• Gordon, Carolyn S.
  • Published : 2001.09.01


Two compact Riemannian manifolds are said to be isospectral if the associated Laplace-Beltrami operators have the same eigenvalue spectrum. We describe a method, based on the used of Riemannian submersions, for constructing isospectral manifolds with different local geometry and survey examples constructed through this method.


isospectral manifolds;Laplacian


  1. Math.Ann v.292 Transplantation et isospectralite I P.Berard
  2. J.London Math.Soc. v.48 Transplantation et isospectralite II
  3. Nagoya Math.J. v.107 Isospectral surfaces of small genus R.Brooks;R.Tse
  4. Ann.Inst.Fourier(Trenoble) v.48 Isospectral deformations of closed Riemannian manifolds with different scalar curvature C.Gordon;R.Gornet;D.Schueth;D.Webb;E.M.Wilson
  5. Ann.of Math.(to appear) Cornocopia of isospectral pairs of metrics constructed on balls and spheres with different local geometries
  6. Ann.of Math. v.112 Varietes Riemanniennes isospectrales et non isometriques M.F.Vigneras
  7. J.Differential Geom. v.8 Mamofold maps commuting with the Laplacian B.Watson
  8. Michigan Math.J. v.43 no.1 The spectrum of the Laplacian on Ricmannian Heisenberg manifolds
  9. Invent.Math.(to appear) Isospectral deformations of metrics on spheres
  10. Ann.Inst.Foruier(Grenoble) v.36 Isospectral Riemann surfaces P.Buser
  11. Perspect.Math. v.8 Riemannian manifolds p-isospectral but not(p+1)-isospectral, Geometry of Manifolds(Matsumoto)
  12. J.Reine Angew.Math.(to appear) Isospectral manifolds with different local geometries
  13. Contemporary Mathematics: Geometry of the Spectrum v.173 Isospectral closed Riemannian manifolds which are not locally isometric II
  14. Invariance theory, the heat equation, and the Atiyah-Singer index theorem P.B.Gilkey
  15. J.Differential Geom. v.19 Isospectral deformations of compact solvmanifolds C.Gordon;E.N.Wilson
  16. Duke Math.J.(to appear) Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric C.S.Gordon;Z.I.Szabo
  17. J.Geom.Anal. v.10 Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms
  18. Amer.Math.Monthly v.73 Can one hear the shape of a drum? M.Kac
  19. Spectra of Riemannian Manifolds Isospectral problem for spherical space forms A.Ikeda;M.Berger;S.Murakami;T.Ochiai(eds.)
  20. J.Geom.Anal.(to appear) Flat monifolds isospectral on p-forms R.Miatello;J.P.Rossetti
  21. Comm.Pur Appl.Math. v.42 Isospectral Deformations II:trace formulas, metrics, and potentials D.DeTurck;C.Gordon;K.B.Lee(appendix)
  22. Geom.Dedicata v.11 Riemannian nilmanifolds attached to Clifford modules A.Kaplan
  23. Ann.of Math. v.149 Continuous families of isospectral metrics on simply connected manifolds D.Schueth
  24. Topology v.19 Some inverse spectral results for negatively curved 2-manifolds V.Guillemin;D.Kazhdan
  25. Ann.of Math. v.121 Riemannian coverings and isospectral minifolds T.Sunada
  26. Bull.London Math.Soc. v.15 On the geometry of groups of Heisenberg type
  27. Proc.Natl.Acad.Sci. v.51 Eigenvalues of the Laplace operator on certain manifolds J.Milnor
  28. Ann.of Math.(to appear) Isospectral balls and spheres
  29. Geom.Funct.Anal. v.9 Locally non-isometric yet super isospectral spaces Z.I.Szabo
  30. Michigan Math.J. v.43 no.1 A new construction of isopectral Riemannian nilmanifolds with examples Ruth Gornet
  31. Topology v.37 Spectral rigidity of a compact negatively curved manifold C.Croke;V.Sharafutdinov
  32. Adv.Math. v.138 Mutually isospectral Riemann surfaces R.Brooks;R.Gornet;W.Gustafson
  33. Invent.Math. v.110 Isospcetral plane domains and surfaces Riemannian orbifolds C.Gordon;D.Webb;S.Wolpert
  34. J.Differential Geom v.47 Continuous families of isospectral Riemannian manifolds which are not locally isometric
  35. J.Differential Geom. v.24 Riemannian manifolds isospectral on functions but not on 1-forms C.S.Gordon
  36. C.R.Acad.Sci. v.3118 Representations relativement equivalentes et varietes Riemanniennes isospectrales Hubert Pesce