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Lens Spaces, Isospectral on Forms but not on Functions

Published online by Cambridge University Press:  01 February 2010

Ruth Gornet  and
Jeffrey McGowan
Ruth Gornet
Affiliation:
Department of Mathematics, University of Texas at Arlington, USA, rgornet@uta.edu
Jeffrey McGowan
Affiliation:
Department of Mathematics, Central Connecticut State University, USA, mcgowan@ccsu.edu

Abstract

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We study the p-form spectrum of the Laplace-Beltrami operator acting on lens spaces as considered by Ikeda [Geometry of manifolds (Academic Press, Boston, MA, 1989) 383–417]. Ikeda gave examples of such spaces that are non-isometric but isospectral for all p ≤ p0. In this paper we exhibit examples of such spaces that are not isometric, and are isospectral for various, but not for all. values of p. In particular, examples are given of non-isometric lens spaces that are isospectral for some values of p but not for the case p = 0.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 9 , 2006 , pp. 270 - 286
Copyright
Copyright © London Mathematical Society 2006

References

1. Bourbaki, Nicolas, Lie groups and Lie algebras, translated from the 1975 and 1982 French originals by Andrew Pressley, Elements of Mathematics (Berlin) (Springer, Berlin, 2005) Chapters 7–9.Google Scholar
2. Caprara, Alberto, Kellerer, Hans and Pferschy, Ulrich, ‘The multiple subset sum problem’, SIAM J. Optim. 11 (2000) 308–319 (electronic).CrossRefGoogle Scholar
3. Caprara, Alberto, Kellerer, Hans and Pferschy, Ulrich, ‘A PTAS for the multiple subset sum problem with different knapsack capacities’, Inform. Process. Lett. 73 (2000) 111–118.CrossRefGoogle Scholar
4. Chavel, Isaac, Eigenvalues in Riemannian geometry, including a chapter by Burton Randol, with an appendix by Jozef Dodziuk, Pure and Applied Mathematics 115 (Academic Press Inc., Orlando, FL, 1984).Google Scholar
5. Donnelly, Harold, ‘G-spaces, the asymptotic splitting of L2(M) into irreducibles’, Math. Ann. 237 (1978) 23–40.CrossRefGoogle Scholar
6. Gordon, Carolyn S., ‘Riemannian manifolds isospectral on functions but not on 1-forms’, J. Differential Geom. 24 (1986) 79–96.CrossRefGoogle Scholar
7. Gordon, Carolyn S., ‘Isospectral closed Riemannian manifolds which are not locally isometric’, J. Differential Geom. 37 (1993) 639–649.CrossRefGoogle Scholar
8. Gordon, Carolyn S., ‘Survey of isospectral manifolds’, Handbook of differential geometry, Vol. I (North-Holland, Amsterdam, 2000) 747–778.CrossRefGoogle Scholar
9. Gordon, Carolyn S., ‘Isospectral deformations of metrics on spheres’, Invent. Math. 145 (2001) 317–331.CrossRefGoogle Scholar
10. Gordon, Carolyn S., ‘Isospectral manifolds with different local geometry’, Mathematics in the new Millennium, Seoul, 2000, J. Korean Math. Soc. 38 (2001) 955–970.Google Scholar
11. Gordon, C. S., Gornet, R., Schueth, D., Webb, D. L. and Wilson, E. N., ‘Isospectral deformations of closed Riemannian manifolds with different scalar curvature’, Ann. Inst. Fourier (Grenoble) 48 (1998) 593–607.CrossRefGoogle Scholar
12. Gordon, C. S. and Rossetti, J. P., ‘Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal’, Ann. Inst. Fourier (Grenoble) 53 (2003) 2297–2314.CrossRefGoogle Scholar
13. Gordon, Carolyn S. and Schueth, Dorothee, ‘Isospectral potentials and conformally equivalent isospectral metrics on spheres, balls and Lie groups’, J. Geom. Anal. 13 (2003) 300–328.CrossRefGoogle Scholar
14. Gornet, Ruth, ‘A new construction of isospectral Riemannian nilmanifolds with examples’, Michigan Math. J. 43 (1996) 159–188.CrossRefGoogle Scholar
15. Gornet, Ruth, ‘Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms’, J. Geom. Anal. 10 (2000) 281–298.CrossRefGoogle Scholar
16. Ikeda, Akira, ‘Riemannian manifolds p-isospectral but not (p + l)-isospectral‘, Geometry of manifolds, Matsumoto, 1988, Perspect. Math. 8 (Academic Press, Boston, MA, 1989) 383–417.Google Scholar
17. Miatello, R. J. and Rossetti, J. P., ‘Flat manifolds isospectral on p-forms’, J. Geom. Anal. 11 (2001) 649–667.CrossRefGoogle Scholar
18. Miatello, R. J. and Rossetti, J. P., ‘Length spectra and p-spectra of compact flat manifolds’, J. Geom. Anal. 13 (2003) 631–657.CrossRefGoogle Scholar
19. Patodi, V. K., ‘Curvature and the eigenforms of the Laplace operator’, J. Differential Geom. 5 (1971) 233’249.CrossRefGoogle Scholar
20. Pesce, Hubert, ‘Représentations relativement équivalentes et variétés riemanniennes isospectrales’, Comment. Math. Helv. 71 (1996) 243–268.CrossRefGoogle Scholar
21. Schueth, Dorothee, ‘Continuous families of isospectral metrics on simply connected manifolds’, Ann. of Math. (2) 149 (1999) 287–308.CrossRefGoogle Scholar
22. Schueth, Dorothee, ‘Isospectral manifolds with different local geometries’, J. Reine Angew. Math. 534 (2001) 41–94.Google Scholar
23. Schueth, Dorothee, ‘Isospectral metrics on five-dimensional spheres’, J. Differential Geom. 58 (2001) 87–111.CrossRefGoogle Scholar
24. Szabó, Z. I., ‘Locally non-isometric yet super isospectral spaces’, Geom. Fund. Anal. 9 (1999) 185–214.CrossRefGoogle Scholar
25. Szabó, Z. I., ‘Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries’, Ann. of Math. (2) 154 (2001) 437–475.CrossRefGoogle Scholar
26. Szabó, Z. I., ‘A cornucopia of isospectral pairs of metrics on spheres with different local geometries’, Ann. of Math. (2) 161 (2005) 343–395.CrossRefGoogle Scholar
27. Szabó, Z. I., ‘Reconstruction of the intertwining operator and new striking examples added to “Isospectral pairs of metrics on balls and spheres with different local geometries”’, arXiv:math.DG/0510202.Google Scholar
Supplementary material: File

JCM 9 Gornet and McGowan Appendix B Part 1

Gornet and McGowan Appendix B Part 1

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JCM 9 Gornet and McGowan Appendix B Part 2

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JCM 9 Gornet and McGowan Appendix B Part 3

Gornet and McGowan Appendix B Part 3

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JCM 9 Gornet and McGowan Appendix C Part 1

Gornet and McGowan Appendix C Part 1

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JCM 9 Gornet and McGowan Appendix C Part 2

Gornet and McGowan Appendix C Part 2

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JCM 9 Gornet and McGowan Appendix C Part 3

Gornet and McGowan Appendix C Part 3

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JCM 9 Gornet and McGowan Appendix D

Gornet and McGowan Appendix D

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