Codimension \(m\) isometric immersions between pseudo-spheres (Q1900072)

The author investigates isometric immersions \(f:S^n_p\to S^{n+m}_{p+1}\), for \(n-p\geq 2m\). Here \(S^m_k\) is the pseudosphere of radius 1 in the pseudo-Euclidean space \(\mathbb{R}^{m+1}_k\). The metric on \(\mathbb{R}^{m+1}_k\) is given by \(\langle x,y\rangle=-\sum^k_{i=1}x_iy_i+ \sum^{m+1}_{i=k+1} x_iy_i\), for \(x=(x_1,\dots,x_{m+1})\), \(y=(y_1,\dots,y_{m+1})\). The main result is the following Theorem: Let \(f:S^n_p\to S^{n+m}_{p+1}\) be an isometric immersion with \(n-p\geq 2m\), \(m\geq 3\). If the set of totally geodesic points does not disconnect \(S^n_p\) and the first normal space of \(f\) is parallel in the set of non-totally geodesic points, then \(f\) is, up to a congruency of \(\mathbb{R}^{n+m+1}_{p+1}\), in the following form: \[ f(x)= (\phi(x),I_1(x),\dots,I_p(x),I_{p+1}(x),\dots, I_{n+m-1}(x)) \] for all \(x\in S^n_p\), where \(\phi: S^n_p\to\mathbb{R}\) is a smooth function and \(I:S^n_p\to S^{n+m-2}_{p+1}\) is a totally geodesic isometric immersion.