Submanifolds of Euclidean space with parallel mean curvature vector

[摘要] The object of the paper is to study some compactsubmanifolds in the Euclidean spaceRnwhose mean curvaturevector is parallel in the normal bundle. First we prove thatthere does not exist ann-dimensional compact simply connectedtotally real submanifold inR2nwhose mean curvature vector isparallel. Then we show that then-dimensional compact totallyreal submanifolds of constant curvature and parallel meancurvature inR2nare flat. Finally we show that compactPositively curved submanifolds inRnwith parallel meancurvature vector are homology spheres. The last result inparticular for even dimensional submanifolds implies that theirEuler poincaré characteristic class is positive, which for theclass of compact positively curved submanifolds admiting isometricimmersion with parallel mean curvature vector inRn, answers theproblem of Chern and Hopf

[学科分类] 数学（综合）