Isometric immersions of \(H_ 1^ n\) into \(H_ 1^{n+1}\) (Q1919917)

Let \(S^n_p\) and \(H^n_p\) be the \(n\)-dimensional hyperquadrics of signature \(p\) and of constant curvatures 1 and \(-1\), respectively. The author proves that the only hyperquadrics which admit a complete totally geodesic foliation of codimension 1 in an open subset are either \(H^n_0\), \(H^n_1\), \(n\geq 1\) or \(S^n_{n-1}\), \(1\leq n\leq 2\). It is also shown that the only isometric immersions \(f:S^n_1 \to S^{n+1}_1\) \((n\geq 3)\) are totally geodesic ones. All complete degenerate totally geodesic foliations in any open subset of \(H^n_1\), \(n\geq 1\), are determined. A characterization of the space of isometric immersions from \(H^n_1\) into \(H^{n+1}_1\) is given.