scholarly journals A Classification of a Totally Umbilical Slant Submanifold of Cosymplectic Manifolds

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Siraj Uddin ◽  
Cenap Ozel ◽  
Viqar Azam Khan
Keyword(s):  
Slant Submanifold ◽  
Totally Umbilical ◽  
Cosymplectic Manifolds

scholarly journals Classification of totally umbilical slant submanifolds of a Kenmotsu manifold

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2405-2412 ◽  
Author(s):  
Siraj Uddin ◽  
Zafar Ahsan ◽  
Yaakub Hadi
Keyword(s):  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Slant Submanifold ◽  
Kenmotsu Manifold ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Slant Submanifolds ◽  
The Mean

The purpose of this paper is to classify totally umbilical slant submanifolds of a Kenmotsu manifold. We prove that a totally umbilical slant submanifold M of a Kenmotsu manifold ?M is either invariant or anti-invariant or dimM = 1 or the mean curvature vector H of M lies in the invariant normal subbundle. Moreover, we find with an example that every totally umbilical proper slant submanifold is totally geodesic.

scholarly journals Pseudo-slant submanifolds in cosymplectic space forms

2016 ◽  
Vol 8 (1) ◽  
pp. 53-74 ◽  
Author(s):  
Süleyman Dirik ◽  
Mehmet Atçeken
Keyword(s):  
Space Form ◽  
Sufficient Conditions ◽  
Slant Submanifold ◽  
Space Forms ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Cosymplectic Manifold ◽  
Slant Submanifolds ◽  
Cosymplectic Manifolds ◽  
Necessary And Sufficient

Abstract In this paper, we study the geometry of the pseudo-slant submanifolds of a cosymplectic space form. Necessary and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold, pseudo-slant product, mixed geodesic and totally geodesic in cosymplectic manifolds. Finally, we give some results for totally umbilical pseudo-slant submanifold in a cosymplectic manifold and cosymplectic space form.

scholarly journals Some Remarks on Quasi-Generalized CR-Null Geometry in Indefinite Nearly Cosymplectic Manifolds

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Fortuné Massamba ◽  
Samuel Ssekajja
Keyword(s):  
Totally Umbilical ◽  
Cosymplectic Manifold ◽  
Cosymplectic Manifolds ◽  
Nearly Cosymplectic ◽  
Geometric Effects

Attention is drawn to some distributions on ascreen Quasi-Generalized Cauchy-Riemannian (QGCR) null submanifolds in an indefinite nearly cosymplectic manifold. We characterize totally umbilical and irrotational ascreen QGCR-null submanifolds. We finally discuss the geometric effects of geodesity conditions on such submanifolds.

scholarly journals Noninvariant Hypersurfaces of a Nearly Trans-Sasakian Manifolds

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Satya Prakash Yadav ◽  
Shyam Kishor
Keyword(s):  
Contact Structure ◽  
Sasakian Manifold ◽  
Necessary Condition ◽  
Sasakian Manifolds ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Cosymplectic Manifold ◽  
Cosymplectic Manifolds ◽  
Nearly Cosymplectic

The present paper focuses on the study of noninvariant hypersurfaces of a nearly trans-Sasakian manifold equipped with(f,g,u,v,λ)-structure. Initially some properties of this structure have been discussed. Further, the second fundamental forms of noninvariant hypersurfaces of nearly trans-Sasakian manifolds and nearly cosymplectic manifolds with(f,g,u,v,λ)-structure have been calculated providedfis parallel. In addition, the eigenvalues offhave been found and proved that a noninvariant hypersurface with(f,g,u,v,λ)-structure of nearly cosymplectic manifold with contact structure becomes totally geodesic. Finally the paper has been concluded by investigating the necessary condition for totally geodesic or totally umbilical noninvariant hypersurface with(f,g,u,v,λ)-structure of a nearly trans-Sasakian manifold.

scholarly journals Totally Umbilical Proper Slant and Hemislant Submanifolds of an LP-Cosymplectic Manifold

2011 ◽  
Vol 2011 ◽  
pp. 1-9
Author(s):  
Siraj Uddin ◽  
Meraj Ali Khan ◽  
Khushwant Singh
Keyword(s):  
Present Note ◽  
Slant Submanifold ◽  
Slant Angle ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Cosymplectic Manifold ◽  
Integrability Conditions

In the present note, we study slant and hemislant submanifolds of an LP-cosymplectic manifold which are totally umbilical. We prove that every totally umbilical proper slant submanifoldMof an LP-cosymplectic manifoldM¯is either totally geodesic or ifMis not totally geodesic inM¯then we derive a formula for slant angle ofM. Also, we obtain the integrability conditions of the distributions of a hemi-slant submanifold, and then we give a result on its classification.

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.

XX.—Differential Geometry on Hypersurfaces in a Cayley Space

1964 ◽  
Vol 66 (4) ◽  
pp. 216-231 ◽  
Author(s):  
K. Yano ◽  
T. Sumitomo
Keyword(s):  
Complex Structure ◽  
Hermitian Structure ◽  
Dimensional Euclidean Space ◽  
Dimensional Sphere ◽  
Almost Complex Structure ◽  
Principal Curvatures ◽  
Totally Umbilical ◽  
Almost Complex ◽  
Necessary And Sufficient

A seven-dimensional Euclidean space considered as the space of purely imaginary Cayley numbers is called a Cayley space. The six-dimensional sphere in a Cayley space admits an almost complex structure which is not integrable. Moreover the algebraic properties of the imaginary Cayley numbers induce an almost complex structure on any oriented differentiable hypersurface in the Cayley space. The Riemannian metric induced on the hypersurface from the metric of the Cayley space is Hermitian with respect to the almost complex structure.It is proved that the induced Hermitian structure of an oriented hypersurface in the Cayley space is almost Kaehlerian if and only if it is Kaehlerian, that a necessary and sufficient condition for a hypersurface in a Cayley space to be an almost Tachibana space is that the hypersurface be totally umbilical, and that a totally umbilical hypersurface in a Cayley space admits a complex structure when and only when it is totally geodesic.For a hypersurface in the Cayley space with the induced Hermitian structure which is an *O-space it is proved that all the principal curvatures of the hypersurface are constant, and from this is deduced a classification of such *O-spaces.

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