Submanifolds with flat normal bundle (Q5906849)

The author studies compact orientable submanifolds \(M\) with flat normal bundle, immersed in real space forms. The main result in this paper states that if \(M\) has non-negative sectional curvature and constant scalar curvature greater than the curvature of the ambient space, then it is either totally umbilical or, locally, the Riemannian product of totally umbilical constantly curved submanifolds. Then, he uses this theorem to obtain some applications to the case when the ambient space is the Euclidean space. In particular, he deals with the problem: is a compact hypersurface of Euclidean space with constant scalar curvature the round sphere? It is known that the answer is affirmative under additional conditions, for instance if its sectional curvature is non-negative [\textit{S.-Y. Cheng} and \textit{S.-T. Yau}, Math. Ann. 225, 195-204 (1977; Zbl 0349.53041)] or if the hypersurface is embedded [\textit{A. Ros}, J. Differ. Geom. 27, 215-220 (1988; Zbl 0638.53051)]. The author obtains that the answer is also yes under the additional condition of non-negative Ricci curvature.