Submanifolds in a hyperbolic space form with flat normal bundle
\textit{S.-Y. Cheng} and \textit{S. T. Yau} [Math. Ann. 225, 195--204 (1977; Zbl 0349.53041)] or \textit{H. Alencar} and \textit{M. P. do Carmo} [Proc. Am. Math. Soc. 120, No. 4, 1223--1229 (1994; Zbl 0802.53017)] dealt with the rigidity problem for a hypersurface with constant scalar curvature in a space form by introducing a selfadjoint second-order differential operator. On availing this technique the author presents and proves the following two noteworthy theorems: (1) Let \(M^n\) be a compact submanifold with nonnegative curvature in the hyperbolic space form \(H^{n+p}\). Suppose that the normal bundle \(N(M)\) is flat and the normalized mean curvature vector is parallel. If the scalar curvature \(R\) is constant and \(R+ 1\geq 0\), then either \(M^n= M_1\times M_2\times\cdots\times M_k\) such that each \(M_i\) is a minimal submanifold of a totally umbilical submanifold \(N_i\) and the \(N_i\)'s are mutually orthogonal along their intersection or \(M^n\) lies in a totally geodesic subspace \(H^{n+1}\) of \(H^{n+p}\). (2) Let \(M^n\) \((n\geq 3)\), be a compact submanifold with parallel normalized mean curvature vector field immersed in \(H^{n+p}\). Suppose that \(R\) is constant and \(\overline R= R+1\geq 0\). If \(N(M)\) is flat and \[ n\overline R\leq S\leq {n\over (n-2)(nR-2)} [n(n- 1)\overline R^2- 4(n- 1)\overline R+ n], \] then \(M^n\) is totally umbilical, where \(S\) denotes the norm square of the second fundamental form of \(M^n\).
scientific article; zbMATH DE number 2031068