Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T04:44:35.704Z Has data issue: false hasContentIssue false
  • English
  • Français

Manifolds with non-positive curvature

Published online by Cambridge University Press:  19 September 2008

K. Burns ,
A. Katok ,
W. Ballman ,
M. Brin ,
P. Eberlein  and
R. Osserman
K. Burns
Affiliation:
University of Maryland, College Park, MD 20742, USA
A. Katok
Affiliation:
California Institute of Technology, Pasadena, CA 91125, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This is an extended report on a problem session about geodesic flows held on May 30, 1984 at MSRI, Berkeley. The list of six authors reflects the principal participants in the discussion. Contributions of other participants are mentioned in the text. The session was chaired by A. Katok; this report was prepared by K. Burns in collaboration with A. Katok.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 5 , Issue 2 , June 1985 , pp. 307 - 317
Copyright
Copyright © Cambridge University Press 1985

References

REFERENCES

[1]Anderson, M. T.. The Dirichlet problem at infinity for manifolds of negative curvature. J. Diff. Geom. 18 (1983), 701–721.Google Scholar
[2]Anderson, M. & Schoen, R.. Positive harmonic functions on complete manifolds of negative curvature. Preprint.Google Scholar
[3]Ballman, W.. Axial isometries of manifolds of nonpositive curvature. Math. Ann. 259 (1982), 131–144.CrossRefGoogle Scholar
[4]Ballman, W. & Brin, M.. On the ergodicity of geodesic flows. Ergod. Th. and Dynam. Sys. 2 (1982), 311–315.Google Scholar
[5]Ballman, W., Brin, M. & Eberlein, P.. Structure of manifolds of non-positive curvature, I. Univ. of Maryland Technical Report TR 84–8 (1984).Google Scholar
[6]Ballman, W., Brin, M. & Spatzier, R.. Structure of manifolds of non-positive curvature, II. Univ. of Maryland Technical Report TR84–31 (1984).Google Scholar
[7]Berger, M.. Geometry of the spectrum, I. AMS Proc. Sympos. in Pure Math. 27 (1975), Part 2, 129–152.CrossRefGoogle Scholar
[8]Bowen, R.. Periodic orbits for hyperbolic flows. Amer. J. Math. 94 (1972), 1–30.Google Scholar
[9]Bowen, R. & Ruelle, D.. The ergodic theory of axiom A flows. Inv. Math. 29 (1975), 181–202.Google Scholar
[10]Bunimovich, L. A.. On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65 (1979), 295–312.CrossRefGoogle Scholar
[11]Burns, K.. Hyperbolic behaviour of geodesic flows on manifolds with no focal points. Ergod. Th. & Dynam. Sys. 3 (1983), 1–12.Google Scholar
[12]Chen, S. S. & Eberlein, P.. Isometry groups of simply connected manifolds of nonpositive curvature. Illinois J. Math. 24 (1980), 73–103.Google Scholar
[13]Eberlein, P.. Geodesic flow in certain manifolds without conjugate points. Trans. Amer. Math. Soc. 167 (1972), 151–170.CrossRefGoogle Scholar
[14]Eberlein, P.. Geodesic rigidity in compact nonpositively curved manifolds. Trans. Amer. Math. Soc. 268 (1981), 411–443.Google Scholar
[15]Eberlein, P.. Rigidity of lattices of non-positive curvature. Ergod. Th. & Dynam. Sys. 3 (1983), 47–85.Google Scholar
[16]Eberlein, P. & O'Neill, B.. Visibility manifolds. Pacific J. Math. 46 (1973), 45–109.Google Scholar
[17]Feldman, J. & Ornstein, D.. Semi-rigidity of horocycle flows over compact surfaces of variable negative curvature. Preprint (1984).Google Scholar
[18]Fricke, R. & Klein, F.. Vorlesungen über die Theorie der Elliptischen Modulfunktionen/ Automorphen funktionen.G. Teubner: Leipzig, 1896/s1912.Google Scholar
[19]Green, L. W.. Remarks on uniformly expanding horocycle parametrizations. J. Diff. Geom. 13 (1978), 263–271.Google Scholar
[20]Gromoll, D. & Wolf, J. A.. Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature. Bull. Amer. Math. Soc. 77 (1971), 545–552.CrossRefGoogle Scholar
[21]Gromov, M.. Three remarks on geodesic dynamics and the fundamental group. Preprint.Google Scholar
[22]Guillemin, V. & Kazhdan, D.. Some inverse spectral results for negatively curved 2-manifolds. Topology 19 (1980), 301–312.Google Scholar
[23]Hurder, S.. Rigidity of group actions and cocycles. MSRI Berkeley Preprint (1984).Google Scholar
To appear in Ergod. Th. & Dynam. Sys (1985)Google Scholar
[24]Katok, A.. Ergodic properties of degenerate integrable Hamiltonian systems. Math. USSR-Izv. 7 (1973), 535–571.Google Scholar
Izv. Akad. nauk. SSSR 37 (1973), 539–576.Google Scholar
[25]Katok, A.. Entropy and closed geodesies. Ergod. Th. & Dynam. Sys. 2 (1982), 339–367.Google Scholar
[26]Kifer, Yu. I.. Brownian motion and harmonic functions on manifolds of negative curvature. Th. Probability Appl. 21 (1976), 81–95.Google Scholar
[27]Knieper, G.. Das Wachstum der Äquivalenzklassen geschlossener Geodätischer in kompakten Mannigfaltigkeiten. Archiv. Math. 40 (1983), 559–568.Google Scholar
[28]Kobayashi, S. & Nomizu, K.. Foundations of Differential Geometry. Vol. 1, Wiley: New York, 1963; 179–193.Google Scholar
[29]Lawson, H. B. & Yau, S. T.. Compact manifolds of non-positive curvature. J. Diff. Geom. 7 (1972), 211–228.Google Scholar
[30]McKean, H. P.. Selberg's trace formula as applied to a compact Riemann surface. Comm. Pure Appl. Math. 25 (1972), 225–246.CrossRefGoogle Scholar
[31]Manning, A.. Topological entropy for geodesic flows. Ann. Math. 110 (1979), 567–573.Google Scholar
[32]Marcus, B.. Unique ergodicity of the horocycle flow: variable negative curvature case. Israel J. Math. 21 (1975), 133–144.Google Scholar
[33]Margalis, G. A.. Applications of ergodic theory to the investigation of manifolds of negative curvature. Functional Anal. Appl. 3 (1969), 335–336.Google Scholar
Funktsional. Anal. i Prilozhen. 3 (1969), 89–90.Google Scholar
[34]Osserman, R. & Sarnak, P.. A new curvature invariant and entropy of geodesic flows. MSRI Berkeley Preprint (1984);Google Scholar
Inv. Math. 77 (1984), 455–462.CrossRefGoogle Scholar
[35]Parry, W. & Pollicott, M.. An analogue of the prime number theorem for closed orbits of axiom A flows. Ann. Math. 118 (1983), 573–591.Google Scholar
[36]Pesin, Ya. B.. Geodesic flows with hyperbolic behaviour of the trajectories and objects connected with them. Russian Math. Surveys 36 (1981), 1–59.CrossRefGoogle Scholar
Uspekhi Mat. nauk. 36 (1981), 3–51.Google Scholar
[37]Ratner, M.. Rigidity of horocycle flows. Ann. of Math. 115 (1982), 587–614.Google Scholar
[38]Schroeder, V.. Rigidity of locally symmetric spaces in the category of spaces of non-positive curvature. (1983).Google Scholar
[39]Sinai, Ja. G.. Gibbs measures in ergodic theory. Russian Math. Surveys 27 (1972), no. 4, 21–69.CrossRefGoogle Scholar
Uspekhi Mat. nauk. SSSR 27 (1972) no. 4, 21–64.Google Scholar
[40]Spatzier, R. J.. The geodesic flow and an approach to the classification of manifolds of non-positive curvature. MSRI Berkeley Preprint (1983).Google Scholar
[41]Stanton, N.. The heat equation in several complex variables. Bull Amer. Math. Soc. 11 (1984), 65–84.Google Scholar
[42]Sullivan, D.. Dirichlet problem for a negatively curved manifold. J. Diff. Geom. 18 (1983), 723–732.Google Scholar
[43]Toll, C.. A multiplicative asymptotic for the prime geodesic theorem. Thesis, Univ. of Maryland. (1984).CrossRefGoogle Scholar
[44]Vignéras, M. F.. Variétés Riemanniennes isopectrales et non isométriques. Ann. Math. 112 (1980), 21–32.Google Scholar
[45]Wojtkowski, M.. Principles for the design of billiards with non-vanishing Lyapunov exponents. Preprint.Google Scholar
[46]Ziller, W.. Geometry of the Katok examples. Ergod. Th. & Dynam. Sys. 3 (1983), 135–157.Google Scholar
[47]Ballman, W.. Non-positively curved manifolds of higher rank. University of Maryland Technical Report TR84–38.Google Scholar