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


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


Zbl 0664.53021
Full Text: DOI


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