Rigidity of submanifolds in the Euclidean and spherical spaces

Abstract

We deal with n-dimensional complete submanifolds $$M^n$$ immersed with nonzero parallel mean curvature vector field $$\mathbf{H}$$ either in the Euclidean space $${\mathbb {R}}^{n+p}$$ or in the Euclidean sphere $${\mathbb {S}}^{n+p}$$. In this setting, we establish sufficient conditions to guarantee that such a submanifold $$M^n$$ must be pseudo-umbilical, which means that $$\mathbf{H}$$ is an umbilical direction. Moreover, assuming a suitable lower bound for the Ricci curvature, we conclude that $$M^n$$ must be isometric to $${\mathbb {S}}^{n}$$, up to scaling.