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Bäcklund transformations, matrix-Riccati systems and isometric immersions of space forms into space forms. (English) Zbl 0798.53057
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 359-385 (1992).
Starting from a local group action of \(Gl(2N,\mathbb{R})\) on the space of \(N \times N\)-matrices the author introduces certain systems of partial differential equations and a Bäcklund transformation of the space of solutions. His investigation includes the generalized wave equation and the generalized sine-Gordon equation. A superposition formula is derived, which describes the commutativity of certain pairs of Bäcklund transformations. The method is applied to construct special maps into Grassmannian manifolds and particular isometric immersions of \(n\)- dimensional spaces of constant curvature into \(S^{2n - 1}\). In \(S^ 3\) a special surface of revolution of constant curvature \(K \in ]0,1[\) is obtained.
For the entire collection see [Zbl 0764.00002].

53C40 Global submanifolds
35A30 Geometric theory, characteristics, transformations in context of PDEs
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds