Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal

Gordon, Carolyn S.; Rossetti, Juan Pablo

Gordon, Carolyn S.; Rossetti, Juan Pablo

Annales de l'Institut Fourier, Volume 53 (2003) no. 7, p. 2297-2314

Abstract
Résumé

Let $M$ be a $2 m$ -dimensional compact Riemannian manifold. We show that the spectrum of the Hodge Laplacian acting on $m$ -forms does not determine whether the manifold has boundary, nor does it determine the lengths of the closed geodesics. Among the many examples are a projective space and a hemisphere that have the same Hodge spectrum on 1- forms, and hyperbolic surfaces, mutually isospectral on 1-forms, with different injectivity radii. The Hodge $m$ -spectrum also does not distinguish orbifolds from manifolds.

Soit $M^{2 m}$ une variété riemannienne compacte. On montre que le spectre du laplacien de Hodge opérant sur les $m$ -formes ne détermine pas si $M$ est à bord, ni les longueurs des géodésiques périodiques. Parmi les nombreux exemples il y a un espace projectif et un hémisphère qui ont le même spectre de Hodge sur les 1-formes, et des espaces hyperboliques, mutuellement isospectraux sur les 1-formes, qui ont des rayons d’injectivité différents. On montre aussi que le $m$ -spectre de Hodge ne distingue pas entre orbifolds et variétés.

MR 2044174 | Zbl 1049.58033

DOI : https://doi.org/10.5802/aif.2007
Classification: 58J53, 53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants

@article{AIF_2003__53_7_2297_0,
     author = {Gordon, Carolyn S. and Rossetti, Juan Pablo},
     title = {Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     pages = {2297-2314},
     doi = {10.5802/aif.2007},
     zbl = {1049.58033},
     mrnumber = {2044174},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2003__53_7_2297_0}
}

Gordon, Carolyn S.; Rossetti, Juan Pablo. Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal. Annales de l'Institut Fourier, Volume 53 (2003) no. 7, pp. 2297-2314. doi : 10.5802/aif.2007. http://www.numdam.org/item/AIF_2003__53_7_2297_0/

References

[B] P. Buser Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston (1992) | MR 1183224 | Zbl 0770.53001

[BBG1] N. Blažić; N. Bokan; P. Gilkey The spectral geometry of the Laplacian and the conformal Laplacian for manifolds with boundary, Global differential geometry and global analysis (Berlin, 1990), Springer, Berlin (Lecture Notes in Math.) Tome No 1481 (1991), pp. 5-17 | Zbl 0755.58047

[BBG2] N. Blažić; N. Bokan; P. Gilkey Spectral geometry of the form valued Laplacian for manifolds with boundary, Indian J. Pure Appl. Math., Tome 23 (1992), pp. 103-120 | MR 1156162 | Zbl 0758.58034

[C] Y. Colin De Verdière Spectre du Laplacien et longeurs des géodésiques périodiques II, Comp. Math., Tome 27 (1973), pp. 159-184 | Numdam | MR 319107 | Zbl 0265.53042

[D1] H. Donnelly Spectrum and the fixed point sets of isometries I, Math. Ann., Tome 224 (1976), pp. 161-170 | Article | MR 420743 | Zbl 0319.53031

[D2] H. Donnelly Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math., Tome 23 (1979), pp. 485-496 | MR 537804 | Zbl 0411.53033

[GGSWW] C.S. Gordon; R. Gornet; D. Schueth; D.L. Webb; E.N. Wilson Isospectral deformations of closed Riemannian manifolds with different scalar curvature, Ann. Inst. Fourier, Tome 48 (1998) no. 2, pp. 593-607 | Article | Numdam | MR 1625586 | Zbl 0922.58083

[Gi] P.B. Gilkey Invariance theory, the heat equation, and the Atiyah-Singer index theorem, Publish or Perish Inc., Wilmington, DE (1984) | MR 783634 | Zbl 0565.58035

[Go1] C.S. Gordon Riemannian manifolds isospectral on functions but not on 1-forms, J. Diff. Geom., Tome 24 (1986), pp. 79-96 | MR 857377 | Zbl 0585.53036

[Go2] C.S. Gordon; R. Brooks, C. Gordon And P. Perry, Eds. Isospectral closed Riemannian manifolds which are not locally isometric II, Contemporary Mathematics: Geometry of the Spectrum, Amer. Math. Soc., Tome vol. 173 (1994), pp. 121-131 | Zbl 0811.58063

[Go3] C.S. Gordon; F.J.E. Dillen, L.C.A. Verstraelen, Eds Survey of isospectral manifolds, Handbook of Differential Geometry I, Elsevier Science B.V. (2000), pp. 747-778 | Zbl 0959.58039

[Go4] C.S. Gordon Isospectral deformations of metrics on spheres, Invent. Math., Tome 145 (2001), pp. 317-331 | Article | MR 1872549 | Zbl 0995.58004

[GSz] C.S. Gordon; Z.I. Szabo Isospectral deformations of negatively curved Riemannian manifolds with boundary which are not locally isometric, Duke Math. J., Tome 113 (2002), pp. 355-383 | Article | MR 1909222 | Zbl 1042.58020

[Gt1] R. Gornet A new construction of isospectral Riemannian nilmanifolds with examples, Mich. Math. J., Tome 43 (1996), pp. 159-188 | Article | MR 1381605 | Zbl 0851.53024

[Gt2] R. Gornet Continuous families of Riemannian manifolds, isospectral on functions but not on $1$ -forms, J. Geom. Anal., Tome 10 (2000), pp. 281-298 | MR 1766484 | Zbl 1009.58023

[GW] C.S. Gordon; E.N. Wilson Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Diff. Geom., Tome 47 (1997), pp. 504-529 | MR 1617640 | Zbl 0915.58104

[Ik] A. Ikeda Riemannian manifolds $p$ -isospectral but not $(p + 1)$ -isospectral, Persp. in Math., Tome 8 (1988), pp. 159-184 | Zbl 0704.53037

[MR1] R. Miatello; J.P. Rossetti Flat manifolds isospectral on $p$ -forms, J. Geom. Anal., Tome 11 (2001), pp. 647-665 | MR 1861302 | Zbl 1040.58014

[MR2] R. Miatello; J.P. Rossetti Comparison of twisted $P$ -form spectra for flat manifolds with diagonal holonomy, Ann. Global Anal. Geom., Tome 21 (2002), pp. 341-376 | Article | MR 1910457 | Zbl 1001.58023

[MR3] R. Miatello; J.P. Rossetti Length spectra and $p$ -spectra of compact flat manifolds, J. Geom. Anal., Tome 13 (2003) no. 4, pp. 631-657 | MR 2005157 | Zbl 1060.58021

[P1] H. Pesce Représentations relativement équivalentes et variétés riemanniennes isospectrales, C. R. Acad. Sci. Paris, Série I, Tome 3118 (1994), pp. 657-659 | MR 1272321 | Zbl 0846.58053

[P2] H. Pesce Quelques applications de la théorie des représentations en géométrie spectrale, Rend. Mat. Appl., Serie VII, Tome 18 (1998), pp. 1-63 | MR 1638226 | Zbl 0923.58056

[S1] D. Schueth Continuous families of isospectral metrics on simply connected manifolds, Ann. Math., Tome 149 (1999), pp. 169-186 | MR 1680563 | Zbl 0964.53027

[S2] D. Schueth Isospectral manifolds with different local geometries, J. reine angew. Math., Tome 534 (2001), pp. 41-94 | Article | MR 1831631 | Zbl 0986.58016

[S3] D. Schueth Isospectral metrics on five-dimensional spheres, J. Diff. Geom., Tome 58 (2001), pp. 87-111 | MR 1895349 | Zbl 1038.58042

[Sc] P. Scott The geometries of 3-manifolds, Bull. London Math. Soc., Tome 15 (1983), pp. 401-487 | Article | MR 705527 | Zbl 0561.57001

[Sun] T. Sunada Riemannian coverings and isospectral manifolds, Annals of Math., Tome 121 (1985), pp. 169-186 | Article | MR 782558 | Zbl 0585.58047

[Sut] C. Sutton Isospectral simply-connected homogeneous spaces and the spectral rigidity of group actions, Comment. Math. Helv., Tome 77 (2002) no. 4, pp. 701-717 | Article | MR 1949110 | Zbl 1018.58025

[Sz1] Z.I. Szabo Locally non-isometric yet super isospectral spaces, Geom. Funct. Anal., Tome 9 (1999), pp. 185-214 | Article | MR 1675894 | Zbl 0964.53026

[Sz2] Z.I. Szabo Isospectral pairs of metrics on balls, spheres, and other manifolds with different local geometries, Ann. of Math., Tome 154 (2001), pp. 437-475 | Article | MR 1865977 | Zbl 1012.53034

[T] W. Thurston The geometry and topology of 3-manifolds, Princeton University Math. Dept., Lecture Notes (1978)