scholarly journals CR-hypersurfaces of the six-dimensional sphere

1994 ◽  
Vol 17 (1) ◽  
pp. 197-200
Author(s):  
M. A. Bashir
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Dimensional Sphere ◽  
Totally Umbilical ◽  
Parallel Second Fundamental Form ◽  
Extrinsic Sphere

We proved that there does not exist a properCR-hypersurface ofS6with parallel second fundamental form. As a result of this we showed thatS6does not admit a properCR-totally umbilical hypersurface. We also proved that an Einstein properCR-hypersurface ofS6is an extrinsic sphere.

Minimal two-spheres in G(2, 4; (C)) with parallel second fundamental form

2017 ◽  
Vol 101 (5-6) ◽  
pp. 899-912
Author(s):  
Wenjuan Zhang ◽  
Xiaoxiang Jiao ◽  
Mingyan Li
Keyword(s):  
Fundamental Form ◽  
Second Fundamental Form ◽  
Parallel Second Fundamental Form

On non-invariant hypersurfaces of a nearly hyperbolic Sasakian manifold

2017 ◽  
Vol 28 (08) ◽  
pp. 1750064
Author(s):  
Mobin Ahmad ◽  
Shadab Ahmad Khan ◽  
Toukeer Khan
Keyword(s):  
Fundamental Form ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Second Fundamental Form ◽  
Text Structure ◽  
Necessary And Sufficient Conditions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Necessary And Sufficient

We consider a nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with [Formula: see text]-structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with [Formula: see text]-structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with [Formula: see text]-structure under the condition when f is parallel.

scholarly journals Classification of totally umbilical submanifolds in symmetric spaces

1980 ◽  
Vol 30 (2) ◽  
pp. 129-136 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Symmetric Space ◽  
Fundamental Form ◽  
Symmetric Spaces ◽  
Second Fundamental Form ◽  
Totally Umbilical ◽  
The Second Fundamental Form ◽  
Totally Umbilical Submanifold ◽  
Totally Umbilical Submanifolds ◽  
First Fundamental Form

AbstractA submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

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