zbMATH — the first resource for mathematics

Geodätische Linien auf Riemannschen Mannigfaltigkeiten. (German) Zbl 0565.53028
This article is an excellent and lucid survey on the global behaviour of geodesics on Riemannian manifolds up to the middle of 1984. The author emphasizes the methods influenced by the calculus of variations. Results from dynamical systems and ergodic theory are included as far as they complete the former ones. The chapter headings are: 1. Fundamental notions, 2. Realm of questions (2.1 Negative curvature, 2.2 Spectrum of the Laplacian, 2.3 Inverse problems), 3. Closed geodesics on compact manifolds (3.1 The variational principle, 3.2 Short closed geodesics on spheres, 3.3 Stability properties of closed geodesics, 3.4 Existence of many closed geodesics on manifolds with finite fundamental group, 3.5 Closed geodesics on manifolds with infinite fundamental group), 4. Geodesics on complete, noncompact manifolds (4.1 Problems, 4.2 Results in dimension \(>2\), 4.3 Results for surfaces). Chapter 2 is devoted to scattered problems which do not fit into the systematic treatment presented in the following chapters. An important aspect of the closed geodesic problem is that closed geodesics become hard to detect when the topology of the underlying manifold is poor. So, Riemannian spheres present the most difficult compact case. In the non-compact case, besides closed geodesics, there is great interest in geodesic lines with a special behaviour at infinity like divergence, boundedness or oscillation. The author discussed a wealth of information and draws a clear picture of leading ideas to which he himself also contributed recently. Many open questions are mentioned and several remarks on the validity and limitations of the methods are made. The bibliography contains 152 items.
Reviewer: R.Walter

53C22 Geodesics in global differential geometry
58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable)
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
53D25 Geodesic flows in symplectic geometry and contact geometry