Quasi-conformal mappings of a surface generated by its isometric transformation, and bendings of the surface onto itself

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.