Chapter 6 Survey of isospectral manifolds

Handbook of Differential Geometry

Volumn 1, Issue , 2000, Pages 747-778

  Gordon, Carolyn S ^a  

^a DARTMOUTH COLLEGE   (United States)

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[No Author keywords available]

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EID: 0002038284     PISSN: 18745741     EISSN: None     Source Type: Book Series    
DOI: 10.1016/S1874-5741(00)80009-6     Document Type: Chapter

References (117)

1. Anderson M. Remarks on the compactness of isospectral sets in low dimensions. Duke Math. J. 63 (1991) 699-711
2. Bérard P. Transplantation et isospectralité I. Math. Ann. 292 (1992) 547-559
3. Bérard P. Transplantation et isospectralité II. J. London Math. Soc. 48 (1993) 565-576
4. Bérard P. Variétés Riemanniennes isospectrales non isométriques. Astérisque 177-178 (1989) 127-154
5. P.Bérard, On the construction of isospectral Riemannian manifolds, Preprint Grenoble (based on a short graduate course)
6. Bérard P. Spectral Geometry: Direct and Inverse Problems. Lecture Notes in Math. vol. 1207 (1980), Springer, New York
7. Bérard P., and Besson G. Spectres et groupes cristallographiques II. Ann. Inst. Fourier (Grenoble) 30 (1980) 237-248
8. Bérard P., and Webb D.L. One cannot hear the orientability of surfaces. C. R. Acad. Sci. Paris 320 1 (1995) 533-536
9. Berger M., Gauduchon P., and Mazet E. Le Spectre d'une Variété Riemannienne. Lecture Notes in Math. vol. 194 (1971), Springer, New York
10. Brooks R. Constructing isospectral manifolds. Amer. Math. Monthly 95 (1988) 823-839
11. Brooks R., and Gordon C. Isospectral families of conformally equivalent Riemannian metrics. Bull. Amer. Math. Soc. (N.S.) 23 (1990) 433-436
12. Brooks R., Gornet R., and Gustafson W. Mutually isospectral Riemann surfaces. Adv. in Math. 138 (1998) 306-322
13. Brooks R., Perry P., and Petersen P. Finiteness of diffeomorphism types of isospectral manifolds. Proc. Symp. Pure Math. Vol. 54 (1993) 89-94
14. Brooks R., Perry P., and Petersen P. Compactness and finiteness theorems for isospectral manifolds. J. Reine Angew. Math. 426 (1992) 67-89
15. Brooks R., Perry P., and Yang P. Isospectral sets of conformally equivalent metrics. Duke Math J. 58 (1989) 131-150
16. Brooks R., and Tse R. Isospectral surfaces of small genus. Nagoya Math. J. 107 (1987) 13-24
17. Brüning J., and Lesch M. On the spectral geometry of algebraic curves. J. Reine Angew. Math. 474 (1996) 25-66
18. Buser P. Isospectral Riemann surfaces. Ann. Inst. Fourier (Grenoble) 36 (1986) 167-192
19. Buser P. Cayley Graphs and Planar Isospectral Domains. Lecture Notes in Math. vol. 1339 (1988), Springer, Berlin
20. Buser P. Geometry and Spectra of Compact Riemann Surfaces (1992), Birkhäuser, Boston
21. 391ff. Buser P., Conway J., Doyle P., and Semmler K.-D. Some planar isospectral domains. Internat. Math. Res. Notices 9 (1994)
22. 3. Comment. Math. Helv. 64 (1989) 363-374
23. Chang S.-Y.A., and Yang P. Isospectral conformal metrics on 3-manifolds. J. Amer. Math. Soc. 3 (1990) 117-145
24. Chavel A. Eigenvalues in Riemannian Geometry (1984), Academic Press, New York
25. Chazarain J. Formule de Poisson pour les variétés Riemanniennes. Invent. Math. 24 (1974) 65-82
26. Verdière Y.C.d. Spectre du Laplacian et longeur des géodesiques periodiques II. Compositio Math. 27 (1973) 159-184
27. Verdière Y.C.d. Sur les longuers des trajectoires periodiques d'un billard. In: Dazord P., and Desolneux-Moulis N. (Eds). Geometrie Symplectique et de Contact: Autour du Theoreme de Poincarè-Birkhoff South Rhone Seminar on Geometry, III (1984), Hermann, Paris 122-139
28. Conway J.H., and Sloane N.J.A. Four-dimensional lattices with the same theta series. Internat. Math. Res. Notices 4 (1992) 93-96
29. Croke C., and Sharafutdinov V. Spectral rigidity of a compact negatively curved manifold. Topology 37 (1998) 1265-1273
30. supplement. DeTurck D. Audible and inaudible geometric properties. Proc. of Conference on Geometry and Topology. Rend. Sim. Fac. Sci. Univ. Cagliari 58 (1988) 1-26
31. Part 3. DeTurck D., Gluck H., Gordon C., and Webb D. The geometry of isospectral deformations. Proc. Symp. Pure Math. Vol. 54 (1993) 135-154
32. DeTurck D., Gluck H., Gordon C., and Webb D. The inaudible geometry of nilmanifolds. Invent. Math. 111 2 (1993) 271-284
33. DeTurck D., and Gordon C. Isospectral deformations II: trace formulas, metrics, and potentials. Comm. Pure Appl. Math. 42 (1989) 1067-1095
34. Duistermaat J., and Guillemin V. The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math. 29 (1975) 30-79
35. 2. Pacific J. Math. 154 (1992) 215-229
36. 2. Manuscripta Math. 93 (1997) 357-368
37. Francoise J.P., and Guillemin V. On the period spectrum of a symplectic mapping. J. Func. Anal. 100 (1991) 317-358
38. Fried D. Cyclic resultants of reciprocal polynomials. Lecture Notes in Math. vol. 1345 (1988), Springer, Berlin 124-128
39. Gel'fand I.M., Graev M.I., and Pyatetskii-Shapiro I.I. Representation Theory and Automorphic Functions (1969), W.M. Saunders Company, Philadelphia
40. Gilkey P.B. The spectral geometry of a Riemannian manifold. J. Differential Geom. 10 (1975) 601-618
41. Gilkey P.B. Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem (1984), Publish or Perish, Wilmington, DE
42. Gilkey P.B., Leahy J.V., and Park J.H. Eigenvalues of the form valued Laplacian for Riemannian submer-sions. Proc. Amer. Math. Soc. 126 (1998) 1845-1850
43. Gilkey P.B., and Park J.H. Riemannian submersions which preserve the eigenforms of the Laplacian. Illi-nois J. Math. 40 (1996) 194-201
44. Gilkey P.B., and Park J.H. Eigenvalues of the Laplacian and Riemannian submersions. Yokohama Math. J. 43 (1995) 7-11
45. Gilkey P.B., and Park J.H. The Bochner Laplacian, Riemannian submersions, heat content asymptotics and heat equation asymptotics. Czech Math. J. 49 124 (1999) 233-240
46. Goldberg S.I., and Ishihara T. Riemannian submersions commuting with the Laplacian. J. Differential Geom. 13 (1978) 139-144
47. Gordon C. The Laplace spectra versus the length spectra of Riemannian manifolds. Contemp. Math. Vol. 51 (1986) 63-79
48. Gordon C. Riemannian manifolds isospectral on functions but not on 1-forms. J. Differential Geom. 24 (1986) 79-96
49. Gordon C. Isospectral closed Riemannian manifolds which are not locally isometric II. In: Brooks R., Gordon C., and Perry P. (Eds). Geometry of the Spectrum. Contemp. Mathematics Vol. 173 (1994), Amer. Math. Soc. 121-131
50. Gordon C., and Gornet R. Spectral geometry on nilmanifolds. In: Andersson S., and Lapidus M. (Eds). Progress in Inverse Spectral Geometry (1997), Birkhäuser, Basel 23-49
51. Gordon C., Gornet R., Schüth D., Webb D., and Wilson E.N. Isospectral deformations of closed Rieman-nian manifolds with different scalar curvature. Ann. Inst. Fourier (Grenoble) 48 (1998) 593-607
52. Gordon C., and Mao Y. Comparisons of Laplace spectra, length spectra and geodesic flows of some Rie-mannian manifolds. Math. Res. Lett. 1 (1994) 677-688
53. Gordon C., Mao Y., and Schüth D. Symplectic rigidity of geodesic flows on two-step nilmanifolds. Ann. Sci. École Norm. Sup. 30 (1997) 417-427
54. Gordon C., and Webb D. Isospectral convex domains in the hyperbolic plane. Proc. Amer. Math. Soc. 120 3 (1994) 981-983
55. Gordon C., and Webb D. Isospectral convex domains in Euclidean space. Math. Res. Lett. 1 (1994) 539-545
56. Gordon C., Webb D., and Wolpert S. Isospectral plane domains and surfaces via Riemannian orbifolds. Invent. Math. 110 (1992) 1-22
57. Gordon C., and Wilson E.N. Isospectral deformations of compact solvmanifolds. J. Differential Geom. 19 (1984) 241-256
58. Gordon C., and Wilson E.N. The spectrum of the Laplacian on Riemannian Heisenberg manifolds. Michi-gan Math. J. 33 (1986) 253-271
59. Gordon C., and Wilson E.N. Continuous families of isospectral Riemannian manifolds which are not locally isometric. J. Differential Geom. 47 (1997) 504-529
60. Gornet R. Equivalence of quasi-regular representations of two and three-step nilpotent Lie groups. J. Func. Anal. 119 (1994) 121-137
61. Gornet R. The length spectrum and representation theory on two and three-step nilpotent Lie groups. In: Brooks R., Gordon C., and Perry P. (Eds). Geometry of the Spectrum. Contemp. Mathematics Vol. 173 (1994), Amer. Math. Soc. 133-156
62. Gornet R. A new construction of isospectral Riemannian nilmanifolds with examples. Michigan Math. J. 43 No. 1 (1996)
63. Gornet R. The marked length spectrum vs. the p-form spectrum of Riemannian nilmanifolds. Comment. Math. Helv. 71 (1996) 297-329
64. R. Gornet, Continuous families of Riemannian manifolds isospectral on functions but not on 1-forms, to appear in J. Geom. Anal.
65. Guillemin V. Wave trace invariants and a theorem of Zelditch. Internat. Math. Res. Notices 12 (1993) 303-308
66. Guillemin V. Wave trace invariants. Duke Math. J. 83 (1996) 287-352
67. Guillemin V., and Kazhdan D. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980) 301-312
68. Hörmander L. The spectral function of an elliptic operator. Acta Math. 121 (1968) 193-218
69. Huber H. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen I. Math. Ann. 138 (1959) 1-26
70. Huber H. Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen II. Math. Ann. 143 (1961) 463-464
71. Ikeda A. Isospectral problem for spherical space forms. In: Berger M., Murakami S., and Ochiai T. (Eds). Spectra of Riemannian Manifolds (1983), Kaigai Publications 57-63
72. Ikeda A. Riemannian manifolds p-isospectral but not (p + 1)-isospectral. Geometry of Manifolds (Matsumoto). Perspect. Math. Vol. 8 (1989), Academic Press, Boston, MA 383-417
73. part II. Kac M. Can one hear the shape of a drum?. Amer. Math. Monthly 73 (1966) 1-23
74. Lee J.M. Hearing the volume of a drum in hyperbolic space. Indiana Univ. Math. J. 39 (1980) 585-615
75. Lee J.M. Finite inverse spectral geometry. Contemp. Math. Vol. 127 (1992) 85-100
76. Lohkamp J. Discontinuity of geometric expansions. Comment. Math. Helv. (1996) 213-228
77. Marvize S., and Melrose R.B. Some spectrally isolated convex planar regions. Proc. Nat. Acad. Sci. USA 79 (1982) 7066-7077
78. Marvize S., and Melrose R.B. Spectral invariants of convex planar regions. J. Differential Geom. 17 (1982) 475-502
79. McKean H. Selberg's trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math. 25 (1972) 225-246
80. R. Melrose, The inverse spectral problem for planar domains, Preprint, gopher at ftp-math-papers.mit.edu
81. Miatello I.D., and Miatello R.J. Isospectral compact flat manifolds. Duke Math. J. 68 (1992) 489-498
82. Milnor J. Eigenvalues of the Laplace operator on certain manifolds. Proc. Nat. Acad. Sci. USA 51 (1964) 542
83. Osgood B., Phillips R., and Sarnak P. Compact isospectral sets of metrics. J. Func. Anal. 80 (1988) 212-234
84. Osgood B., Phillips R., and Sarnak P. Moduli space heights and isospectral sets of plane domains. Ann. of Math. 129 (1989) 293-362
85. Patodi V.K. Curvature and the fundamental solution of the heat equation. J. Indian Math. Soc. 34 (1970) 269-285
86. Ouyang H., and Pesce H. Déformations isospectrales sur les nilvariétés de rang deux. C. R. Acad. Sci. Paris, Série I 314 (1992) 621-623
87. Pesce H. Calcul du spectre d'une nilvariété de rang deux et applications. Trans. Amer. Math. Soc. 339 (1993) 433-461
88. Pesce H. Représentations relativement équivalentes et variétés Riemanniennes isospectrales. C. R. Acad. Sci. Paris Série I 3118 (1994) 657-659
89. Pesce H. Representations de groupes et variétés riemanniennes isospectrales. In: Brooks R., Gordon C., and Perry P. (Eds). Geometry of the Spectrum. Contemp. Mathematics Vol. 173 (1994), Amer. Math. Soc. 231-240
90. Pesce H. Une réciproque générique du théorème de Sunada. Compositio Math. 109 (1997) 357-365
91. Pesce H. Borne explicite du nombre de tores plats isospectraux à un tore donné. Manuscripta Math. 75 (1992) 211-223
92. Pesce H. Déformations isospectrales de certaines nilvariétés et finitude spectrale des variétés de Heisen-berg. Ann. Sci. École Norm. Sup. 25 (1992) 515-538
93. Popov G. Length spectrum invariants of Riemannian manifolds. Math. Z. 213 (1993) 311-351
94. Reid A. Isospectrality and commensurability of arithmetic hyperbolic 2-and 3-manifolds. Duke Math. J. 65 (1992) 215-228
95. Sarnak P. The Schur lectures (1992). In: Piatetsky-Shapiro I., and Gelbart S. (Eds). IMCP 8 (1995) 183-236
96. Schüth D. Continuous families of quasi-regular representations of solvable Lie groups. J. Func. Anal. 134 No. 1 (1995) 247-259
97. Schüth D. Line bundle Laplacians over isospectral nilmanifolds. Trans. Amer. Math. Soc. 349 (1997) 3787-3802
98. Schüth D. Continuous families of isospectral metrics on simply connected manifolds. Ann. Math. 149 (1999) 287-308
99. Spatzier R.J. On isospectral locally symmetric spaces and a theorem of von Neumann. Duke Math. J. 59 (1989) 289-294
100. Sternberg S. On the structure of local homeomorphisms of Euclidean n-space II. Amer. J. Math. 80 (1958) 623-631
101. Sternberg S. The structure of local homeomorphisms III. Amer. J. Math. 81 (1959) 578-604
102. Sunada T. Riemannian coverings and isospectral manifolds. Ann. of Math. 121 (1985) 169-186
103. Part 3. Szabó Z. Spectral theory for operator families on Riemannian manifolds. Differential Geometry: Riemannian Geometry. Proc. Symp. Pure Math. Vol. 54 (1993), Amer. Math. Soc., Providence, RI 615-665
104. Szabó Z. Locally nonisometric yet super isospectral spaces. Geom. Funct. Anal. 9 (1999) 185-214
105. Tanno S. Eigenvalues of the Laplacian of Riemannian manifolds. Tôhoku Math. J. 25 (1973) 391-403
106. Urakawa H. Bounded domains which are isospectral but not congruent. Ann. Sci. École Norm. Sup. 15 (1982) 441-456
107. Vignéras M.F. Variétés Riemanniennes isospectrales et non isométriques. Ann. of Math. 112 (1980) 21-32
108. Vijayan C.R. Non-homeomorphic nilmanifolds with identical unitary spectrum. J. Func. Anal. 144 (1997) 424-447
109. Watson B. Manifold maps commuting with the Laplacian. J. Differential Geom. 8 (1973) 85-94
110. Wolpert S. The eigenvalue spectrum as moduli for compact Riemann surfaces. Bull. Amer. Math. Soc. 83 (1977) 1306-1308
111. Wolpert S. The eigenvalue spectrum as moduli for flat tori. Trans. Amer. Math. Soc. 244 (1978) 313-321
112. Zelditch S. Isospectrality in the FIO category. J. Differential Geom. 35 (1992) 689-710
113. Zelditch S. On the generic spectrum of a Riemannian cover. Ann. Inst. Fourier (Grenoble) 40 2 (1990) 407-442
114. Zelditch S. Wave invariants at elliptic closed geodesics. Geom. Func. Anal. 7 (1997) 145-213
115. Zelditch S. Wave invariants for nondegenerate closed geodesics. Geom. Func. Anal. 8 (1998) 179-217
116. Zelditch S. The inverse spectral problem for surfaces of revolution. J. Differential Geom. 49 (1998) 207-264
117. Zelditch S. Fine structure of Zoll spectra. J. Funct. Anal. 143 (1997) 415-460