Seminar talk, 23 November 2016

From Geometry of Differential Equations
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Speaker: Alexey Remizov

Title: Singularities of geodesic flows in two-dimensional signature varying metrics

The talk will review recent results on singularities of geodesic flows in smooth two-dimensional signature varying metrics (such metrics we call pseudo-Riemannian). In general position there exists a curve on which the pseudo-Riemannian metric degenerates. The points of degeneration are the singular points of the corresponding geodesic flow. This implies that contrary to the standard existence and uniqueness theorem the geodesics cannot pass through points of degeneration in an arbitrary direction, but only in certain directions said to be admissible. In general case, the number of admissible directions is finite and in almost all points of degeneration curve is equal to 1 or 3, and in some points of degeneration curve is equal to 2. Also qualitatively, the behavior of geodesics in points of degeneration of pseudo-Riemannian metric rather significantly differs from that of Riemannian metric. All these will discussed in the talk.