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

Gordon, Carolyn S.; Rossetti, Juan Pablo

doi:10.5802/aif.2007

Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
[Volume du bord et spectre de longueurs des variétés riemanniennes : les invariants que le spectre de Hodge de degré moyen ne révèle pas]

Gordon, Carolyn S.¹ ; Rossetti, Juan Pablo ²

¹ Dartmouth College, Hanover, N.H. 03755 (USA)
² Universidad Nacional de Córdoba, FAMAF, Ciudad Universitaria, 5000 Córdoba (Argentine)

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

Résumé
Abstract

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.

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.

MR Zbl

DOI : 10.5802/aif.2007

Classification : 58J53, 53C20
Keywords: spectral geometry, Hodge Laplacian, isospectral manifolds, heat invariants
Mot clés : géométrie spectrale, laplacien de Hodge, variétés isospectrales, invariants de la chaleur

Affiliations des auteurs :

Gordon, Carolyn S. ¹ ; Rossetti, Juan Pablo ²

¹ Dartmouth College, Hanover, N.H. 03755 (USA)
² Universidad Nacional de Córdoba, FAMAF, Ciudad Universitaria, 5000 Córdoba (Argentine)

@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},
     pages = {2297--2314},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {53},
     number = {7},
     year = {2003},
     doi = {10.5802/aif.2007},
     zbl = {1049.58033},
     mrnumber = {2044174},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2007/}
}

TY  - JOUR
AU  - Gordon, Carolyn S.
AU  - Rossetti, Juan Pablo
TI  - Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
JO  - Annales de l'Institut Fourier
PY  - 2003
SP  - 2297
EP  - 2314
VL  - 53
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2007/
DO  - 10.5802/aif.2007
LA  - en
ID  - AIF_2003__53_7_2297_0
ER  -

%0 Journal Article
%A Gordon, Carolyn S.
%A Rossetti, Juan Pablo
%T Boundary volume and length spectra of Riemannian manifolds: what the middle degree Hodge spectrum doesn't reveal
%J Annales de l'Institut Fourier
%D 2003
%P 2297-2314
%V 53
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2007/
%R 10.5802/aif.2007
%G en
%F 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, Tome 53 (2003) no. 7, pp. 2297-2314. doi : 10.5802/aif.2007. https://aif.centre-mersenne.org/articles/10.5802/aif.2007/

Bibliographie
Cité par

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

[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) (Lecture Notes in Math.), Volume No 1481 (1991), pp. 5-17 | Zbl

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

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

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

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

[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, Volume 48 (1998) no. 2, pp. 593-607 | DOI | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT