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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].

##### MSC:
 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