scholarly journals COMPLETE CLASSIFICATION OF LORENTZ SURFACES WITH PARALLEL MEAN CURVATURE VECTOR IN ARBITRARY PSEUDO-EUCLIDEAN SPACE

2010 ◽  
Vol 64 (2) ◽  
pp. 261-279 ◽  
Author(s):  
Bang-Yen CHEN
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Complete Classification ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Parallel Mean Curvature

scholarly journals Submanifolds of Euclidean space with parallel mean curvature vector

1991 ◽  
Vol 14 (3) ◽  
pp. 533-536
Author(s):  
Tahsin Ghazal ◽  
Sharief Deshmukh
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Characteristic Class ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Parallel Mean Curvature ◽  
Real Submanifold ◽  
Totally Real ◽  
Positively Curved

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

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