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CLASSIFICATION OF TOTALLY UMBILICAL SLANT SUBMANIFOLDS OF A (k,µ) -CONTACT MANIFOLD
Abstract
The objective of this paper is to classify totally umbilical slant submanifolds of a (k, µ)-contact manifold. We prove that a totally umbilical slant sub-manifold M of a (k, µ)-contact manifold M is either invariant or anti-invariant or dim M=1 or the mean curvature vector H of M lies in the invariant normal sub bundle. 2010 Mathematics Subject Classification: 53C40, 53C42.
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In this paper, we study the possibility of obtaining an induced contact metric structure on a slant submanifold of a contact metric manifold. We also give a characterization theorem for three-dimensional slant submanifolds.
A slant immersion is an isometric immersion from a Riemannian manifold into an almost Hermitian manifold with constant slant angle. In this article we study and characterized slant surfaces into the complex 2-plane C^2 via its Gauss ma. We also prove that every surface without complex tangent points in a 4-dimensional almost Hermitian manifold M_ is slant with respect to a suitable chosen almost complex structure on M_.
An immersion of a differentiable manifold into an almost Hermitian manifold is called a \textit{general slant immersion} if it has constant Wirtinger angle ([3, 6]).
A general slant immersion which is neither holomorphic nor totally real is called a proper slant immersion.
In the first part of this article, we prove that every general slant immersion of a compact manifold into the complex Euclidean m-space is totally real.
This result generalizes the well-known fact that there exist no compact holomorphic submanifolds in any complex Euclidean space.
In the second part, we classify proper slant surfaces in when they are contained in a hypersphere , or contained in a hyperplane , or when their Gauss maps have rank $
A 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.
The intrinsic geometry of 3 dimensional non anti invariant slant submanifolds of K-contact manifolds is studied.
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