On a class of vector fields with discontinuity of divide-by-zero type and its applications

TL;DR

This paper investigates a class of vector fields with discontinuities caused by division-by-zero on hypersurfaces, analyzing their phase portraits and singular points, with applications to geodesic flows in singular metrics on surfaces.

Contribution

It introduces a detailed analysis of vector fields with denominator-induced discontinuities and explores their applications in differential geometry, particularly in geodesic flow singularities.

Findings

Characterization of phase portraits near discontinuities

Conditions under which singular points occur

Applications to geodesic flows in singular metrics

Abstract

We study phase portraits and singular points of vector fields of a special type, that is, vector fields whose components are fractions with a common denominator vanishing on a smooth regular hypersurface in the phase space. We assume also some additional conditions, which are fulfilled, for instance, if the vector field is divergence-free. This problem is motivated by a large number of applications. In this paper, we consider three of them in the framework of differential geometry: singularities of geodesic flows in various singular metrics on surfaces.