Euclidean submanifolds with nonparallel normal space (Q752488)

The author deals with the local theory of isometric immersions with nonparallel first normal spaces \(N_ 1\) of higher dimension. As an application of his general results, the author proves: Let f: \(M^ n\to {\mathbb{R}}^ N\) be an isometric immersion with dim \(N_ 1=2\) everywhere. Then M contains an open and dense subset \(M'\) such that \(M'=U_ 1\cup U_ 2\cup U_ 3,\) where \(U_ 1\), \(U_ 2\), \(U_ 3\) are open and disjoint subsets satisfying the following conditions: (1) The connected components of \(U_ 1\) have substantial codimension 2. (2) \(f(U_ 2)\) is foliated by totally geodesic (n-2)-dimensional affine subspaces. (3) For each connected component \(U_{\lambda}\) of \(U_ 3\) there exists an open subset \(V_{\lambda}\) of \({\mathbb{R}}^{n+1}\) and isometric immersions \(g_{\lambda}: U_{\lambda}\to {\mathbb{R}}^{n+1}\), \(h_{\lambda}: V_{\lambda}\to {\mathbb{R}}^ N\), both with 1-dimensional first normal spaces such that \(g_{\lambda}(U_{\lambda})\subset V_{\lambda}\) and \(f|_{U_{\lambda}}=h_{\lambda}\circ g_{\lambda}.\) This result is an improvement of Theorem 4 of \textit{J. Grifone} and \textit{J. M. Morvan} in J. Differ. Geom. 16, 351-371 (1981; Zbl 0492.53036).