A brief survey on singularities of geodesic flows in smooth signature changing metrics on 2-surfaces

Abstract

We present a survey on generic singularities of geodesic flows in smooth signature changing metrics (often called pseudo-Riemannian) in dimension 2. Generically, a pseudo-Riemannian metric on a 2-manifold S changes its signature (degenerates) along a curve S0, which locally separates S into a Riemannian (R) and a Lorentzian (L) domain. The geodesic flow does not have singularities over R and L, and for any point q∈ R∪ L and every tangential direction p∈ ℝℙ there exists a unique geodesic passing through the point q with the direction p. On the contrary, geodesics cannot pass through a point q∈ S0 in arbitrary tangential directions, but only in some admissible directions; the number of admissible directions is 1 or 2 or 3. We study this phenomenon and the local properties of geodesics near q∈ S0.