Geodesic
Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 281-288

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It is proved that any surface $S^{*}$ isometric to a given compact surface $S$ and disposed sufficiently close to $S$ generates a quasi-conformal mapping of $S$ onto itself. On the base of this result it is proved that a compact surface admitting sliding bendings onto itself is topologically a sphere or a torus and its intrinsic metric is of rotation type. 
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     title = {Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself},
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I. Kh. Sabitov. Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 1, pp. 281-288. http://geodesic.mathdoc.fr/item/FPM_1995_1_1_a15/