Geodesic
Mappings that preserve cones in Lobachevskii space
Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 687-694 
A. K. Guts. Mappings that preserve cones in Lobachevskii space. Matematičeskie zametki, Tome 13 (1973) no. 5, pp. 687-694. http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a6/
@article{MZM_1973_13_5_a6,
     author = {A. K. Guts},
     title = {Mappings that preserve cones in {Lobachevskii} space},
     journal = {Matemati\v{c}eskie zametki},
     pages = {687--694},
     year = {1973},
     volume = {13},
     number = {5},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1973_13_5_a6/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\textЛ^n$ be $n$-dimensional Lobachevskii space, and $\{l_x:X\in\textЛ^n\}$ be a family of lines, parallel to a line $l_o$, $o\in\textЛ^n$ (in a given direction). Let $\{C_x:X\in\textЛ^n\}$ be a family of circular cones in $\textЛ^n$ of opening $\alpha$ with axes $l_X$ and vertex $X$. Then, if $f:\textЛ^n\to\textЛ^n$ ($n>2$) is a bijective mapping and $f(Cx)=C_{f(x)}$, it follows thatf is a motion in the space $\textЛ^n$.