BONNET SURFACES IN FOUR-DIMENSIONAL SPACE FORMS

2004 ◽  
Vol 15 (10) ◽  
pp. 981-985
Author(s):  
ATSUSHI FUJIOKA
Keyword(s):  
Mean Curvature ◽  
Dimensional Space ◽  
Curvature Vector ◽  
Reduction Theorem ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Space Forms ◽  
3 Dimensional ◽  
Flat Normal Bundle ◽  
Parallel Mean Curvature

We study isometric deformations of surfaces in four-dimensional space forms preserving the length of the mean curvature vector. In particular we consider the natural condition, called to be simple, and show that such surfaces with flat normal bundle are Bonnet surfaces in totally geodesic or umbilic 3-dimensional space forms, which is regarded as a generalization of Chen–Yau's reduction theorem for surfaces with parallel mean curvature vector.

scholarly journals Classification of f-biharmonic submanifolds in Lorentz space forms

2021 ◽  
Vol 19 (1) ◽  
pp. 1299-1314
Author(s):  
Li Du
Keyword(s):  
Vector Field ◽  
Mean Curvature ◽  
Lorentz Space ◽  
Curvature Vector ◽  
Shape Operator ◽  
Mean Curvature Vector ◽  
Biharmonic Submanifolds ◽  
Parallel Mean Curvature Vector ◽  
Space Forms ◽  
Parallel Mean Curvature

Abstract In this paper, f-biharmonic submanifolds with parallel normalized mean curvature vector field in Lorentz space forms are discussed. When f f is a constant, we prove that such submanifolds have parallel mean curvature vector field with the minimal polynomial of the shape operator of degree ≤ 2 \le 2 . When f f is a function, we completely classify such pseudo-umbilical submanifolds.

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