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

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. © Springer International Publishing AG, part of Springer Nature 2018.

Авторы
Pavlova N.G. 1 , Remizov A.O.2, 3
Сборник материалов конференции
Издательство
Springer New York LLC
Язык
Английский
Страницы
135-155
Статус
Опубликовано
Том
222
Год
2018
Организации
  • 1 Department of Nonlinear Analysis and Optimization, RUDN University, Moscow, Russian Federation
  • 2 V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russian Federation
  • 3 Laboratory of Dynamical Systems, National Research University Higher School of Economics, Moscow, Russian Federation
Ключевые слова
Geodesics; Normal forms; Pseudo-Riemannian metrics; Singular points
Дата создания
19.10.2018
Дата изменения
19.10.2018
Постоянная ссылка
https://repository.rudn.ru/ru/records/article/record/7154/