The length spectrum of riemannian two-step nilmanifolds
Annales scientifiques de l'École Normale Supérieure (2000)
- Volume: 33, Issue: 2, page 181-209
- ISSN: 0012-9593
Access Full Article
topHow to cite
topGornet, Ruth, and Mast, Maura B.. "The length spectrum of riemannian two-step nilmanifolds." Annales scientifiques de l'École Normale Supérieure 33.2 (2000): 181-209. <http://eudml.org/doc/82513>.
@article{Gornet2000,
author = {Gornet, Ruth, Mast, Maura B.},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Heisenberg-like group; Heisenberg type group; length spectrum; Laplace spectrum; isospectral; two-step nilmanifold},
language = {eng},
number = {2},
pages = {181-209},
publisher = {Elsevier},
title = {The length spectrum of riemannian two-step nilmanifolds},
url = {http://eudml.org/doc/82513},
volume = {33},
year = {2000},
}
TY - JOUR
AU - Gornet, Ruth
AU - Mast, Maura B.
TI - The length spectrum of riemannian two-step nilmanifolds
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2000
PB - Elsevier
VL - 33
IS - 2
SP - 181
EP - 209
LA - eng
KW - Heisenberg-like group; Heisenberg type group; length spectrum; Laplace spectrum; isospectral; two-step nilmanifold
UR - http://eudml.org/doc/82513
ER -
References
top- [1] Anselone P., Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice Hall, 1971. Zbl0228.47001MR56 #1753
- [2] Bérard P., Spectral Geometry : Direct and Inverse Problems, Lecture Notes in Mathematics, Vol. 1207, Springer, Berlin, 1980. Zbl0608.58001MR88f:58146
- [3] Berger M., Le spectre des variétés Riemanniennes, Rev. Roum. Math. Pures et Appl. 13 (1968) 915-931. Zbl0181.49603MR39 #892
- [4] Berger M., Gauduchon P., Mazet E., Le Spectre d'Une Variété Riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer, Berlin, 1971. Zbl0223.53034MR43 #8025
- [5] Berndt J., Tricerri F., Vanhecke L., Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes in Mathematics, Vol. 1598, Springer, Berlin, 1995. Zbl0818.53067MR97a:53068
- [6] Besse A.L., Manifolds all of Whose Geodesics are Closed, Springer, Berlin, 1978. Zbl0387.53010MR80c:53044
- [7] Blanchard H., Private communication.
- [8] Buser P., Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Vol. 106, Birkhäuser, 1992. Zbl0770.53001MR93g:58149
- [9] Colin de Verdière Y., Spectre du Laplacian et longueur des géodesiques periodiques I, II, Compositio Math. 27 (1973) 83-106, 159-184. Zbl0272.53034MR50 #1293
- [10] Damek E., Ricci F., A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992) 139-142. Zbl0755.53032MR93b:53043
- [11] Duistermaat J.J., Guillemin V.W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1977) 39-79. Zbl0307.35071MR53 #9307
- [12] Eberlein P., Geometry of two-step nilpotent groups with a left invariant metric, Ann. Scien. de l'Ecole Norm. Sup. 27 (1994) 611-660. Zbl0820.53047MR95m:53059
- [13] Eberlein P., Geometry of two-step nilpotent groups with a left invariant metric II, Trans. Amer. Math. Soc. 343 (1994) 805-828. Zbl0830.53039MR95b:53061
- [14] Fanaï H.-R., Rigidité du flot géodésique de certaines nilvariétés de rang deux, in : Séminaire de Théorie Spectrale et Géométrie (Grenoble), 1996-1997, pp. 25-36. Zbl0902.58027MR99c:53036
- [15] Fanaï H.-R., Conjugaison géodésique des nilvariétés de rang deux, Journal of Lie Theory (1999) (to appear). Zbl0990.53084
- [16] Gohberg I., Lancaster P., Rodman L., Matrix Polynomials, Academic Press, 1982. Zbl0482.15001MR84c:15012
- [17] Gordon C.S., The Laplace spectra versus the length spectra of Riemannian manifolds, Contemporary Math. 51 (1986) 63-79. Zbl0591.53042MR87i:58170
- [18] Gordon C.S., Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom. 37 (1993) 639-649. Zbl0792.53037MR94b:58098
- [19] Gordon C.S., Isospectral closed Riemannian manifolds which are not locally isometric : II, in : Brooks R., Gordon C.S., Perry P.(Eds.), Contemporary Mathematics : Geometry of the Spectrum, Vol. 173, Amer. Math. Soc., 1994, pp. 121-131. Zbl0811.58063MR95k:58166
- [20] Gordon C.S., Gornet R., Schueth D., Webb D.L., Wilson E.N., Isospectral deformations of closed Riemannian manifolds with different scalar curvature, Ann. Inst. Fourier (Grenoble) 48 (1998) 593-607. Zbl0922.58083MR99b:53049
- [21] Gordon C.S., Wilson E.N., The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986) 253-271. Zbl0599.53038MR87k:58275
- [22] Gordon C.S., Wilson E.N., Continuous families of isospectral Riemannian manifolds which are not locally isometric, J. Differential Geom. 47 (1997) 504-529. Zbl0915.58104MR99a:58159
- [23] Gornet R., The length spectrum and representation theory on two and three-step nilpotent Lie groups, in : Brooks R., Gordon C.S., Perry P. (Eds.), Contemporary Mathematics : Geometry of the Spectrum, Vol. 173, Amer. Math. Soc., 1994, pp. 133-156. Zbl0842.22008MR95i:22015
- [24] Gornet R., A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996) 159-188. Zbl0851.53024MR97b:58143
- [25] Gornet R., The marked length spectrum vs. the p-form spectrum of Riemannian nilmanifolds, Comment. Math. Helv. 71 (1996) 297-329. Zbl0861.58043MR97e:58222
- [26] Huber H., Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv. 30 (1955) 20-62. Zbl0065.31603MR17,603b
- [27] Huber H., Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen I, Math. Ann. 138 (1959) 1-26 ; II, Math. Ann. 142 (1961) 385-398 ; Nachtrag zu II, Math. Ann. 143 (1961) 463-464. Zbl0101.05702MR27 #4923
- [28] Kaplan A., Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981) 127-136. Zbl0495.53046MR82h:22008
- [29] Kaplan A., On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983) 35-42. Zbl0521.53048MR84h:53063
- [30] Kaplan A., Lie groups of Heisenberg type, in : Conference on Differential Geometry on Homogeneous Spaces (Turin 1983), Rend. Sem. Mat. Univ. Politec. Torino (Special Issue), 1984, pp. 117-130. Zbl0634.53033MR87d:53092
- [31] Kato T., Perturbation Theory for Linear Operators, Classics in Mathematics, Springer, Berlin, 1984.
- [32] Kato T., A Short Introduction to Perturbation Theory for Linear Operators, Springer, Berlin, 1982. Zbl0493.47008MR83m:47015
- [33] Lee K.B., Park K., Smoothly closed geodesics in 2-step nilmanifolds, Indiana Univ. Math. J. 45 (1996) 1-14. Zbl0862.53037MR97h:53044
- [34] Mast M., Closed geodesics in 2-step nilmanifolds, Indiana Univ. Math. J. 43 (1994) 885-911. Zbl0818.53065MR96a:53057
- [35] Milnor J., Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. USA 51 (1964) 542. Zbl0124.31202MR28 #5403
- [36] Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293-329. Zbl0341.53030MR54 #12970
- [37] Pesce H., Déformations L-isospectrales sur les nilvariétés de rang deux, C. R. Acad. Sci. Paris, Série I 315 (1992) 821-823. Zbl0761.53023MR94b:58101
- [38] Pesce H., Calcul du spectre d'une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339 (1993) 433-461. Zbl0791.58099MR93k:58227
- [39] Pesce H., Une formule de Poisson pour les variétés de Heisenberg, Duke Math. J. 73 (1994) 79-95. Zbl0803.58054MR95i:58183
- [40] Raghunathan M.S., Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebeite, Vol. 68, Springer, Berlin, 1972. Zbl0254.22005MR58 #22394a
- [41] Schueth D., Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. (2) 149 (1) (1999) 287-308. Zbl0964.53027MR2000c:58063
- [42] Szabó Z., Locally nonisometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1) (1999) 185-214. Zbl0964.53026MR2000a:58089
- [43] Tanno S., Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. 25 (1973) 391-403. Zbl0266.53033MR48 #12405
- [44] Varadarajan V.S., Lie Groups, Lie Algebras, and their Representations, Graduate Texts in Mathematics, Vol. 102, Springer, Berlin, 1984. Zbl0955.22500MR85e:22001
- [45] Wolf J., Curvature in nilpotent Lie groups, Proc. Amer. Math. Soc. 15 (1964) 271-274. Zbl0134.17905MR28 #5405
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.