Geodesic
Isometric transformations of a~surface inducing conformal maps of the~surface onto itself
Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 115-127

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We consider isometries of a surface inducing, via a specific scheme, conformal maps of the surface onto itself. We give a complete description of the surfaces that admit isometries corresponding to identity conformal maps. For the general case we provide several results and state some unsolved problems. 
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     author = {I. Kh. Sabitov},
     title = {Isometric transformations of a~surface inducing  conformal maps of the~surface onto itself},
     journal = {Sbornik. Mathematics},
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     volume = {189},
     number = {1},
     year = {1998},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1998_189_1_a5/}
}
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I. Kh. Sabitov. Isometric transformations of a~surface inducing  conformal maps of the~surface onto itself. Sbornik. Mathematics, Tome 189 (1998) no. 1, pp. 115-127. http://geodesic.mathdoc.fr/item/SM_1998_189_1_a5/