International Journal of Mathematics and Mathematical Sciences
Volume 14 (1991), Issue 3, Pages 533-536
doi:10.1155/S0161171291000728
Submanifolds of Euclidean space with parallel mean curvature vector
Department of Mathematics, College of Science, P.O. Box 2455, King Saud University, Riyadh 11451, Saudi Arabia
Received 21 November 1989; Revised 19 October 1990
Copyright © 1991 Tahsin Ghazal and Sharief Deshmukh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The object of the paper is to study some compact
submanifolds in the Euclidean space Rn 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 R2n 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 R2n are flat. Finally we show that compact
Positively curved submanifolds in Rn 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 Rn, answers the
problem of Chern and Hopf