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup., (4) 27 (1994), 611-660. Zbl0830.53039MR95m:53059
  9. [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. Zbl0772.58062MR93m:58113
  10. [G2] C.S. GORDON, Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom., 37 (1993), 639-649. Zbl0792.53037MR94b:58098
  11. [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. Zbl0811.58063MR95k:58166
  12. [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. Zbl0892.58076MR2000m:58058
  13. [GWW] C.S. GORDON, D. WEBB, and S. WOLPERT, Isospectral plane domains and surfaces via Riemannian orbifolds, Invent. Math., 110 (1992), 1-22. Zbl0778.58068MR93h:58172
  14. [GW1] C.S. GORDON and E.N. WILSON, Isospectral deformations of compact solvmanifolds, J. Differential Geom., 19 (1984), 241-256. Zbl0523.58043MR85j:58143
  15. [GW2] C.S. GORDON and E.N. WILSON, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J., 33 (1986), 253-271. Zbl0599.53038MR87k:58275
  16. [GW3] C.S. GORDON and E.N. WILSON, Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differential Geom., to appear. Zbl0915.58104
  17. [Gt1] R. GORNET, A new construction of isospectral Riemannian manifolds with examples, Michigan Math. J., 43 (1996), 159-188. Zbl0851.53024MR97b:58143
  18. [Gt2] R. GORNET, Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, J. Geom. Anal., to appear. Zbl1009.58023
  19. [I] A. IKEDA, On lens spaces which are isospectral but not isometric, Ann. Sci. École Norm. Sup., (4) 13 (1980), 303-315. Zbl0451.58037MR83a:58091
  20. [M] J. MILNOR, Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. U.S.A., 51 (1964), 542. Zbl0124.31202MR28 #5403
  21. [Sch] D. SCHUETH, Isospectral deformations on Riemannian manifolds which are diffeomorphic to compact Heisenberg manifolds, Comment. Math. Helv., 70 (1995), 434-454. Zbl0847.58078MR96g:58200
  22. [Su] T. SUNADA, Riemannian coverings and isospectral manifolds, Ann. of Math., (2) 121 (1985), 169-186. Zbl0585.58047MR86h:58141
  23. [Sz] Z. SZABO, Locally nonisometric yet super isospectral spaces, preprint. Zbl0964.53026
  24. [V] M.F. VIGNÉRAS, Variétés riemanniennes isospectrales et non isométriques, Ann. of Math., (2) 112 (1980)? 21-32. Zbl0445.53026MR82b:58102
  25. [W] E.N. WILSON, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata, 12 (1982), 337-346. Zbl0489.53045MR84a:53048

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.