Geodesic
Equivalence of Paths in Galilean Geometry
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 3-16

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An explicit description of finite transcendence bases in differential fields of differential rational functions that are invariant under the action of Galilean transformation group in a real finite-dimensional space is presented. Necessary and sufficient conditions of the equivalence of paths in the $n$-dimensional Galilean space are obtained.
Keywords: Galilean space, differential invariant, transcendence basis, path in a finite-dimensional space. 
@article{INTO_2018_144_a0,
     author = {V. I. Chilin and K. K. Muminov},
     title = {Equivalence of {Paths} in {Galilean} {Geometry}},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {3--16},
     publisher = {mathdoc},
     volume = {144},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2018_144_a0/}
}
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V. I. Chilin; K. K. Muminov. Equivalence of Paths in Galilean Geometry. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the International Conference «Problems of Modern Topology and its Applications», Tome 144 (2018), pp. 3-16. http://geodesic.mathdoc.fr/item/INTO_2018_144_a0/