On the local and global properties of geodesics in pseudo-Riemannian metrics. (English) Zbl 1323.53036

Remizov, A. O.

Differ. Geom. Appl. 39, 36-58 (2015).

Summary: The paper is a study of geodesics in two-dimensional pseudo-Riemannian metrics. Firstly, the local properties of geodesics in a neighborhood of generic parabolic points are investigated. The equation of the geodesic flow has singularities at such points that leads to a curious phenomenon: geodesics cannot pass through such a point in arbitrary tangential directions, but only in certain directions said to be admissible (the number of admissible directions is generically 1 or 3). Secondly, we study the global properties of geodesics in pseudo-Riemannian metrics possessing differentiable groups of symmetries. At the end of the paper, two special types of discontinuous metrics are considered.

MSC:

53C22	Geodesics in global differential geometry
34C05	Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
53C50	Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics

Keywords:

pseudo-Riemannian metrics; geodesics; implicit differential equations; singular points; envelops