Geodesic
Homogeneous regions on the conformal sphere
Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 321-328 Cet article a éte moissonné depuis la source Math-Net.Ru

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All regions on a sphere which are similar with respect to a group of conformal transformations of this sphere are determined. Such regions are: the whole sphere, a hemisphere, the complement of a sphere of lower dimension, and the complement of a point. 
@article{MZM_1970_8_3_a4,
     author = {B. N. Kimel'fel'd},
     title = {Homogeneous regions on the conformal sphere},
     journal = {Matemati\v{c}eskie zametki},
     pages = {321--328},
     year = {1970},
     volume = {8},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a4/}
}
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B. N. Kimel'fel'd. Homogeneous regions on the conformal sphere. Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 321-328. http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a4/