Geometry of -step nilpotent groups with a left invariant metric
Annales scientifiques de l'École Normale Supérieure (1994)
- Volume: 27, Issue: 5, page 611-660
- ISSN: 0012-9593
Access Full Article
topHow to cite
topEberlein, Patrick. "Geometry of $2$-step nilpotent groups with a left invariant metric." Annales scientifiques de l'École Normale Supérieure 27.5 (1994): 611-660. <http://eudml.org/doc/82371>.
@article{Eberlein1994,
author = {Eberlein, Patrick},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {2-step nilpotent Lie groups; 2-step compact nilmanifolds; closed geodesics; maximal length spectrum},
language = {eng},
number = {5},
pages = {611-660},
publisher = {Elsevier},
title = {Geometry of $2$-step nilpotent groups with a left invariant metric},
url = {http://eudml.org/doc/82371},
volume = {27},
year = {1994},
}
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
PB - Elsevier
VL - 27
IS - 5
SP - 611
EP - 660
LA - eng
KW - 2-step nilpotent Lie groups; 2-step compact nilmanifolds; closed geodesics; maximal length spectrum
UR - http://eudml.org/doc/82371
ER -
References
top- [A] V. ARNOLD, Mathematical Methods of classical Mechanics (Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, 1980). Zbl0386.70001
- [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). Zbl0722.58045MR91a:58188
- [BGM] M. BERGER, P. GAUDUCHON and E. MAZET, Le spectre d'une variété Riemannienne (Springer Lecture Notes, Vol. 194, New York, 1971). Zbl0223.53034MR43 #8025
- [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). Zbl0761.22010MR92e:22017
- [CE] J. CHEEGER and D. EBIN, Comparison Theorems in Riemannian geometry, North-Holland, Amsterdam, 1975. Zbl0309.53035MR56 #16538
- [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). MR92i:58197
- [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). Zbl0649.53025MR88m:58186
- [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). Zbl0709.53030MR91e:58197
- [DG3] D. DE TURCK and C. GORDON, Isospectral Metrics and Finite Riemannian Coverings (Contemp. Math., Vol. 64, 1987, pp. 79-92). Zbl0646.58030MR88d:58124
- [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). Zbl0694.53037MR90k:58233
- [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). Zbl0728.58039MR93f:58236
- [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. Zbl0728.58040MR93f:58237
- [DGGW4] D. DE TURCK, H. GLUCK, C. GORDON and D. WEBB, The Inaudible Geometry of Nilmanifolds (Invent. Math., Vol. 111, 1993, pp. 271-284). Zbl0779.53026MR93k:58222
- [DGGW5] D. DE TURCK, H. GLUCK, C. GORDON and D. WEBB, Conformal Isospectral Deformations (Indiana Univ. Math. Jour., Vol. 41, 1992, pp. 99-107). Zbl0742.58055MR93c:58218
- [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). Zbl0830.53039MR95b:53061
- [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). Zbl0591.53042MR87i:58170
- [G2] C. GORDON, Riemannian Manifolds Isospectral on Functions but not on 1-forms (J. Diff. Geom., Vol. 24, 1986, pp. 79-96). Zbl0585.53036MR87m:58174
- [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). Zbl0523.58043MR85j:58143
- [GW2] C. GORDON and E. WILSON, The Spectrum of the Laplacian on Riemannian Heisenberg manifolds (Michigan Math. Jour., Vol. 33, 1986, pp. 253-271). Zbl0599.53038MR87k:58275
- [Hei] E. HEINTZE, On Homogeneous Manifolds of Negative Curvature, (Math. Annalen, Vol. 211, 1974, pp. 23-34). Zbl0273.53042MR50 #5695
- [Hel] S. HELGASON, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. Zbl0111.18101MR26 #2986
- [K1] A. KAPLAN, Riemannian Nilmanifolds Attached to Clifford Modules, (Geom. Dedicata, Vol. 11, 1981, pp. 127-136). Zbl0495.53046MR82h:22008
- [K2] A. KAPLAN, On the Geometry of the Groups of Heisenberg Type (Bull. London Math. Soc., Vol. 15, 1983, pp. 35-42). Zbl0521.53048MR84h:53063
- [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). Zbl0164.22202MR37 #6876
- [Ko] A. KORANYI, Geometric Properties of Heisenberg Type Groups, (Advances in Math., Vol. 56, 1985, pp. 28-38). Zbl0589.53053MR86h:53050
- [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). Zbl0341.53030MR54 #12970
- [PS] R. PALAIS and T. STEWART, Torus Bundles Over a Torus (Proc. Amer. Math. Soc., Vol. 12, 1961, pp. 26-29). Zbl0102.38702MR23 #A963
- [R] M. S. RAGHUNATHAN, Discrete Subgroups of Lie Groups, Springer, New York, 1972. Zbl0254.22005MR58 #22394a
- [St] N. STEENROD, Topology of Fibre Bundles, Princeton Univ. Press, Princeton, 1951. Zbl0054.07103MR12,522b
- [Wi] E. WILSON, Isometry Groups on Homogeneous Nilmanifolds (Geom. Dedicata, Vol. 12, 1982, pp. 337-346). Zbl0489.53045MR84a:53048
- [Wo1] J. WOLF, Curvature in Nilpotent Lie Groups (Proc. Amer. Math. Soc., Vol. 15, 1964, pp. 271-274). Zbl0134.17905MR28 #5405
- [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). Zbl0113.37101
Citations in EuDML Documents
top- Hamid-Reza Fanaï, Rigidité du flot géodésique de certaines nilvariétés de rang deux
- Carolyn S. Gordon, Yiping Mao, Dorothee Schueth, Symplectic rigidity of geodesic flows on two-step nilmanifolds
- Mohammed Guediri, Mona Bin-Asfour, Ricci-flat left-invariant Lorentzian metrics on 2-step nilpotent Lie groups
- Naceurdine Bensalem, Fernand Pelletier, Some geometrical properties of infinite-dimensional bilinear controlled systems
- Carolyn S. Gordon, Ruth Gornet, Dorothee Schueth, David L. Webb, Edward N. Wilson, Isospectral deformations of closed riemannian manifolds with different scalar curvature
- Hamid-Reza Fanaï, Atefeh Hasan-Zadeh, An application of Lie groupoids to a rigidity problem of 2-step nilmanifolds
- Ruth Gornet, Maura B. Mast, The length spectrum of riemannian two-step nilmanifolds
NotesEmbed ?
topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.