Isometric immersions of pseudo-Riemannian space forms
Let \(f: M^n_s(K)\to N^{2n-1}_{s+q}(c)\) be a local isometric immersion of a time-like submanifold \(M^n_s(K)\) with constant sectional curvature \(K\) and index \(s\) into a pseudo-Riemannian space form \(N^{2n-1}_{s+q}(c)\) with constant sectional curvature \(c\) and index \(s+ q\), where \(q\geq 0\), \(1\leq s\leq n-1\) and \(K\neq c\). These immersions are studied by means of the moving frames adapted to \(M^n_s(K)\) in \(N^{2n-1}_{s+q}(c)\) using Cartan's method, structural equations etc. A theorem by \textit{J. L. Barbosa}, \textit{W. Ferreira} and \textit{K. Tenenblat} [Ann. Global Anal. Geom. 14, No. 4, 381--401 (1996; Zbl 0866.53049)] is generalized to this case. Namely, assume that the second fundamental form is orthogonally diagonalizable and there exists a point \(p\) of \(M\) where the principal normal curvatures are different from \(K- c\). Then it is established that on an open contractible region \(U\) of \(p\), there exist line of curvature coordinates \(\{x_1,\dots, x_n\}\) and local parallel normal frame fields, for which the first and second fundamental forms have the diagonal forms. Also, the corresponding Gauss-Codazzi-Ricci equations are derived. The same is done for the particular cases of \(K< c\) and \(q= 0\), and of \(K> c\) and \(q= 0\), where the normal bundle is flat. The higher-dimensional generalizations of the classical Bäcklund theorem are given. By introducing the Chebyshev coordinates and using the Bäcklund theorem an explicit Bäcklund transformation is given, as well as a permutability formula between the generalized sine-Laplace equation and the generalized sinh-Laplace equation.
scientific article; zbMATH DE number 2175254
scientific article