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References (15)
-
Harold Morse
A fundamental class of geodesics on any closed surface of genus greater than oneTransactions of the American Mathematical Society, 26
-
P. Eberlein (1972)
Geodesic flow in certain manifolds without conjugate pointsTransactions of the American Mathematical Society, 167
-
J. Mather (1988)
Destruction of invariant circlesErgodic Theory and Dynamical Systems, 8
-
G. Birkhoff (1922)
Surface transformations and their dynamical applicationsActa Mathematica, 43
-
W. Ballmann, M. Brin, R. Spatzier (1985)
Structure of manifolds with nonpositive curvature IIAnnals of Math., 122
-
M. Bonk (1996)
Quasi-geodesic segments and Gromov hyperbolic spacesGeometriae Dedicata, 62
-
C. Siegel, J. Moser (1971)
Lectures on Celestial Mechanics -
J. Mather (1991)
Variational construction of orbits of twist diffeomorphismsJournal of the American Mathematical Society, 4
-
V. Bangert, V. Schroeder (1991)
Existence of flat tori in analytic manifolds of nonpositive curvatureAnnales Scientifiques De L Ecole Normale Superieure, 24
-
V. Bangert (1988)
Mather Sets for Twist Maps and Geodesics on Tori -
G. Hedlund (1932)
Geodesics on a Two-Dimensional Riemannian Manifold With Periodic CoefficientsAnnals of Mathematics, 33
-
R. Ruggiero (1999)
On Manifolds Admitting Continuous Foliations by GeodesicsGeometriae Dedicata, 78
-
R. Ruggiero (1991)
On the creation of conjugate pointsMathematische Zeitschrift, 208
-
R. Ruggiero (1999)
Topological stability and Gromov hyperbolicityErgodic Theory and Dynamical Systems, 19
-
V. Arnold (1974)
Mathematical Methods of Classical Mechanics
- Publisher
- Springer Journals
- Copyright
- Copyright © 2000 by Sociedade Brasileira de Matemática
- Subject
- Mathematics; Mathematics, general; Theoretical, Mathematical and Computational Physics
- ISSN
- 1678-7544
- eISSN
- 1678-7714
- DOI
- 10.1007/BF01377597
- Publisher site
- See Article on Publisher Site
Abstract
Given a rational homology classh in a two dimensional torusT 2, we show that the set of Riemannian metrics inT 2 with no geodesic foliations having rotation numberh isC k dense for everyk ∈ N. We also show that, generically in theC 2 topology, there are no geodesic foliations with rational rotation number. We apply these results and Mather's theory to show the following: let (M, g) be a compact, differentiable Riemannian manifold with nonpositive curvature, if (M, g) satisfies the shadowing property, then (M, g) has no flat, totally geodesic, immersed tori. In particular,M has rank one and the Pesin set of the geodesic flow has positive Lebesgue measure. Moreover, if (M, g) is analytic, the universal covering ofM is a Gromov hyperbolic space.
Journal
Bulletin of the Brazilian Mathematical Society, New Series – Springer Journals
Published: Feb 1, 2000