Singularities of geodesic flows in $2D$ metrics with varying signature

The talk is devoted to singularities of geodesic flows in smooth 2-dimensional metrics with varying signature (such metrics are also called pseudo-Riemannian). Generically, a pseudo-Riemannian metric degenerated on a smooth curve, whose points are singular points of the corresponding geodesic flow. The existence and uniqueness theorem at such points fails, whence geodesics cannot pass through a degenerate point in all possible directions, but in some “admissible” directions only. For generic 2- dimensional pseudo-Riemannian metric the number of admissible directions is 1 or 3, or 2 (at some isolated points). This explains by the fact that the admissible directions correspond to real roots of a certain cubic polynomial. The investigation of the behavior of geodesics at degenerate points is based on the theory of local normal forms of vector fields with non-isolated singular points. A brief survey can be found here: https://arxiv.org/pdf/1801.09815.pdf