Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-04-30T12:31:20.438Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructing Isospectral But Non-Isometric Riemannian Manifolds

Published online by Cambridge University Press:  20 November 2018

Sheng Chen
Sheng Chen*
Affiliation:
Department of Mathematics Southwest Texas State University San Marcos, Texas 78666 U.S.A.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper we examine the examples of isospectral but non-isometric Riemannian manifolds given by Milnor, Ikeda, and Vignéras. Of these, only Milnor's example is accounted for by Sunada's method of constructing isospectral manifolds, and even then only as an "unnatural" construction.

Keywords

53C2020F3411R52
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 35 , Issue 3 , 01 September 1992 , pp. 303 - 310
Copyright
Copyright © Canadian Mathematical Society 1992

References

1. Boothby, W. M., An introduction to differentiable manifolds and Riemannian geometry , Academic Press, New York, 1975.Google Scholar
2. Brooks, R., Constructing isospectral manifolds, Amer. Math. Monthly, Vol. 95,8(1988), 823–839.Google Scholar
3. Buser, P., Isospectral Riemann surfaces , Ann. Inst. Fourier XXXVI( 1986), 167–192.Google Scholar
4. DeTurck, D. and Gordon, C., Isospectral deformations: Parti: Riemannian structures on two-step nilspaces, Comm. Pure. Appl. Math., 40(1987),367–387.Google Scholar
5. De Turck, D. and Gordon, C., Isospectral metric and finite Riemannian covering, Contemp. Math., 64(1987), 79–92.Google Scholar
6. Gordon, C. and Wilson, E., Isospectral deformations of compact solvmanifolds, J. Diff. Geom., 19(1984), 241–256.Google Scholar
7. Ikeda, A., On lens spaces which are isospectral but not isometric, Ann. Scient. E. Norm. Sup 13(1980), 303–315.Google Scholar
8. Milnor, J., Eigenvalues of the Laplace operators on certain manifolds , Proc. Nat. Acad. Sci. USA. 51(1964), 542.Google Scholar
9. O'Meara, O. T., Introduction to quadratic forms , Springer, New York, (1973).Google Scholar
10. Perlis, R., On the equation ζk(s) = ζK′(s), J. NumberTh. (3) 9(1977), 342–360.Google Scholar
11. Serre, J. -P., A course in arithmetic , Springer, New York, (1973).Google Scholar
12. Sunada, T., Riemannian coverings and isospectral manifolds, Ann. Math. 121(1985), 169–186.Google Scholar
13. Vignéras, M. F., Arithmétique des algèbres de quaternions , Springer-Verlag Lecture Notes 800(1980).Google Scholar
14. Vignéras, M. F., Variétés riemaniennes isospectrales et non isométriques, Ann. of Math. 112(1980), 21–32.Google Scholar