We study the geometry of the contact pseudo-slant submanifolds of a Kenmotsu manifold. Necessary and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold, contact pseudo-slant product, mixed geodesic and totally geodesic in Kenmotsu manifolds. c ©2016 All rights reserved.

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Contact pseudo-slant submanifolds of a Kenmotsu manifold

In this paper we study the warped product submanifolds of a Lorentzian paracosymplectic manifold and obtain some nonexistence results. We show that a warped product semi-invariant submanifold in the form {$M=M_{T}\times_{f}M_{\bot}$} of Lorentzian paracosymplectic manifold such that the characteristic vector field is normal to $M$ is an usual Riemannian product manifold where totally geodesic and totally umbilical submanifolds of warped product are invariant and anti-invariant, respectively. We prove that the distributions involved in the definition of a warped product semi-invariant submanifold are always integrable. A necessary and sufficient condition for a semi-invariant submanifold of a Lorentzian paracosymplectic manifold to be warped product semi-invariant submanifold is obtained. We also investigate the existence and nonexistence of warped product semi-slant and warped product anti-slant submanifolds in a Lorentzian paracosymplectic manifold.

Warped product submanifolds of Lorentzian paracosymplectic manifolds

The present thesis entitled “The Geometry of Slant Lightlike Submanifolds” comprises certain investigations carried out by me at the School of Mathematics (SOM), Thapar University, Patiala, under the supervision of Dr. S. S. Bhatia, Professor, School of Mathematics, Thapar University, Patiala and Dr. Rakesh Kumar, Assistant Professor, Department of Basic and Applied Sciences, Punjabi University, Patiala. The core of differential geometry and modern geometrical dynamics represents the concept of a manifold. Manifolds are the higher-dimensional analogues of surfaces. The local and global properties of smooth manifolds equipped with a metric tensor encodes its geometry. To study the geometric aspects of a manifold, it is more convenient to first embed it into a known manifold and then study the geometry which is induced on it. This approach gives impetus to the study of submanifolds which later developed into a fascinating study of the theory of submanifolds. The geometry of submanifolds of an almost Hermitian manifold depends upon the behaviour of the tangent bundle of the submanifold with respect to the almost complex structure J̄ of the manifold. The action of the almost complex structure give rise to the two well known classes of submanifolds namely, the holomorphic submanifolds and the totally real submanifolds. Let M̄ be an almost Hermitian manifold with almost complex structure J̄ . Consider M as a submanifold of M̄ and let the tangent space and normal space of M at p ∈ M be denoted by TpM and TpM, respectively. If TpM is invariant under the action of J̄ for each p ∈M , that is, if J̄TpM = TpM, for each p ∈ M , then M is called an invariant (or holomorphic) submanifold of M̄ . On the other hand, if J̄TpM is contained in the normal space TpM ⊥, for each p ∈M ,

http://tudr.thapar.edu:8080/jspui/bitstream/10266/3876/4/3876.pdf

The geometry of slant lightlike submanifolds

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The geometry of slant lightlike submanifolds