Abstract
We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on p-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean \({\mathbb{R}^3}\) and are shown to be isospectral on functions using dimension formulas for the eigenspaces developed in [12]. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bérard, P.: Transplantation et isospectralité. I. Math. Ann. 292(3), 547–559 (1992), MR 1152950
Chen, W., Ruan, Y.: Orbifold Gromov-Witten theory (Orbifolds in mathematics and physics). Contemp. Math. 310, 25–85 (2002), MR 1950941
Chiang, Y.-J.: Harmonic maps of V-manifolds. Ann. Global Anal. Geom. 8(3), 315–344 (1990), MR 1089240
Conway, J.H., Sloane, N.J.A.: Four-dimensional lattices with the same theta series. Internat. Math. Res. Notices 4, 93–96 (1992), MR 1177121
DeTurck, D., Gordon, C.S.: Isospectral deformations. II. Trace formulas, metrics, and potentials. Comm. Pure Appl. Math. 42(8), 1067–1095 (1989), MR 1029118
Dryden, E., Gordon, C.S., Greenwald, S.: Asymptotic expansion of the heat kernel for orbifolds. Michigan Math. J. (to appear)
Doyle, P.G., Rossetti, J.P.: Tetra and Didi, the cosmic spectral twins. Geom. Topol. 8, 1227–1242 (2004), MR 2087082
Doyle, P.G., Rossetti, J.P.: Isospectral hyperbolic surfaces have matching geodesics. Preprint. arXiv:math.DG/0605765v1. New York J. Math. (to appear)
Dryden, E., Strohmaier, A.: Huber’s theorem for hyperbolic orbisurfaces. Canad. Math. Bull. (to appear)
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(7), 2297–2314 (2003), MR 2044174
Ikeda, A.: On space forms of real Grassmann manifolds which are isospectral but not isometric. Kodai Math. J. 20(1), 1–7 (1997), MR 1443360
Miatello, R.J., Rossetti, J.P.: Flat manifolds isospectral on p-forms. J. Geom. Anal. 11(4), 649–667 (2001), MR 1861302
Miatello, R.J., Rossetti, J.P.: Comparison of twisted p-form spectra for flat manifolds with different holonomy. Ann. Global Anal. Geom. 21(4), 341–376 (2002), MR 1910457
Pesce, H.: Variétés isospectrales et représentations de groupes. (Geometry of the spectrum). Contemp. Math. 173, 231–240 (1994), MR 1298208
Rossetti, J.P., Conway, J.H.: Hearing the platycosms. Math. Res. Lett. 13(2–3), 475–494 (2006), MR 2231133
Satake, I.: On a generalization of the notion of manifold. Proc. Nat. Acad. Sci. USA 42, 359–363 (1956), MR 0079769
Shams, N., Stanhope, E., Webb, D.L.: One cannot hear orbifold isotropy type. Arch. Math. (Basel) 87(4), 375–384 (2006), MR 2263484
Stanhope, E.: Spectral bounds on orbifold isotropy. Ann. Global Anal. Geom. 27(4), 355–375 (2005), MR 2155380
Sunada, T.: Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 121(1), 169–185 (1985), MR 0782558
Thurston, W.: The geometry and topology of three-manifolds. Lecture notes (1980), Princeton University. Chapter 13. http://msri.org/publications/books/gt3m/
Weilandt M. Isospectral orbifolds with different isotropy orders. Diploma thesis (2007), Humboldt-Univ., Berlin
Wolf, J.A.: Spaces of Constant Curvature, 5th edn. Publish or Perish, Houston, (1984), MR 0928600
Author information
Authors and Affiliations
Corresponding author
Additional information
All three authors were partially supported by DFG Sonderforschungsbereich 647.
Rights and permissions
About this article
Cite this article
Rossetti, J.P., Schueth, D. & Weilandt, M. Isospectral orbifolds with different maximal isotropy orders. Ann Glob Anal Geom 34, 351–366 (2008). https://doi.org/10.1007/s10455-008-9110-3
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10455-008-9110-3