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Invariant Submanifolds of Generalized Sasakian-Space-Forms
Abstract
The new characterizations of invariant submanifolds of generalized Sasakian-space forms (SSF) in terms of their behavior with respect to the various curvature tensors are obtained in this work. By examining the connections between these submanifolds' second fundamental form σ and certain curvature tensors Wi with i = 2, 3, 4, 6, 7, we derive necessary and sufficient conditions for their geodesicity. We show that total geodesicity corresponds to the claim that tensor products vanish, Q(σ, Wi) = 0, subject to different non-degeneracy conditions for each curvature tensor. The essential characterization comes from the W6 curvature tensor, which suffices to fulfil 2n(f1−f3)≠0, and all other tensors lead to complementary constraints concerning the structural functions f1, f2, and f3. These findings offer various avenues for studying the geometric nature of invariant submanifolds and enhance our insights into the behaviour of generalized Sasakian-space-forms.
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We study the geometry of the contact pseudo-slant submanifolds of a Kenmotsu manifold. Necessary
and su�cient conditions are given for a submanifold to be a pseudo-slant submanifold, contact
pseudo-slant product, mixed geodesic and totally geodesic in Kenmotsu manifolds.
In this paper we introduce a new curvature tensor named as the T-curvature tensor. We show that the quasi-conformal, conformal, conharmonic, concircular, pseudo-projective, projective, M-projective, Wi-curvature tensors (i = 0,. .. , 9), W * j-curvature tensors (j = 0, 1) are the particular cases of the T-curvature tensor. Some properties for the T-curvature tensor are given. We obtain the results for T-conservative and T-flat semi-Riemannian manifolds.
In this paper, we consider doubly warped product mani-folds and we establish a general inequality for doubly warped products isometrically immersed in arbitrary Riemannian manifolds. Some apli-cations are derived.
The only pseudo-symmetric Kählerian manifolds of dimension n ≥ 6 are the semi-symmetric ones ([5], [6]). A similar result was obtained for Ricci pseudo-symmetry ([10]). Therefore Olszak introduced a variant of pseudo-symmetric Kählerian manifolds [10]. For dimension 4 Olszak gave an example of non semi-symmetric pseudo-symmetric Kählerian manifold [11]. In contrast however there do exist pseudo-symmetric Sasakian manifolds which are not semi-symmetric. In the present paper, we investigate the pseudo-symmetry of Sasakian space forms.
The present volume is the written version of a series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's recent work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.
Recently B.Y. Chen (4) studied warped products which are CR- submanifolds in Kaehler manifolds. B. Sahin (8) extended the above result for warped product semi-slant submanifolds Nµ £ fNT of a Kaehler manifold. In the present paper we have investigated the existence of doubly as well as CR-warped product submanifolds of Kenmotsu manifolds.
Recently, K. Matsumoto and I. Mihai established a sharp in-equality for warped products isometrically immersed in Sasakian space forms. As applications, they obtained obstructions to minimal isometric immersions of warped products into Sasakian space forms. P. Alegre, D.E. Blair and A. Carriazo have introduced the notion of generalized Sasakian space form. In the present paper, we obtain a sharp inequality for warped products isometrically immersed in generalized Sasakian space forms. Some applica-tions are derived.
Generalized Sasakian-space-forms are introduced and studied. Many examples of these manifolds are presented, by using some
different geometric techniques such as Riemannian submersions, warped products or conformal and related transformations. New
results on generalized complex-space-forms are also obtained.
In this paper we study the geometry of distributions of semi-slant submanifolds of (α,β) trans-Sasakian manifolds, some problems concerning the stability of slant submanifolds of generalized Sasakian space-forms and we also investigate the first normal Chern class for integral submanifolds of (α,β) trans-Sasakian generalized Sasakian space-forms.
In 1978, Bejancu introduced and studied CR-submanifolds of a Kaehler manifold [2]. Since then many papers appeared on this topic with ambient manifold as Sasakian space form [7], cosymplectic space form [12], Kenmotsu space form [10] etc. Recently Alegre et al. [1] introduced generalized Sasakian space form which generalizes above mentioned space forms. The aim of this paper is to study CR-submanifolds of generalized Sasakian space form.
In this paper, we investigate some fundamental properties of contact CR-submanifolds of a Kenmotsu manifold. We show that the anti-invariant distribution is always integrable and give a necessary and sufficient condition for the invariant distribution to be integrable. Then, properties of the induced structures on submanifold by almost contact metric structure on the ambient manifold are categorized. Finally, we give some results for contact CR-products and totally umbilical contact CR-submanifolds in a Kenmotsu manifold and in a Kenmotsu space form.
The present paper deals with warped product submanifolds of (LCS) n -manifolds and warped product semi-slant submanifolds of (LCS) n -manifolds. It is shown that there exists no proper warped product submanifold of (LCS) n -manifolds. However, we obtain some results for the existence or non-existence of warped product semi-slant submanifolds of (LCS) n -manifolds.
In the present paper submanifolds of generalized Sasakian-space-forms are studied. We focus on almost semi-invariant submanifolds, these generalize invariant, anti-invariant, and slant submanifolds. Sectional curva-tures, Ricci tensor and scalar curvature are also studied. The paper finishes with some results about totally umbilical submanifolds.
We study the τ-curvature tensor in a (k,μ) manifold. We study τ-flat and ξ-τ-flat (k,μ)-contact metric manifolds and obtain conditions for a τ-flat (k,μ)-contact metric manifold to be ϕ-symmetric. We consider ϕ-τ-symmetric and ϕ-τ-Ricci recurrent (k,μ)-contact metric manifolds and (k,μ)-contact metric manifolds satisfying the semi-symmetry condition τ·S=0.
In this article, we establish some inequalities for invariant submanifolds involving totally real sectional curvature and the scalar curvature. The equality cases are also discussed.
In this paper, contact metric and trans-Sasakian generalized Sasakian-space-forms are deeply studied. We present some general results for manifolds with dimension greater than or equal to 5, and we also pay a special attention to the 3-dimensional cases.
P. Alegre, D. Blair and A. Carriazo introduced and studied generalized Sasakian-space-forms. In this paper we classify conformally flat generalized Sasakian-space-forms under the assumption that the characteristic vector field is Killing. Also we classify locally symmetric generalized Sasakian-space-forms.
Contact manifolds.- Almost contact manifolds.- Geometric interpretation of the contact condition.- K-contact and sasakian structures.- Sasakian space forms.- Non-existence of flat contact metric structures.- The tangent sphere bundle.
Generalized Sasakian-space-forms and Conformal Changes of the Metric
- P Alegre
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Contact CR-warped product submanifolds in cosymplectic space forms
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On generalized Sasakian-space-forms satisfying certain conditions
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Narain, D., Yadav, S. and Dwivedi, P. K., On generalized Sasakian-space-forms satisfying certain conditions, Int. J.
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Some results on (LCS)2n+1-manifolds
- G T Sreenivasa
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Sreenivasa, G. T., Venkatesha and Bagewadi, C. S., Some results on (LCS)2n+1-manifolds, Bull. Math. Analysis and
Appl., 1(3) (2009), 64-70.
Some results on ( 1 , 2 , 3 ) 2 +1 -manifolds
- S Yadav
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Yadav, S., Suthar, D. L. and Srivastava, A. K., Some results on ( 1, 2, 3 ) 2 +1 -manifolds, Int. J. Pure and Appl.
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On the T curvature Tensor of Generalized Sasakian-Space-Forms
- A Kumari
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Kumari, A. and Chanyal, S. K., On the T curvature Tensor of Generalized Sasakian-Space-Forms, IOSR Journal of
Mathematics, 11(3) (2015) 61-68.