Abstract
For hyperbolic manifolds a single the relation of a broken line with vertices on the given rays and a given rotation volume, the existence of a broken line with vertices on the given rays and given turns, etc. are proving. One affirms that if in the three-dimensional Lobachevsky space an orientable polyhedral surface with smooth metric and edge, which consists of the geodesic cycles ℎ1, ℎ2, ..., ℎ𝑘, then this surface can be smoothly extended by tapes 𝑣1, 𝑣2, ..., 𝑣𝑘, for which vertex curvaturesof of the outer edges are given.
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