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

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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 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.