A geometric interpretation of the spectral problem for the generalized sine-Gordon system (Q2716558)

The authors of this interesting paper study the generalized sine-Gordon system, NEWLINE\[NEWLINE \partial \alpha_{ik}/\partial x_j=\alpha_{ij}\beta_{jk} \quad (k\neq j), \qquad \partial \beta_{ik}/\partial x_j=\beta_{ij}\beta_{jk} \quad (i\neq j\neq k\neq i), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \partial \beta_{jk}/\partial x_j+\partial \beta_{kj}/\partial x_k+\sum\limits_{i=1}^n\beta_{ij}\beta_{ik}=\alpha_{nj}\alpha_{nk} \quad (j\neq k), NEWLINE\]NEWLINE where indices \(i,j,k\) run from 1 to \(n\geq 2\), \((\alpha_{ik})=A\) is \(n\times n\) orthogonal matrix, \(\beta_{jk}\) are defined (\(j\neq k\)) by the first equation of the considered system. It is assumed that \(\beta_{kk}=0\). This system implicitly describes \(n\)-dimensional submanifolds of constant sectional curvature \(K=-1\) (Lobachevsky spaces) immersed in a Riemannian space form of constant curvature \(\widetilde K>-1\). In particular, it describes immersions of \(n\)-dimensional Lobachevsky space \(L^n\) of curvature \(K=-1\) in the Euclidean space \(E^{2n-1}\) and in spheres \(S^{2n-1}\) of any radius. To obtain the associated spectral problem one considers immersions of these submanifolds in spheres. It is discussed the so-called Sym-Tafel formula which yields the radius vector of negative constant curvature submanifolds in Euclidean spaces.