Geometric aspects of Bäcklund transformations of Weingarten submanifolds (Q1891184)

If \(f_ 1\) and \(f_ 2\) are immersions of a surface \(M\) into \(\mathbb{R}^ 3\) such that their induced frame bundles \({\mathcal F}_ 1\) and \({\mathcal F}_ 2\) differ by a constant right action, then \(f_ 1\) and \(f_ 2\) are both linear Weingarten immersions and the right action corresponds to a generalization of the classical Bäcklund transformation. These facts were known to Darboux and to R. Bryant. In this paper the results are extended to immersions of \(n\)-dimensional manifolds into \(\mathbb{R}^{2n-1}\) satisfying a Weingarten condition on the normal bundle. The work may be regarded as a generalization of the Bäcklund transformation developed by \textit{K. Tenenblat} and \textit{C.-L. Terng} [Ann. Math., II. Ser. 111, 477-490 (1980; Zbl 0462.35079)] with the mutual tangency restriction lifted.