Riemannian nilmanifolds and the trace formula

Gornet, Ruth

doi:10.1090/S0002-9947-05-03965-6

Riemannian nilmanifolds and the trace formula
HTML articles powered by AMS MathViewer

by Ruth Gornet

Trans. Amer. Math. Soc. 357 (2005), 4445-4479

DOI: https://doi.org/10.1090/S0002-9947-05-03965-6

Published electronically: June 10, 2005

PDF | Request permission

Abstract:

This paper examines the clean intersection hypothesis required for the expression of the wave invariants, computed from the asymptotic expansion of the classical wave trace by Duistermaat and Guillemin. The main result of this paper is the calculation of a necessary and sufficient condition for an arbitrary Riemannian two-step nilmanifold to satisfy the clean intersection hypothesis. This condition is stated in terms of metric Lie algebra data. We use the calculation to show that generic two-step nilmanifolds satisfy the clean intersection hypothesis. In contrast, we also show that the family of two-step nilmanifolds that fail the clean intersection hypothesis are dense in the family of two-step nilmanifolds. Finally, we give examples of nilmanifolds that fail the clean intersection hypothesis.

References

Philip M. Anselone, Collectively compact operator approximation theory and applications to integral equations, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. With an appendix by Joel Davis. MR 0443383
Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313, DOI 10.1007/BFb0064643
Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885, DOI 10.1007/978-3-642-61876-5
Peter Buser, Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1183224
Isaac Chavel, Eigenvalues in Riemannian geometry, Pure and Applied Mathematics, vol. 115, Academic Press, Inc., Orlando, FL, 1984. Including a chapter by Burton Randol; With an appendix by Jozef Dodziuk. MR 768584
Yves Colin de Verdière, Spectre du laplacien et longueurs des géodésiques périodiques. I, II, Compositio Math. 27 (1973), 83–106; ibid. 27 (1973), 159–184 (French). MR 348798
Dennis M. DeTurck and Carolyn S. Gordon, Isospectral deformations. I. Riemannian structures on two-step nilspaces, Comm. Pure Appl. Math. 40 (1987), no. 3, 367–387. MR 882070, DOI 10.1002/cpa.3160400306
Dennis M. DeTurck and Carolyn S. Gordon, Isospectral deformations. II. Trace formulas, metrics, and potentials, Comm. Pure Appl. Math. 42 (1989), no. 8, 1067–1095. With an appendix by Kyung Bai Lee. MR 1029118, DOI 10.1002/cpa.3160420803
J. J. Duistermaat and V. W. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), no. 1, 39–79. MR 405514, DOI 10.1007/BF01405172
Patrick Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric, Ann. Sci. École Norm. Sup. (4) 27 (1994), no. 5, 611–660. MR 1296558, DOI 10.24033/asens.1702
Patrick Eberlein, Geometry of $2$-step nilpotent groups with a left invariant metric. II, Trans. Amer. Math. Soc. 343 (1994), no. 2, 805–828. MR 1250818, DOI 10.1090/S0002-9947-1994-1250818-2
Carolyn S. Gordon, The Laplace spectra versus the length spectra of Riemannian manifolds, Nonlinear problems in geometry (Mobile, Ala., 1985) Contemp. Math., vol. 51, Amer. Math. Soc., Providence, RI, 1986, pp. 63–80. MR 848934, DOI 10.1090/conm/051/848934
Carolyn S. Gordon, Riemannian manifolds isospectral on functions but not on $1$-forms, J. Differential Geom. 24 (1986), no. 1, 79–96. MR 857377
Carolyn Gordon, Isospectral closed Riemannian manifolds which are not locally isometric, J. Differential Geom. 37 (1993), no. 3, 639–649. MR 1217163
Carolyn S. Gordon, Isospectral deformations of metrics on spheres, Invent. Math. 145 (2001), no. 2, 317–331. MR 1872549, DOI 10.1007/s002220100150
Carolyn S. Gordon, Isospectral manifolds with different local geometry, J. Korean Math. Soc. 38 (2001), no. 5, 955–970. Mathematics in the new millennium (Seoul, 2000). MR 1849334
Carolyn S. Gordon and Edward N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241–256. MR 739790
Carolyn S. Gordon and Edward N. Wilson, The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), no. 2, 253–271. MR 837583, DOI 10.1307/mmj/1029003354
Carolyn S. Gordon and Edward N. Wilson, Continuous families of isospectral Riemannian metrics which are not locally isometric, J. Differential Geom. 47 (1997), no. 3, 504–529. MR 1617640
Ruth Gornet, Equivalence of quasi-regular representations of two- and three-step nilpotent Lie groups, J. Funct. Anal. 119 (1994), no. 1, 121–137. MR 1255275, DOI 10.1006/jfan.1994.1005
Ruth Gornet, The length spectrum and representation theory on two- and three-step nilpotent Lie groups, Geometry of the spectrum (Seattle, WA, 1993) Contemp. Math., vol. 173, Amer. Math. Soc., Providence, RI, 1994, pp. 133–155. MR 1298202, DOI 10.1090/conm/173/01822
Ruth Gornet, A new construction of isospectral Riemannian nilmanifolds with examples, Michigan Math. J. 43 (1996), no. 1, 159–188. MR 1381605, DOI 10.1307/mmj/1029005395
Ruth Gornet, The marked length spectrum vs. the Laplace spectrum on forms on Riemannian nilmanifolds, Comment. Math. Helv. 71 (1996), no. 2, 297–329. MR 1396677, DOI 10.1007/BF02566421
Ruth Gornet, Continuous families of Riemannian manifolds, isospectral on functions but not on 1-forms, J. Geom. Anal. 10 (2000), no. 2, 281–298. MR 1766484, DOI 10.1007/BF02921826
—, The wave invariants of Riemannian two-step nilmanifolds, in preparation (2005).
Ruth Gornet and Maura B. Mast, The length spectrum of Riemannian two-step nilmanifolds, Ann. Sci. École Norm. Sup. (4) 33 (2000), no. 2, 181–209 (English, with English and French summaries). MR 1755115, DOI 10.1016/S0012-9593(00)00111-7
Ruth Gornet and Maura B. Mast, Length minimizing geodesics and the length spectrum of Riemannian two-step nilmanifolds, J. Geom. Anal. 13 (2003), no. 1, 107–143. MR 1967040, DOI 10.1007/BF02931000
Victor Guillemin, Wave-trace invariants, Duke Math. J. 83 (1996), no. 2, 287–352. MR 1390650, DOI 10.1215/S0012-7094-96-08311-8
V. Guillemin and A. Uribe, On the de Haas-van Alphen effect, Asymptotic Anal. 6 (1993), no. 3, 205–217. MR 1201193, DOI 10.3233/ASY-1993-6301
Aroldo Kaplan, Riemannian nilmanifolds attached to Clifford modules, Geom. Dedicata 11 (1981), no. 2, 127–136. MR 621376, DOI 10.1007/BF00147615
Tosio Kato, A short introduction to perturbation theory for linear operators, Springer-Verlag, New York-Berlin, 1982. MR 678094, DOI 10.1007/978-1-4612-5700-4
Kyung Bai Lee and Keun Park, Smoothly closed geodesics in $2$-step nilmanifolds, Indiana Univ. Math. J. 45 (1996), no. 1, 1–14. MR 1406681, DOI 10.1512/iumj.1996.45.1077
R. J. Miatello and J. P. Rossetti, Length spectra and $p$-spectra of compact flat manifolds, J. Geom. Anal. 13 (2003), no. 4, 631–657. MR 2005157, DOI 10.1007/BF02921882
He Ouyang, On isospectral rigidity of deformations on two-step nilmanifolds, Ph.D. Thesis, Washington University in St. Louis.
Hubert Pesce, Déformations isospectrales sur certaines nilvariétés et finitude spectrale des variétés de Heisenberg, Ann. Sci. École Norm. Sup. (4) 25 (1992), no. 5, 515–538 (French, with French summary). MR 1191734, DOI 10.24033/asens.1657
Hubert Pesce, Déformations $L$-isospectrales sur les nilvariétés de rang deux, C. R. Acad. Sci. Paris Sér. I Math. 315 (1992), no. 7, 821–823 (French, with English and French summaries). MR 1184910
Hubert Pesce, Calcul du spectre d’une nilvariété de rang deux et applications, Trans. Amer. Math. Soc. 339 (1993), no. 1, 433–461 (French). MR 1145731, DOI 10.1090/S0002-9947-1993-1145731-4
Hubert Pesce, Une formule de Poisson pour les variétés de Heisenberg, Duke Math. J. 73 (1994), no. 1, 79–95 (French). MR 1257278, DOI 10.1215/S0012-7094-94-07303-1
M. S. Raghunathan, Discrete subgroups of Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 68, Springer-Verlag, New York-Heidelberg, 1972. MR 0507234, DOI 10.1007/978-3-642-86426-1
A. Selberg, Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series, J. Indian Math. Soc. (N.S.) 20 (1956), 47–87. MR 88511
Robert S. Strichartz, A guide to distribution theory and Fourier transforms, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1994. MR 1276724
V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Graduate Texts in Mathematics, vol. 102, Springer-Verlag, New York, 1984. Reprint of the 1974 edition. MR 746308, DOI 10.1007/978-1-4612-1126-6
Steve Zelditch, The inverse spectral problem for surfaces of revolution, J. Differential Geom. 49 (1998), no. 2, 207–264. MR 1664907
S. Zelditch, Spectral determination of analytic bi-axisymmetric plane domains, Geom. Funct. Anal. 10 (2000), no. 3, 628–677. MR 1779616, DOI 10.1007/PL00001633