Rigidity of codimension two indefinite spheres
Let \(S^ n_ s\) denote the n-dimensional pseudosphere of signature \(s\in \{0,...,n-1\}\) and curvature 1 and let \(f: S^ n_ s\to S^{n+p}_{s+q}\) be an isometric immersion with \(0\leq q\) and \(1\leq p\). It is known that f is totally geodesic, if \(q=0\) and \(p<n-s\) [see \textit{D. Ferus}, Math. Ann. 217, 155-156 (1975; Zbl 0295.53026), and \textit{M. Dajczer} and \textit{S. Fornari}, Yokohama Math. J. 35, No.1/2, 61-69 (1987; Zbl 0644.53058)]. However there are other examples if \(q>0\) [see \textit{M. Dajczer} and \textit{P. Dombrowski} in Global differential geometry and global analysis 1984, Proc. Conf., Berlin 1984, Lect. Notes Math. 1156, 59-73 (1985; Zbl 0571.53042)] and Dajczer/Fornari [loc. cit.]. In the paper under review the authors give an exact description of all such immersions for the case that \(q=1\), \(p=2\) and \(n-s>p+q\), if the (closed) set of totally geodesic points does not disconnect \(S^ n_ s\).
scientific article; zbMATH DE number 4151294