Geodesic
Singularities of a~geodesic flow on surfaces with a~cuspidal edge
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. I, Tome 268 (2010), pp. 258-267

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This paper is a study of singularities of geodesic flows on surfaces with nonisolated singular points that form a smooth curve (like a cuspidal edge). The main results of the paper are normal forms of the corresponding direction field on the tangent bundle of the plane of local coordinates and the projection of its trajectories to the surface. 
@article{TM_2010_268_a16,
     author = {A. O. Remizov},
     title = {Singularities of a~geodesic flow on surfaces with a~cuspidal edge},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {258--267},
     publisher = {mathdoc},
     volume = {268},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2010_268_a16/}
}
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A. O. Remizov. Singularities of a~geodesic flow on surfaces with a~cuspidal edge. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Differential equations and topology. I, Tome 268 (2010), pp. 258-267. http://geodesic.mathdoc.fr/item/TM_2010_268_a16/