The length spectrum of riemannian two-step nilmanifolds
Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 33 (2000) no. 2, p. 181-209
@article{ASENS_2000_4_33_2_181_0,
     author = {Gornet, Ruth and Mast, Maura B.},
     title = {The length spectrum of riemannian two-step nilmanifolds},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Elsevier},
     volume = {Ser. 4, 33},
     number = {2},
     year = {2000},
     pages = {181-209},
     doi = {10.1016/s0012-9593(00)00111-7},
     zbl = {0968.53036},
     mrnumber = {2001d:58042},
     language = {en},
     url = {http://www.numdam.org/item/ASENS_2000_4_33_2_181_0}
}
Gornet, Ruth; Mast, Maura B. The length spectrum of riemannian two-step nilmanifolds. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 33 (2000) no. 2, pp. 181-209. doi : 10.1016/s0012-9593(00)00111-7. http://www.numdam.org/item/ASENS_2000_4_33_2_181_0/

[1] Anselone P., Collectively Compact Operator Approximation Theory and Applications to Integral Equations, Prentice Hall, 1971. | MR 56 #1753 | Zbl 0228.47001

[2] Bérard P., Spectral Geometry : Direct and Inverse Problems, Lecture Notes in Mathematics, Vol. 1207, Springer, Berlin, 1980. | MR 88f:58146 | Zbl 0608.58001

[3] Berger M., Le spectre des variétés Riemanniennes, Rev. Roum. Math. Pures et Appl. 13 (1968) 915-931. | MR 39 #892 | Zbl 0181.49603

[4] Berger M., Gauduchon P., Mazet E., Le Spectre d'Une Variété Riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer, Berlin, 1971. | MR 43 #8025 | Zbl 0223.53034

[5] Berndt J., Tricerri F., Vanhecke L., Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces, Lecture Notes in Mathematics, Vol. 1598, Springer, Berlin, 1995. | MR 97a:53068 | Zbl 0818.53067

[6] Besse A.L., Manifolds all of Whose Geodesics are Closed, Springer, Berlin, 1978. | MR 80c:53044 | Zbl 0387.53010

[7] Blanchard H., Private communication.

[8] Buser P., Geometry and Spectra of Compact Riemann Surfaces, Progress in Mathematics, Vol. 106, Birkhäuser, 1992. | MR 93g:58149 | Zbl 0770.53001

[9] Colin De Verdière Y., Spectre du Laplacian et longueur des géodesiques periodiques I, II, Compositio Math. 27 (1973) 83-106, 159-184. | Numdam | MR 50 #1293 | Zbl 0281.53036 | Zbl 0272.53034

[10] Damek E., Ricci F., A class of nonsymmetric harmonic Riemannian spaces, Bull. Amer. Math. Soc. (N.S.) 27 (1992) 139-142. | MR 93b:53043 | Zbl 0755.53032

[11] Duistermaat J.J., Guillemin V.W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1977) 39-79. | MR 53 #9307 | Zbl 0307.35071

[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. | Numdam | MR 95m:53059 | Zbl 0820.53047

[13] Eberlein P., Geometry of two-step nilpotent groups with a left invariant metric II, Trans. Amer. Math. Soc. 343 (1994) 805-828. | MR 95b:53061 | Zbl 0830.53039

[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. | Numdam | MR 99c:53036 | Zbl 0902.58027

[15] Fanaï H.-R., Conjugaison géodésique des nilvariétés de rang deux, Journal of Lie Theory (1999) (to appear). | Zbl 0990.53084

[16] Gohberg I., Lancaster P., Rodman L., Matrix Polynomials, Academic Press, 1982. | MR 84c:15012 | Zbl 0482.15001

[17] Gordon C.S., The Laplace spectra versus the length spectra of Riemannian manifolds, Contemporary Math. 51 (1986) 63-79. | MR 87i:58170 | Zbl 0591.53042

[18] Gordon C.S., Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom. 37 (1993) 639-649. | MR 94b:58098 | Zbl 0792.53037

[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. | MR 95k:58166 | Zbl 0811.58063

[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. | Numdam | MR 99b:53049 | Zbl 0922.58083

[21] Gordon C.S., Wilson E.N., The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986) 253-271. | MR 87k:58275 | Zbl 0599.53038

[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. | MR 99a:58159 | Zbl 0915.58104

[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. | MR 95i:22015 | Zbl 0842.22008

[24] Gornet R., A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996) 159-188. | MR 97b:58143 | Zbl 0851.53024

[25] Gornet R., The marked length spectrum vs. the p-form spectrum of Riemannian nilmanifolds, Comment. Math. Helv. 71 (1996) 297-329. | MR 97e:58222 | Zbl 0861.58043

[26] Huber H., Über eine neue Klasse automorpher Funktionen und ein Gitterpunktproblem in der hyperbolischen Ebene, Comment. Math. Helv. 30 (1955) 20-62. | MR 17,603b | Zbl 0065.31603

[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. | MR 27 #4923 | Zbl 0101.05702

[28] Kaplan A., Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981) 127-136. | MR 82h:22008 | Zbl 0495.53046

[29] Kaplan A., On the geometry of groups of Heisenberg type, Bull. London Math. Soc. 15 (1983) 35-42. | MR 84h:53063 | Zbl 0521.53048

[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. | MR 87d:53092 | Zbl 0634.53033

[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. | MR 83m:47015 | Zbl 0493.47008

[33] Lee K.B., Park K., Smoothly closed geodesics in 2-step nilmanifolds, Indiana Univ. Math. J. 45 (1996) 1-14. | MR 97h:53044 | Zbl 0862.53037

[34] Mast M., Closed geodesics in 2-step nilmanifolds, Indiana Univ. Math. J. 43 (1994) 885-911. | MR 96a:53057 | Zbl 0818.53065

[35] Milnor J., Eigenvalues of the Laplace operator on certain manifolds, Proc. Nat. Acad. Sci. USA 51 (1964) 542. | MR 28 #5403 | Zbl 0124.31202

[36] Milnor J., Curvatures of left invariant metrics on Lie groups, Adv. Math. 21 (1976) 293-329. | MR 54 #12970 | Zbl 0341.53030

[37] Pesce H., Déformations L-isospectrales sur les nilvariétés de rang deux, C. R. Acad. Sci. Paris, Série I 315 (1992) 821-823. | MR 94b:58101 | Zbl 0761.53023

[38] Pesce H., Calcul du spectre d'une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339 (1993) 433-461. | MR 93k:58227 | Zbl 0791.58099

[39] Pesce H., Une formule de Poisson pour les variétés de Heisenberg, Duke Math. J. 73 (1994) 79-95. | MR 95i:58183 | Zbl 0803.58054

[40] Raghunathan M.S., Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebeite, Vol. 68, Springer, Berlin, 1972. | MR 58 #22394a | Zbl 0254.22005

[41] Schueth D., Continuous families of isospectral metrics on simply connected manifolds, Ann. of Math. (2) 149 (1) (1999) 287-308. | MR 2000c:58063 | Zbl 0964.53027

[42] Szabó Z., Locally nonisometric yet super isospectral spaces, Geom. Funct. Anal. 9 (1) (1999) 185-214. | MR 2000a:58089 | Zbl 0964.53026

[43] Tanno S., Eigenvalues of the Laplacian of Riemannian manifolds, Tohoku Math. J. 25 (1973) 391-403. | MR 48 #12405 | Zbl 0266.53033

[44] Varadarajan V.S., Lie Groups, Lie Algebras, and their Representations, Graduate Texts in Mathematics, Vol. 102, Springer, Berlin, 1984. | MR 85e:22001 | Zbl 0955.22500

[45] Wolf J., Curvature in nilpotent Lie groups, Proc. Amer. Math. Soc. 15 (1964) 271-274. | MR 28 #5405 | Zbl 0134.17905