Archiv der Mathematik

, Volume 95, Issue 1, pp 75–85 | Cite as

Equivariant isospectrality and Sunada’s method



We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold \({\mathcal{O}}\) and a smooth manifold M are isospectral, then they cannot admit non-trivial finite Riemannian covers \({M_1 \to\mathcal{O}}\) and M2M where M1 and M2 are isospectral manifolds.

Mathematics Subject Classification (2000)

Primary 53C20 58J50 


Laplacian Eigenvalue spectrum Orbifolds 


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  1. 1.
    Y.-J. Chiang, Spectral geometry of V-manifolds and its application to harmonic maps, Differential Geometry: partial differential equations on manifolds, (1993), 93–99.Google Scholar
  2. 2.
    Choi S.: Geometric structures on orbifolds and holonomy representations. Geom. Dedicata 104, 161–199 (2004)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Donnelly H.: The asymptotic splitting of L 2(M) into irreducibles. Math. Ann. 237, 23–40 (1978)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Donnelly H.: Asymptotic expansions for the compact quotients of properly discontinuous group actions. Illinois J. Math. 23, 485–496 (1979)MATHMathSciNetGoogle Scholar
  5. 5.
    Dryden E.B. et al.: Asymptotic expansion of the heat kernel for orbifolds. Michigan Math. J. 56, 205–238 (2008)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Eberlein P.: Geometry of nonpositively curved manifolds, Chicago Lectures in Mathematics. The Chicago University Press, Chicago (1996)Google Scholar
  7. 7.
    Folland G.: A course in abstract harmonic analysis. CRC Press, Boca Raton (1995)MATHGoogle Scholar
  8. 8.
    Gordon C.S.: Isospectral deformations of metrics on spheres. Invent. Math. 145, 317–331 (2001)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Gordon C.S., 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, 2297–2314 (2003)MATHMathSciNetGoogle Scholar
  10. 10.
    Pesce H.: Représentations relativement équivalentes et variétés riemanniennes isospectrales. Comment. Math. Helv. 71, 243–268 (1996)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Rossetti J.P., Schueth D., Weilandt M.: Isospectral orbifolds with different maximal isotropy orders. Ann. Glob. Anal. Geom 34, 351–366 (2008)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Schueth D.: Isospectral manifolds with different local geometries. J. reine angew. Math. 534, 41–94 (2001)MATHMathSciNetGoogle Scholar
  13. 13.
    D. Schueth, Constructing isospectral metrics via principal connections, Geometric Analysis and Nonlinear Partial Differential Equations (S. Hildebrandt, H. Karcher eds.) Springer-Verlag (2003), 69–79.Google Scholar
  14. 14.
    Schueth D.: Isospectral metrics on five-dimensional spheres. J. Differential Geom. 58, 87–111 (2001)MATHMathSciNetGoogle Scholar
  15. 15.
    Sunada T.: Riemannian coverings and isospectral manifolds. Ann. of Math. 121, 169–186 (1985)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Sutton C.J.: Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions. Comment. Math. Helv. 77, 701–717 (2002)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    W. Thurston, The geometry and topology of 3-manifolds, Princeton University, 1976–79.Google Scholar
  18. 18.
    M. Weilandt, Isospectral orbifolds with different isotropy orders, Diplom Thesis, Humboldt-Universität zu Berlin, 2007.

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© Springer Basel AG 2010

Authors and Affiliations

  1. 1.Department of MathematicsDartmouth CollegeHanoverUSA

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