×

Minimal geodesics. (English) Zbl 0676.53055

A geodesic c: \({\mathbb{R}}\to M\) in a Riemannian manifold M is minimal if a lift \(\tilde c\) of c to the universal Riemannian cover \(\tilde M\) of M has the following property: Every compact segment of \(\tilde c\) is a shortest connection between its endpoints. If M is a 2-torus there is a rich theory of minimal geodesics which is closely related to the Aubry- Mather theory on invariant sets of monotone twist maps of the annulus [see e.g. the author’s survey in Dyn. Rep. 1, 1-56 (1988; Zbl 0664.53021)]. The present paper studies the existence and properties of minimal geodesics on compact Riemannian manifolds of dimension \(\geq 3\). The results provide a weak generalization of the 2-dimensional theory. The main difference is that minimal geodesics may become very rare in dimensions \(\geq 3\). A detailed discussion of the minimal geodesics of a class of Riemannian metrics on a 3-torus, the so-called Hedlund examples, shows that the obtained results are - to a certain extent - optimal.
Reviewer: V.Bangert

MSC:

53C22 Geodesics in global differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry

Citations:

Zbl 0664.53021
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Rinow, Die innere Geometrie der metrischen Räume. Grundlehren der math. Wiss. 105 (1961) · Zbl 0096.16302
[2] DOI: 10.2307/1968215 · Zbl 0006.32601
[3] Gromov, J. Differential Geom. 18 pp 1– (1983)
[4] Gromov, Structures métriques pour les Variétés Riemanniennes (1981)
[5] DOI: 10.1512/iumj.1974.24.24031 · Zbl 0289.49044
[6] DOI: 10.1007/BF00945819 · Zbl 0641.70014
[7] none, Anal. i Prilozhen. 20 pp 9– (1986)
[8] DOI: 10.1007/BF01083491 · Zbl 0641.58032
[9] DOI: 10.1007/BF01388907 · Zbl 0642.58040
[10] Bangert, Dynamics Reported 1 pp 1– (1988)
[11] Aubry, Physica 8D pp 381– (1983)
[12] Moser, Erg. Th. & Dynam. Sys. 6 pp 325– (1986)
[13] DOI: 10.2307/1989225 · JFM 50.0466.04
[14] DOI: 10.1007/BF02564683 · Zbl 0689.58025
[15] Mather, Erg. Th. & Dynatn. Sys. 8 pp 199– (1988)
[16] Mather, Publ. Math. IHES 63 pp 153– (1986) · Zbl 0603.58028
[17] Leichtwei?, Konvexe Mengen (1980)
[18] DOI: 10.1007/BF01418743 · Zbl 0219.53037
[19] Katok, Minimal orbits for small perturbations of completely integrable Hamiltonian systems (1989) · Zbl 0762.58024
[20] Herman, Existence et non existence de tores invariants par des diffeomorphismes symplectiques (1988)
[21] DOI: 10.1137/1028153 · Zbl 0606.58022
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.