Nevmerzhitska O. Bianchi-Backlund transformation in many-dimensional Riemannian spaces

The thesis concerns the development of a generalized theory of Bianchi-Backlund transformations for pseudo-spherical submanifolds in many-dimensional spaces of con-stant curvature and in Riemannian product spaces. Given a two-dimensional ruled surfaces in one of spaces En, Sn, Hn, SnxR1, HnxR1, shift mappings at constant distance along rulings are considered. It is shown, that no ruled surface in En, Hn, HnR1 generates pseudo-spherical congruencies (Bianchi-Backlund transformations) with the help of the shift mappings. On the other hand, it is proved that a shift mapping of a ruled surface in either Sn or SnxR1 represents a pseudo-spherical congruence if and only if the surface in question is intrinsically flat and has a constant negative extrinsic curvature. Moreover, an analytical description of the ruled surfaces with vanishing intrinsic curvature and with constant negative extrin-sic curvature is discussed. A natural generalization of the Bianchi transformation to the case of general pseudo-spherical surfaces in S3xR1 is proposed. A particular class of pseudo-spherical surfaces referred to as standard is introduced. It is demonstrated that a pseudo-spherical surface in S3xR1 allows a generalized Bianchi transformation if and only if it is stan-dard, and in this case the transformed surface appears to be standard pseudo-spherical too. Moreover, it is shown that generalized Bianchi transformations of standard pseudo-spherical surfaces in S3xR1 is generated by classical Bianchi transformations of pseudo-spherical surfaces in S3. A complete descriptions of pseudo-spherical submanifolds in Euclidean space with Bianchi transformation degenerated into curve is obtained