Abstract
We give an explicit formula for the multiplicities of the eigenvalues ofthe Laplacian acting on sections of natural vector bundles over acompact flat Riemannian manifold M Γ = Γ\ℝn,Γ a Bieberbach group. In the case of the Laplacian acting onp-forms, twisted by a unitary character of Γ, when Γ hasdiagonal holonomy group F ≃ ℤ2 k, these multiplicities have acombinatorial expression in terms of integral values of Krawtchoukpolynomials and the so called Sunada numbers. If the Krawtchoukpolynomial K p n(x)does not have an integral root, this expressioncan be inverted and conversely, the presence of such roots allows toproduce many examples of p-isospectral manifolds that are notisospectral on functions. We compare the notions of twistedp-isospectrality, twisted Sunada isospectrality and twisted finitep-isospectrality, a condition having to do with a finite part of thespectrum, proving several implications among them and getting a converseto Sunada's theorem in our context, for n ≤ 8. Furthermore, a finitepart of the spectrum determines the full spectrum. We give new pairs ofnonhomeomorphic flat manifolds satisfying some kind of isospectralityand not another. For instance: (a) manifolds which are isospectral onp-forms for only a few values of p ≠ 0, (b) manifolds which aretwisted isospectral for every χ, a nontrivial character of F, and(c) large twisted isospectral sets.
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References
Brown H., Bülow R., Neubüser J., Wondratschok H. and Zassenhaus H.: Crystallographic Groups of Four-Dimensional Space, Wiley, New York, 1978.
Bérard, P.: FrTransplantation et isospectralité I, Math. Ann. 292 (1992), 547–559.
Büser, P. and Courtois, G.: Finite parts of the spectrum of a Riemann surface, Math. Ann. 287 (1990), 523–530.
Cobb, P.: Manifolds with holonomy group ℤ2⊕ ℤ2and first Betti number zero, J. Differential Geom. 10 (1975), 221–224.
Charlap, L.: Bieberbach Groups and Flat Manifolds, Springer, New York, 1988.
Chihara, L. and Stanton, D.: Zeros of generalized Krawtchouk polynomials, J. Approx. Theory 60 (1990), 43–57.
Dai, X. and Wei, G.: Finite part of the spectrum and isospectrality, Contemp. Math. 173 (1994), 99–107.
DeTurck, D. and Gordon, C.: Isospectral deformations I: Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math. 40 (1987), 367–387.
DeTurck, D. and Gordon, C.: Isospectral deformations II: Trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), 1067–1095.
Dotti, I. and Miatello, R.: Isospectral compact flat manifolds, Duke Math. J. 68 (1992), 489–498.
Eskin, G., Ralston, J. and Trubowitz, E.: On isospectral periodic potentials in ℝn, Comm. Pure Appl. Math. 37 (1984), 647–676 (part I), 715-753 (part II).
Gilkey, P.: On spherical space forms with metacyclic fundamental group which are isospectral but not equivarianly cobordant Compositio Math. 56 (1985), 171–200.
Gordon, C.: Riemannian manifolds isospectral on functions but not on 1-forms, J. Differential Geom. 24 (1986), 79–96.
Gordon, C.: Survey of isospectral manifolds, in: F. J. E. Dillen and L. C. A. Verstraeten (eds), Handbook of Differential Geometry, Vol. 1, Elsevier, Amsterdam, 2000, pp. 747–778.
Gordon, C., Ouyang, H. and Schueth, D.: Distinguishing isospectral nilmanifolds by bundle Laplacians, Math. Res. Lett. 4 (1997), 23–33.
Gornet, R.: Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms, J. Geom. Anal. 10(2) (2000), 281–298.
Ikeda, A.: Riemannian manifolds p-isospectral but not p + 1-isospectral, Perspect. Math. 8 (1988), 159–184.
Krasikov, I. and Litsyn, S.: On integral zeros of Krawtchouk polynomials J. Combin. Theory A 74 (1996), 71–99.
Miatello, R. and Rossetti, J. P.: Isospectral Hantzsche-Wendt manifolds, J. Reine Angew. Math. 515 (1999), 1–23.
Miatello, R. and Rossetti, J. P.: Hantzsche-Wendt manifolds of dimension 7, in: I. Kolár, O. Kowalski, D. Krupka and J. Slovak (eds), Differential Geometry and Applications, Proc. 7th Internat. Conference, Masaryk Univ., Brno, 1999, pp. 379-390.
Miatello, R. and Rossetti, J. P.: Flat manifolds isospectral on p-forms, J. Geom. Anal. 11 (2001), 647–665.
Pesce, H.: FrUne reciproque générique du theorème de Sunada, Compositio Math. 109 (1997), 357–365.
Pesce, H.: FrVariétés hyperboliques et elliptiques fortement isospectrales, J. Funct. Anal. 134 (1995), 363–391.
Roman, S.: Coding and Information Theory, Springer, New York, 1992.
Rossetti, J. P. and Tirao, P.: Five-dimensional Bieberbach groups with holonomy group ℤ2⊕ ℤ2, Geom. Dedicata 77 (1999), 149–172.
Schueth, D.: Line bundle Laplacians over isospectral nilmanifolds, Trans. Amer. Math. Soc. 349 (1997), 3787–3802.
Schueth, D.: Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. 149 (1999), 287–308.
Sunada, T.: Riemannian coverings and isospectral manifolds, Ann. of Math. 121 (1985), 169–186.
Van Lint, J. H. and Wilson, R. M.: A Course in Combinatorics, Cambridge Univ. Press, Cambridge, 1992.
Wolf, J.: Spaces of Constant Curvature, McGraw-Hill, New York, 1967.
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Miatello, R.J., Rossetti, J.P. Comparison of Twisted P-Form Spectra for Flat Manifolds with Diagonal Holonomy. Annals of Global Analysis and Geometry 21, 341–376 (2002). https://doi.org/10.1023/A:1015651821995
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DOI: https://doi.org/10.1023/A:1015651821995