Viqar Azam Khan

Viqar Azam Khan

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71
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Publications

Publications (71)
Article
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A new class of warped product manifolds which is known as sequential warped product manifolds have been defined in [15] and studied in detail in [9]. This article is dedicated to study sequential warped product submanifolds having factors holomorphic, totally real and pointwise slant submanifolds of nearly Kaehler manifolds. We obtained Chen?s ineq...
Preprint
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We study immersions of a hemi-slant submanifold of lcK manifolds as a warped product with the leaves of the holomorphic (respectively slant) distribution warped and establish characterisation theorems and estimations for the squared length of the second fundamental form in both cases.
Chapter
In an almost Hermitian manifold (M¯,J,g), the almost complex structure J turns a vector field to another vector field perpendicular to it. The impact of this property onto a submanifold M of M¯ yields invariant (complex or holomorphic) and anti invariant (totally real) distributions on M, where a distribution D on M is holomorphic if JDx=Dx for eac...
Article
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Many differential geometric properties of submanifolds of a Kaehler manifold are looked into via canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product (i.e. locally a Riemannian product of a holomorphic and a totally real submanifold) if and only if the canonical tensor P is par...
Article
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J. L. Cabrerizo et al. [5] studied slant submanifolds of Sasakian and Kcontact manifolds. Semi-slant submanifolds were introduced as a generalized version of CR-submanifold. Cabrerizo et al. [4] obtained interesting results for the semi-slant submanifold of Sasakian manifolds. The purpose of the present paper is to study slant and semi-slant subman...
Article
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Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in Sahin (Geom Dedic 117:195–202, 2006), it is interesting to find their existence in a more general setting, e.g., nearly Kaehler manifolds. In this paper, we obtain a necessary and sufficient condition for a semi-slant submanifold of a nearly Kaehler manifold...
Article
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In the present article, we have investigated pointwise pseudo-slant submanifolds of Kenmotsu manifolds and have sought conditions under which these submanifolds are warped products. To this end first, it is shown that these submanifolds can not be expressed as non-trivial doubly warped product submanifolds. However, as there exist non-trivial (sing...
Article
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Warped product manifolds provide excellent setting to model space-time near black holes or bodies with large gravitational force (cf. [1], [2], [14]). Recently, results are published exploring the existence (or non-existence) of warped product submanifolds in Kaehle- rian and contact settings (cf. [6], [17], [20]). To continue the sequel, we have c...
Article
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In [6], Cabras, Ianus and Pitis proved that in a cosymplectic manifold there does not exist any extrinsic sphere tangent to the structure vector field ξ. We consider the structure vector field ξ normal to the submanifold in the sense of Papaghiuc [12] and derive that a totally umbilical CR-submanifold of a cosymplectic manifold is either (i) totall...
Article
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Many differential geometric properties of a submanifold of a Kaehler manifold are conceived via canonical structure tensors T and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product if and only if T is parallel on the submanifold (c.f. [2]). Warped product submanifolds are generalized version of CR-product sub...
Article
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The present article is devoted to the study of conditions on a hemi-slant submanifold of a nearly Kaehler manifold under which the submanifold is a warped product submanifold.
Article
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Non-existence of warped product semi-slant submanifolds of Kaehler manifolds was proved in [17], it is interesting to find their existence. In this paper, we prove the existence of warped product semi-slant submanifolds of nearly Kaehler manifolds by a characterization. To this end we obtain an inequality for the squared norm of second fundamental...
Article
Full-text available
Many differential geometric properties of a submanifold of a Kaehler manifold are conceived via the canonical structure tensors P and F on the submanifold. For instance, a CR-submanifold of a Kaehler manifold is a CR-product if and only if P is parallel on the submanifold (cf. [6]). Warped product submanifolds are generalized versions of CR-product...
Article
Full-text available
Recently, many authors studied the relations between the squared norm of the second fundamental form (extrinsic invariant) and the warping func-tion (intrinsic invariant) for warped product submanifolds (see [1, 7, 14]). In-spired by those relations we establish a general sharp inequality, namely h 2 ≥ 2s[∇lnf 2 + α 2 − β 2 ], for contact CR-warped...
Article
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In the present paper we have study totally umbilical hemi-slant sub-manifolds of Cosymplectic manifolds via Riemannian curvature tensor and finally obtained a classification for the Totally umbilical hemi-slant submanifolds of Cosymplectic manifolds.
Article
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Abstract.We study the geometry of warped product submanifolds of Lorentz-ian훽-Kenmotsu manifolds. We obtain a characterization result for CR-warpedproducts
Article
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In this article, we obtain the necessary and sufficient conditions that the semi-invariant submanifold to be a locally warped product submanifold of invariant and anti-invariant submanifolds of a cosymplectic manifold in terms of canonical structures T and F. The inequality and equality cases are also discussed for the squared norm of the second fu...
Article
Warped product manifolds provide a natural frame work for time dependent mechanical system and have applications in Physics. The studies on warped product manifolds with extrinsic geometric point of view intensified after B.Y.Chen's work on CR-warped product submanifolds of Kaehler manifolds. He investigated the existence of CR-submanifolds of a Ka...
Article
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In the present paper, we study warped product semi-slant submanifolds of a Kenmotsu manifold. We obtain some results on the existence of such type warped product submanifolds of a Kenmotsu manifold with an example.
Article
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We study slant submanifolds of a cosymplectic manifold. It is shown that a totally umbilical slant submanifold of a cosymplectic manifold is either an anti-invariant submanifold or a 1−dimensional submanifold. We show that every totally umbilical proper slant submanifold of a cosymplectic manifold is totally geodesic.
Article
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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.
Article
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We study the geometry of warped product submanifolds of Lorentz-ian íµí»½-Kenmotsu manifolds. We obtain a characterization result for CR-warped products.
Article
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We study the geometry of warped product submanifolds of Lorentz-ian í µí»½-Kenmotsu manifolds. We obtain a characterization result for CR-warped products.
Article
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Warped product manifolds are known to have applications in Physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [HONG, S. T.: Warped products and black holes, Nuovo Cimento Soc. Ital. Fis. B 120 (2005), 1227–1234]). The studies on warped product manifolds with extrinsic geometric poin...
Article
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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 submanifold M of an LP-cosymplectic manifold M is either totally geodesic or if M is not totally geodesic in M then we derive a formula for slant angle of M. Also, we obtain th...
Article
Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a mas-sive star (cf. [9]). The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen's work on CR-warped product submanifolds of Kaehle...
Article
Full-text available
Warped product manifolds are known to have applications in physics. For instance, they provide an excellent setting to model space-time near a black hole or a massive star (cf. [9]). The studies on warped product manifolds with extrinsic geometric point of view were intensified after the B.Y. Chen's work on CR-warped product submanifolds of Kaehler...
Article
Full-text available
We study warped product anti-slant submanifolds of cosymplectic manifolds. It is shown that the cosymplectic manifold do not admit non trivial warped product submanifolds in the form N ⊥ × f N θ and then we obtain some results for the existence of warped products of the type N θ × f N ⊥ , where N ⊥ and N θ are anti-invariant and proper slant subman...
Article
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We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structure F.
Article
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Warped product manifolds provide excellent setting to model space-time near black holes or bodies with large gravitational force. Recently, there have been studies to explore the existence, of warped products in certain settings [B. Y. Chen, Monatsh. Math. 133, No. 3, 177–195 (2001; Zbl 0996.53044); B. Sahin, Geom. Dedicata 117, 195–202 (2006; Zbl...
Article
Full-text available
We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structure F.
Article
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The geometry of slant submanifolds of a nearly trans-Sasakian manifold is studied when the tensor field Q is parallel. It is proved that Q is not parallel on the submanifold unless it is anti-invariant and thus the result of [CABRERIZO, J. L.—CARRIAZO, A.—FERNANDEZ, L. M.—FERNANDEZ, M.: Slant submanifolds in Sasakian manifolds, Glasg. Math. J. 42 (...
Article
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As warped product manifolds provide an excellent setting to model space time near black holes or bodies with large gravitational field, the study of these manifolds assumes significance in general. B. Y. Chen [4] initiated the study of CR-warped product submanifolds in a Kaehler manifold. He obtained a characterization for a CR-submanifold to be lo...
Article
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Warped product manifolds provide excellent setting to model space-time near black holes or bodies with large gravitational force (cf. [J. K. Beem, P. E. Ehrlich and K. Easley, Pure and Applied Mathematics. NY: Marcel Dekker (1996; Zbl 0846.53001), R. L. Bishop, Trans. Am. Math. Soc. 145, 1–49 (1969; Zbl 0191.52002) and S. Hiepko, Math. Ann. 241, 20...
Article
We give a characterization for a CR-submanifold of a Kähler manifold to be a CR-warped product submanifold in terms of the canonical structures P and F.
Article
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Warped product manifolds are studied from an extrinsic geometry point of view with respect to their applications in physics. The existence of warped product submanifolds in various ambient spaces is an important issue as these manifolds provide an excellent setting to model space-time near bodies with strong gravitational fields. In the present not...
Article
Cosymplectic manifolds provide a natural setting for time de-pendent mechanical systems as they are locally product of a Kaehler man-ifold and a one dimensional manifold. Thus study of warped product sub-manifolds of cosymplectic manifolds is significant. In this paper we have proved results on the non-existence of warped product submanifolds of ce...
Article
In the present note, it is proved that there donot exist warped product semi-slant submanifolds in a Sasakian manifold other than contact CR-warped product submanifolds and thus the results obtained in [8] are generalized.
Article
Recently, B.-Y. Chen [Monatsh. Math. 133, No. 3, 177–195 (2001; Zbl 0996.53044)] studied CR-submanifolds which are CR-submanifolds in Kähler manifolds. B. Sahin [Geom. Dedicata 117, 195–202 (2006; Zbl 1093.53059)] extended the above result for warped product semi-slant submanifolds N θ × f N T of a Kähler manifold. In the present paper we investiga...
Article
We study warped product Pseudo-slant submanifolds of Sasakian manifolds. We prove a theorem for the existence of warped product submanifolds of a Sasakian manifold in terms of the canonical structure F.
Article
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B.Y. Chen [4] showed that there exists no proper warped CR-submanifolds N ⊥ × f N T of a Kaehler manifold and obtained many results on CR-warped products N T × f N ⊥ . Contact CR-warped prod-uct submanifolds in Sasakian manifold were studied by I. Hasegawa and I. Mihai [6]. In this paper we have investigated the existence of contact CR-warped produ...
Article
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In the present note we have obtained some basic results pertaining to the geometry of slant and semi-slant submanifolds of a Kenmotsu manifold.
Article
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The aim in the present paper is to study some basic geometric properties of semi-slant submanifolds of a nearly Kaehler manifold.
Article
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Carriazo [2] defined the notion of bi-slant submanifolds of an almost Hermitian manifold. As a special case of these submanifolds, he introduced anti-slant submanifolds which he, later called as pseudo-slant submanifolds [2]. The purpose of the present paper is to study the pseudo-slant submanifolds of a Sasakian manifold. In this paper we work out...
Article
In this paper we study warped product CR-submanifolds in a nearly Kaehler manifold and extend the results of B.Y. Chen [7] concerning warped product CR-submanifolds in Kaehler manifolds to this more general setting.
Article
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We have studied some differential geometric aspects of semi-invariant submanifolds of a nearly trans-Sasakian manifold. That have led to the classification of totally umbilical semi-invariant submanifolds of a nearly trans-Sasakian manifold.
Article
The purpose of present note is to study the totally umbilical semi-invariant submanifolds of a nearly Kenmotsu manifold. In this note we have discussed the integrability of invariant and anti-invariant distributions and conse-quently obtained a classification for the totally umbilical semi-invariant submanifold of a nearly Kenmotsu manifold.
Article
The aim in the present paper is to study some basic geometric properties of semi-slant submanifolds of a nearly Kaehler manifold.
Article
The aim in the present paper is to study some basic geometric properties of semi-slant submanifolds of a nearly Kaehler manifold.
Article
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The purpose of the present paper is to study semi-slant submanifolds of a trans-Sasakian manifold.
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The purpose of the present paper is to study slant submanifolds in the setting of LP -contact manifolds. In particular, we have studied the slant submanifolds of an LP -Sasakian manifold via some curvature tensors.
Article
A classification theorem for the totally umbilical CR-submanifold of a nearly Khler manifold is proved.
Article
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In the present paper, a classification theorem for totally um- bilical semi-invariant submanifold is established. CR-submanifolds of a Sasakian space form are studied in detail, and finally a theorem for a CR- submanifold of a Sasakian manifold to be a proper contact CR-product is proved.
Article
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In the present paper we study totally umbilical CR-submanifolds of a Kaehler manifold. A classification theorem for a $D^\perp$-totally umbilical CR-submanifold is proved. The conditions under which a CR- submanifold becomes a CR-product are obtained, and finally a theorem for a CR-submanifold to be a proper CR-product is also established.
Article
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TOTALLY UMBILICAL SUBMANIFOLDS OF A COMPLEX SPACE FORM

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