Submanifolds of Euclidean space with parallel mean curvature vector

Tahsin Ghazal,Sharief Deshmukh

doi:10.1155/s0161171291000728

Abstract

The object of the paper is to study some compact submanifolds in the Euclidean spaceRnwhose mean curvature vector is parallel in the normal bundle. First we prove that there does not exist ann-dimensional compact simply connected totally real submanifold inR2nwhose mean curvature vector is parallel. Then we show that then-dimensional compact totally real submanifolds of constant curvature and parallel mean curvature inR2nare flat. Finally we show that compact Positively curved submanifolds inRnwith 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 inRn, answers the problem of Chern and Hopf

Highlights

The object of the paper is to study some compact submaniforlds in the Euclidean space Rn whose mean curvature vector is parallel in the normal bundle
Euler poincare’ characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric n, immersion with parallel mean curvature vector in R answers the problem of Chern and Hopf
If M is a submanifold of Rn with normal bundle 9, the connection V induces the Riemannian connection

Summary

Introduction

The object of the paper is to study some compact submaniforlds in the Euclidean space Rn whose mean curvature vector is parallel in the normal bundle. Euler poincare’ characteristic class is positive, which for the class of compact positively curved submanifolds admiting isometric n, immersion with parallel mean curvature vector in R answers the problem of Chern and Hopf. The mean curvature vector H of M is given by H The mean curvature vector H is said to be parallel

Results

Conclusion