Submanifolds of Euclidean space with parallel mean curvature vector

The object of the paper is to study some compact submanifolds in the Euclidean space R n whose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist an n -dimensional compact simply connected totally real submanifold in R 2 n whose mean curvature vector is parallel. Then we show that the n -dimensional compact totally real submanifolds of constant curvature and parallel mean curvature in R 2 n are flat. Finally we show that compact Positively curved submanifolds in R n with parallel mean curvature vector are homology spheres. The last result in particular for even dimensional submanifolds implies that their Euler poincaré characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric immersion with parallel mean curvature vector in R n , answers the problem of Chern and Hopf