Isotropic immersions and parallel immersions of space forms into space forms (Q1769859)

The author proves the following theorem. Let \(f: M^n(c)\to\widetilde M^{n+p}(\widetilde c)\) be an iotropic immersion between space forms. Assume that (i) \(H^2\leq{2(n+1)\over n} c-\widetilde c\), (ii) \(0\leq (1-n)\Delta H^2+ n\langle {\mathfrak y},\Delta {\mathfrak y}\rangle\), where \(\sigma\) is the second fundamental form of \(f\), \({\mathfrak y}={1\over n}\text{ trace\,}\sigma\), and \(H=\| {\mathfrak y}\|\). Then \(f\) is parallel. In addition, \(f\) is locally equivalent to a totally umbilic imbedding or a second standard minimal immersion followed by a totally umbilic imbedding.