Geometry of 2-step nilpotent groups with a left invariant metric
Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 27 (1994) no. 5, pp. 611-660.
@article{ASENS_1994_4_27_5_611_0,
     author = {Eberlein, Patrick},
     title = {Geometry of $2$-step nilpotent groups with a left invariant metric},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {611--660},
     publisher = {Elsevier},
     volume = {Ser. 4, 27},
     number = {5},
     year = {1994},
     doi = {10.24033/asens.1702},
     mrnumber = {95m:53059},
     zbl = {0820.53047},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.1702/}
}
TY  - JOUR
AU  - Eberlein, Patrick
TI  - Geometry of $2$-step nilpotent groups with a left invariant metric
JO  - Annales scientifiques de l'École Normale Supérieure
PY  - 1994
SP  - 611
EP  - 660
VL  - 27
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.24033/asens.1702/
DO  - 10.24033/asens.1702
LA  - en
ID  - ASENS_1994_4_27_5_611_0
ER  - 
%0 Journal Article
%A Eberlein, Patrick
%T Geometry of $2$-step nilpotent groups with a left invariant metric
%J Annales scientifiques de l'École Normale Supérieure
%D 1994
%P 611-660
%V 27
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.24033/asens.1702/
%R 10.24033/asens.1702
%G en
%F ASENS_1994_4_27_5_611_0
Eberlein, Patrick. Geometry of $2$-step nilpotent groups with a left invariant metric. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 27 (1994) no. 5, pp. 611-660. doi : 10.24033/asens.1702. http://www.numdam.org/articles/10.24033/asens.1702/

[A] V. Arnold, Mathematical Methods of classical Mechanics (Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, 1980). | Zbl

[Ba] W. Ballmann, personal communication.

[BG] R. Brooks and C. Gordon, Isospectral Famillies of Conformally Equivalent Riemannian Metrics (Bull. Amer. Math. Soc., Vol. 23, 1990, pp. 433-436). | MR | Zbl

[BGM] M. Berger, P. Gauduchon and E. Mazet, Le spectre d'une variété Riemannienne (Springer Lecture Notes, Vol. 194, New York, 1971). | MR | Zbl

[CDKR] M. Cowling, A. Dooley, A. Koranyi and F. Ricci, H-type Groups and Iwasawa Decompositions (Advances in Math., Vol. 87, 1991, pp. 1-41). | MR | Zbl

[CE] J. Cheeger and D. Ebin, Comparison Theorems in Riemannian geometry, North-Holland, Amsterdam, 1975. | MR | Zbl

[D1] D. De Turck, Audible and Inaudible Geometric Properties (Proc. of Conference on Differential Geometry and Topology, Rendiconti Sem. della Facolta Sci. dell' Univ. di Cagliari, 58, 1988-Supplement, pp. 1-26). | MR

[D2] D. De Turck, The Geometry of Isospectral Deformations (to appear in Proc. Sympos. Pure Math., Summer Geomety Institute, July 1990).

[DG1] D. De Turck and C. Gordon, Isospectral Deformations I : Riemannian Structures on 2-step Nilspaces (Comm. Pure and Applied Math., Vol. 40, 1987, pp. 367-387). | MR | Zbl

[DG2] D. De Turck and C. Gordon, Isospectral Deformations II : Trace Formulas, Metrics and Potentials (Comm. Pure and Applied Math., Vol. 42, 1989, pp. 1067-1095). | MR | Zbl

[DG3] D. De Turck and C. Gordon, Isospectral Metrics and Finite Riemannian Coverings (Contemp. Math., Vol. 64, 1987, pp. 79-92). | MR | Zbl

[DGGW1] D. De Turck, H. Gluck, C. Gordon and D. Webb, You Cannot Hear the Mass of a Homology Class (Comment. Math. Helvetici, Vol. 64, 1989, pp. 589-617). | MR | Zbl

[DGGW2] D. De Turck, H. Gluck, C. Gordon and D. Webb, How Can a Drum Change Shape While Sounding the Same ?, in Differential Geometry : Symposium in Honor of Manfredo do Carmo, ed. B. Lawson and K. Tenenblatt (Pitman Surveys in Pure and Applied Math., Vol. 52, 1991, pp. 111-122). | MR | Zbl

[DGGW3] D. De Turck, H. Gluck, C. Gordon and D. Webb, How Can a Drum Change Shape While Sounding the Same ?, Part 2, in Mechanics, Analysis and Geometry, 200 Years After Lagrange, ed. M. Francaviglia, Elsevier Press, 1991, pp. 335-358. | MR | Zbl

[DGGW4] D. De Turck, H. Gluck, C. Gordon and D. Webb, The Inaudible Geometry of Nilmanifolds (Invent. Math., Vol. 111, 1993, pp. 271-284). | MR | Zbl

[DGGW5] D. De Turck, H. Gluck, C. Gordon and D. Webb, Conformal Isospectral Deformations (Indiana Univ. Math. Jour., Vol. 41, 1992, pp. 99-107). | MR | Zbl

[E1] P. Eberlein, Geometry of 2-step Nilpotent Groups with a Left Invariant Metric, II (Trans. Amer. Math. Soc., Vol. 343, 1994, pp. 805-828). | MR | Zbl

[E2] P. Eberlein, Geometry of Nonpositively Curved Manifolds lecture notes, to appear, in Univ. Chicago Press.

[G1] C. Gordon, The Laplace Spectra Versus the Length Spectral of Riemannian Manifolds (Contemp. Math., Vol. 51, 1986, pp. 63-80). | MR | Zbl

[G2] C. Gordon, Riemannian Manifolds Isospectral on Functions but not on 1-forms (J. Diff. Geom., Vol. 24, 1986, pp. 79-96). | MR | Zbl

[G3] C. Gordon, personal communication.

[GW1] C. Gordon and E. Wilson, Isospectral Deformations of Compact Solvmanifolds (J. Diff. Geom., Vol. 19, 1984, pp. 241-256). | MR | Zbl

[GW2] C. Gordon and E. Wilson, The Spectrum of the Laplacian on Riemannian Heisenberg manifolds (Michigan Math. Jour., Vol. 33, 1986, pp. 253-271). | MR | Zbl

[Hei] E. Heintze, On Homogeneous Manifolds of Negative Curvature, (Math. Annalen, Vol. 211, 1974, pp. 23-34). | MR | Zbl

[Hel] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. | MR | Zbl

[K1] A. Kaplan, Riemannian Nilmanifolds Attached to Clifford Modules, (Geom. Dedicata, Vol. 11, 1981, pp. 127-136). | MR | Zbl

[K2] A. Kaplan, On the Geometry of the Groups of Heisenberg Type (Bull. London Math. Soc., Vol. 15, 1983, pp. 35-42). | MR | Zbl

[Ka] F. Karpelevic, The Geometry of Geodesics and the Eigenfunctions of the Beltrami-Laplace Operator on Symmetric Spaces (Trans. Moscow Math. Soc., Tome 14, 1965, AMS Translation, pp. 51-199). | MR | Zbl

[Ko] A. Koranyi, Geometric Properties of Heisenberg Type Groups, (Advances in Math., Vol. 56, 1985, pp. 28-38). | MR | Zbl

[Ma] M. Mast, Closed Geodesics in 2-step Nilmanifolds, dissertation, Univ. of N. Carolina, 1992.

[Mi] J. Milnor, Curvatures of Left Invariant Metrics on Lie Groups, (Advances in Math., Vol. 21, 1976, pp. 293-329). | MR | Zbl

[PS] R. Palais and T. Stewart, Torus Bundles Over a Torus (Proc. Amer. Math. Soc., Vol. 12, 1961, pp. 26-29). | MR | Zbl

[R] M. S. Raghunathan, Discrete Subgroups of Lie Groups, Springer, New York, 1972. | MR | Zbl

[St] N. Steenrod, Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. | MR | Zbl

[Wi] E. Wilson, Isometry Groups on Homogeneous Nilmanifolds (Geom. Dedicata, Vol. 12, 1982, pp. 337-346). | MR | Zbl

[Wo1] J. Wolf, Curvature in Nilpotent Lie Groups (Proc. Amer. Math. Soc., Vol. 15, 1964, pp. 271-274). | MR | Zbl

[Wo2] J. Wolf, On locally Symmetric Spaces of Nonnegative Curvature and Certain Other Locally Symmetric Spaces (Comm. Math. Helv., Vol. 37, 1963, p. 265-295, plus typewritten erratum). | Zbl

Cité par Sources :