In this paper, we study -dimensional complete immersed submanifolds in a Euclidean space . We prove that if is an -dimensional compact connected immersed submanifold with nonzero mean curvature in and satisfies either:
then is diffeomorphic to a standard -sphere, where and denote the squared norm of the second fundamental form of and the scalar curvature of , respectively.
On the other hand, in the case of constant mean curvature, we generalized results of Klotz and Osserman  to arbitrary dimensions and codimensions; that is, we proved that the totally umbilical sphere , the totally geodesic Euclidean space , and the generalized cylinder are only -dimensional complete connected submanifolds with constant mean curvature in if holds.
"Spherical rigidities of submanifolds in Euclidean spaces." J. Math. Soc. Japan 56 (2) 475 - 487, April, 2004. https://doi.org/10.2969/jmsj/1191418640