Minimal geodesics. (English) Zbl 0676.53055

Bangert, Victor

Ergodic Theory Dyn. Syst. 10, No. 2, 263-286 (1990).

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

Keywords:

3-manifold; minimal geodesics; 3-torus

Citations:

Zbl 0664.53021

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