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

Keywords:

3-manifold; minimal geodesics; 3-torus

Zbl 0664.53021
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References:

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