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The study of generalized Sasakian space forms is continued in this paper. The behavior of such spaces under generalized D-conformal deformations is analyzed. As a consequence, new examples of generalized Sasakian space forms are given.
Mathematics Subject Classification (2010)Primary 53C25–Secondary 53D15

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... In 1992, J. C. Marrero [7] proved that with certain deformation, we can get a trans-Sasakian structure starting from a Sasakian one. In [1], generalized Dconformal deformations are applied to trans-Sasakian manifolds where the covariant derivatives of the deformed metric is evaluated under the condition that the functions used in deformation depend only on the direction of the characteristic vector field of the trans-Sasakian structure. Other similar deformations are studied in [2,3,6]. ...

... This deformation appeared in [1]. In addition, if f = 1 then we have D-isometric deformation [3], but for h = f we get the deformation of Blair [6] and for h = f = a where a is a positive constant, we obtain D-homothetic deformation [12]. ...

... We denote the tensor field of type (1,2) ...

In this work, we have investigated a new deformation of almost contact metric manifolds. New relations between classes of 3-dimensional almost contact metric have been discovered. Several concrete examples are discussed.

... This deformation appeared in [1] and [13]. In addition, if f = 1 then we have D-isometric deformation [3], but for h = f we get the deformation of Blair [6] and for h = f = a where a is a positive constant we obtain D-homothetic deformation [17]. ...

... We denote byÑ (1) the tensor field of type (1,2) on M defined for all X and Y vector fields on M bỹ ...

... We denote byÑ (1) the tensor field of type (1,2) on M defined for all X and Y vector fields on M bỹ ...

It’s shown that for some changes of metrics and structural tensors, the product of two transSasakian manifolds is a Kählerian manifold. This gives new positive answer and more generally to Blair-Oubina’s open question. Concrete examples are given.

... This deformation appeared in [1] and [13]. In addition, if f = 1 then we have D-isometric deformation [3], but for h = f we get the deformation of Blair [6] and for h = f = a where a is a positive constant we obtain D-homothetic deformation [17]. ...

... We denote byÑ (1) the tensor field of type (1,2) on M defined for all X and Y vector fields on M bỹ ...

It’s shown that for some changes of metrics and structural tensors, the product of two transSasakian manifolds is a Kählerian manifold. This gives new positive answer and more generally to Blair-Oubina’s open question. (See [7] and [17]). Concrete examples are given.

... For example conformal deformations, D-homothetic deformations etc. These deformations were studied by several authors [1][2][3][4]. In [4], generalized D-conformal deformations are applied to trans-Sasakian manifolds where the covariant derivatives of the deformed metric is evaluated under the condition that the functions used in deformation depend only on the direction of the characteristic vector field of the trans-Sasakian structure. ...

... These deformations were studied by several authors [1][2][3][4]. In [4], generalized D-conformal deformations are applied to trans-Sasakian manifolds where the covariant derivatives of the deformed metric is evaluated under the condition that the functions used in deformation depend only on the direction of the characteristic vector field of the trans-Sasakian structure. In this study, in order to simplify tedious calculations, we obtain the new covariant derivatives of deformed almost contact metric structures seperately for the cases where the characteristic vector field is parallel, Killing and the one form dual to the characteristic vector field is closed. ...

... where a and b are positive functions on M, one can easily check that M,φ,ξ,η,g is an almost contact metric manifold too. This deformation is called a generalized D-conformal deformation [4]. After this deformation, the derivation of the new fundamental 2-formΦ is ...

In this work, we consider almost contact metric manifolds. We investigate the generalized D-conformal deformations of nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds. The new Levi–Civita covariant derivative of the new metric corresponding to deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Under some restrictions, deformed nearly K-cosymplectic, quasi-Sasakian and β -Kenmotsu manifolds are obtained. Then, the scalar curvature of these three classes of deformed manifolds are analyzed.

... By this type of a change, we get a new almost contact metric structure induced by the new vector cross product on the manifold. Thus we get a new deformation of the almost contact metric structure, different than deformations of the almost contact metric structures studied so far, such as conformal deformations [20,9] and D-homothetic deformations [19,1]. Our further aim is to understand this new deformation of the almost contact metric structures. ...

... This deformation is different than a conformal deformation [20,9] or a Dhomothetic deformation [19,1] of the a. c. m. s. ...

... For ω = kξ, however, we get a (generalized) D-homothetic deformation [1] (3) ...

It is known that manifolds with G2 structures have almost contact metric structures [3, 15]. In this manuscript, we deform a parallel G2 structure by a parallel vector field and investigate the properties of the almost contact metric structure obtained by the deformation of the G2 structure.

... Given an almost contact metric structure (M, g, φ, ξ, η), the generalized D-conformal deformation [1] on M is given by ...

... where a and b are two positive functions on M . Then it is easy to see that (M, g * , φ * , ξ * , η * ) is also an almost contact metric manifold [1]. We notice that, the transformation (5.1) reduces to 1. D-homothetic deformation [19] for a = b = constant and 2. conformal deformation [16] for a 2 = b. ...

The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal $\eta$-Ricci soliton. Here, we have investigated the nature of the conformal $\eta$-Ricci soliton within the framework of Kenmotsu manifolds. It is shown that an $\eta$-Einstein Kenmotsu manifold admitting conformal $\eta$-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal $\eta$-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal $\eta$-Ricci soliton on Kenmotu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal $\eta$-Ricci soliton on Kenmotsu manifold.

... We can also present examples of * -slant submanifolds of an α-Sasakian manifold using D-homothetic deformations. It was shown in [2] that given a Sasakian space form (M (c), φ, η, g), a D-homothetic deformation in the sense of Olszak ([20]) given by ...

... , m is an orthonormal frame of M in ( M, g * ). Moreover, in [2] it is obtained that ...

In this paper, we introduce the notion of ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-slant submanifold as that slant submanifold whose second fundamental form satisfies the equality case of an inequality between its mean curvature and its scalar curvature. In addition to that, we give several interesting examples about these submanifolds. Finally, we obtain the Ricci curvature for a ∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$*$$\end{document}-slant submanifold depending on the mean curvature vector and we give lower and upper bounds for the Ricci curvature.

... Their conformal deformations as well as their so-called D-homothetic deformations are also generalized Sasakian space forms (see [1]). Other examples can be found in [2]. Now, let (M, g) be a submanifold of an almost contact metric manifold ( M, g, ξ, η, φ). ...

... In [22], Marrero proved that every trans-Sasakian manifold of dimension greater or equal to 5 is either α-Sasakian, β-Kenmotsu or cosymplectic. Moreover, Alegre and Carriazo [2] Alegre and Carriazo also showed in [3] that if M (f 1 , f 2 , f 3 ) is a β-Sasakian generalized Sasakian space form, then the functions f 1 , f 2 and f 3 depend only on the direction of ξ. Examples are given in [1]. ...

We study f-biharmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient conditions for f-biharmonicity in the general case and many particular cases. Some geometric estimates as well as non-existence results are also obtained.

... D-CONFORMAL DEFORMATIONS Let (M, φ, ξ, η, g) be an almost contact metric manifold, where g is a Ricci soliton. The generalized D-conformal deformation [1] on M is given by ...

... where a and b are two positive functions on M . It is well known that (M, φ * , ξ * , η * , g * ) is also an almost contact metric manifold [1]. We note that the transformation (4) reduces to D-homothetic [15] or conformal according as a = b =constant or a 2 = b [13]. ...

In this paper we study Ricci solitons in generalized D-conformally deformed Kenmotsu manifold and we analyzed the nature of Ricci solitons when associated vector field is orthagonal to Reeb vector field.

... +í µí± 3 [í µí¼ í µí± í µí¼ í µí± í µí±-í µí¼ í µí± í µí¼ í µí± í µí± + í µí± í µí±, í µí± í µí¼ í µí± í µí¼ − í µí±(í µí±, í µí±)í µí¼(í µí±)í µí¼] (2.6) for any vector fields X,Y,Z on M, where R denotes the curvature tensor of M and í µí± 1 , í µí± 2 , í µí± 3 are smooth functions on the manifold. The Ricci operator Q, Ricci tensor S and the scalar curvature r of the manifold of dimension (2n+1) are respectively given by [15] QX= (2ní µí± 1 + 3í µí± 2 − í µí± 3 )X-(3í µí± 2 + (2í µí± − 1)í µí± 3 )í µí¼ í µí± í µí¼, (2.7) ...

... Now, interchanging X and Y in equation (3.9),we get í µí± í µí±, í µí±, í µí±, í µí± = í µí± í µí±, í µí±, í µí±, í µí± + í µí± í µí¼í µí±, í µí± í µí±(í µí¼í µí±, í µí±) − í µí± í µí¼í µí±, í µí± í µí±(í µí¼í µí±, í µí±) + í µí± 1 − í µí± 3 [ í µí¼ í µí± í µí± í µí±, í µí± − í µí¼ í µí± í µí± í µí±, í µí± í µí¼ í µí± ...

... In such case we will write the manifold as M (f 1 , f 2 , f 3 ). This kind of manifolds appears as a natural generalization of the Sasakian-space-forms by taking: f 1 = c+3 4 and f 2 = f 3 = c− 1 4 , where c denotes constant φ-sectional curvature. The φ-sectional curvature of generalized Sasakian-space-form M (f 1 , f 2 , f 3 ) is f 1 + 3f 2 . ...

... The φ-sectional curvature of generalized Sasakian-space-form M (f 1 , f 2 , f 3 ) is f 1 + 3f 2 . Moreover, cosymplectic space-form and Kenmotsu space-form are also considered as particular types of generalized Sasakian-space-form. Generalized Sasakian-space-forms have been studied in a number of papers from several points of view (for instance, [2]- [4], [6]- [8], [9]- [11], [13]- [17], etc). ...

The object of the paper is to characterize generalized Sasakian-space-forms satisfying certain curvature conditions on W 5-curvature tensor. We characterize W 5-flat, φ-W 5-flat and φ-W 5-semisymmetric generalized Sasakian-space-forms.

... where f 1 , f 2 , f 3 are differentiable functions on M and X , Y , Z are vector fields on M. In such a case, we will write the manifold as M( f 1 , f 2 , f 3 ). This kind of manifolds appears as a natural generalization of Sasakian space forms by taking: f 1 = c+3 4 and f 2 = f 3 = c− 1 4 , where c denotes a constant φ-sectional curvature. The φ-sectional curvature of a generalized Sasakian space form M( f 1 ...

... Example 5.5. [4] Let N(c) is a complex space form, and by the warped product M = (− π 2 , π 2 ) × f N endowed with the almost contact metric structure (φ, ξ, η, g f ) is a generalized Sasakian space form with functions ...

Generalized Sasakian space forms have become today a rather specialized subject, but many contemporary works are concerned with the study of their properties and of their related curvature tensors. The goal of this paper is to study the E-Bochner curvature tensor on generalized Sasakian space forms, and to characterize the situations when it is, respectively: E-Bochner symmetric (∇Be=0); E-Bochner semisymmetric (R(dot operator)Be=0); E-Bochner recurrent; E-Bochner pseudosymmetric; such that Be(ξ,X)(dot operator)S=0; such that Be(ξ,X)(dot operator)R=0.

... That [35]. The properties of M (f 1 , f 2 , f 3 ) in different contexts have been studied by several geometers but few are ( [2][3][4], [12], [13], [22], [27], [44]). ...

Our aim is to study the properties of Fischer-Marsden conjecture and Ricci-Bourguignon solitons within the framework of generalized Sasakian-space-forms with β-Kenmotsu structure. It is proven that a (2n + 1)-dimensional generalized Sasakian-space-form with β-Kenmotsu structure satisfying the Fischer-Marsden equation is a conformal gradient soliton. Also, it is shown that a generalized Sasakian-space-form with β-Kenmotsu structure admitting a gradient Ricci-Bourguignon soliton is either Ψ\T k × M 2n+1−k or gradient η-Yamabe soliton.

... where f 1 , f 2 , f 3 are differentiable functions and X, Y, Z for vector fields on M 2n+1 ( f 1 , f 2 , f 3 ). The Sasakian manifold with constant φ-sectional curvature is a Sasakian-space-form, and cosymplectic and Kenmotsu space-forms are also considered particular types of generalized Sasakian-space-forms. Additionally, the generalized Sasakian-space-forms have been investigated in [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21] and many others. In Riemannian geometry, numerous researchers have studied curvature properties and how much they affected the manifold itself. ...

The aim of the present paper is to study and investigate the geometrical properties of a concircular curvature tensor on generalized Sasakian-space-forms. In this manner, we obtained results for ϕ-concircularly flat, ϕ-semisymmetric, locally concircularly symmetric and locally concircularly ϕ-symmetric generalized Sasakian-space-forms. Finally, we construct examples of the generalized Sasakian-space-forms to verify some results.

... A submanifold of a contact manifold is said to be totally geodesic if every geodesic in that submanifold is also geodesic in the ambient manifold. The generalised Sasakian space forms (G.S.S.F.) have been investigated by numerous researchers like Alegre and Carriazo [1], [2], [3]. Thereafter, (G.S.S.F.) have been study by many authors [4], [9], [10], [14], [16], [19]. ...

In this paper, we obtain necessary and sufficient condition for an invariant submanifold of generalised sasakian space form with semi-symmetric metric connections to be totally geodesic.

... In differential geometry, The theory of Invariant submanifold has been alluring field of research for a long time. The generalized Sasakian space forms (G.S.S.F.) have been investigated by numerous researchers like Alegre and Carriazo [1,2,3]. Thereafter generalized Sasakian spaceform have been study by many authors [4,10,15]. ...

In this paper, we obtain necessary and sufficient condition for an Invariant submanifold of generalized Sasakian space form with semi-symmetric metric connections to be totally geodesic.

... In ( [3]), generalized D-conformal deformations are applied to trans-Sasakian manifolds where the covariant derivatives of the deformed metric is investigated under the condition that the functions used in deformation depend only on the direction of the characteristic vector field of the trans-Sasakian structure. ...

The present paper attempts to study the almost Ricci solitons in a generalized D-conformally deformed (LCS)n-manifold considering potential vector field as a solenoidal vector field, a gradient vector field or the Reeb vector field of the deformed structure and explicitly investigate the Ricci and scalar curvatures for some cases. We also determine some inequalities for the Ricci curvature of the deformed (LCS)n-manifold when it admits a gradient almost Ricci soliton.

... For example, it is known that [26] any 3-dimensional (α, β)-trans Sasakian manifolds with α, β depending on ξ are generalized Sasakian space-forms. Such type of manifolds have been studied by several authors, likes Alegre and Carriazo ([2], [3], [4]) Belkhelfa et al. [7], Carriazo [13], Al-Ghefari et al. [5], Gherib et al. [19], Kim [21] and many others. It is noted that a (2n+1)-dimensional (n > 1) generalized Sasakian space-formM 2n+1 (f 1 , f 2 , f 3 ) is conformally flat if and only if f 2 = 0 [21]. ...

... For example, it is known that [26] any 3-dimensional (α, β)-trans Sasakian manifolds with α, β depending on ξ are generalized Sasakian space-forms. Such type of manifolds have been studied by several authors, likes Alegre and Carriazo ([2], [3], [4]) Belkhelfa et al. [7], Carriazo [13], Al-Ghefari et al. [5], Gherib et al. [19], Kim [21] and many others. It is noted that a (2n+1)-dimensional (n > 1) generalized Sasakian space-formM 2n+1 (f 1 , f 2 , f 3 ) is conformally flat if and only if f 2 = 0 [21]. ...

The object of the present paper is to study certain geometrical properties of the submanifolds of generalized Sasakian
space-forms. We deduce some results related to the invariant and
anti-invariant slant submanifolds of the generalized Sasakian space-forms. Finally, we study the properties of the sectional curvature, totally geodesic and umbilical submanifolds of the generalized Sasakian space-forms. To prove the existence of almost semiinvariant and anti-invariant submanifolds, we provide the non-trivial
examples.

... Notice that several examples of non-trivial generalized Sasakian space-forms are given in [16] using different geometric constructions, such as Riemannian submersions, warped products, and D-conformal deformations. Afterwards, many interesting results have been proved in these ambient spaces (see, e.g., [17][18][19][20][21][22][23][24][25][26][27]). We only recall that, very recently, Bejan and Güler [28] obtained an unexpected link between the class of generalized Sasakian space-forms and the class of Kähler manifolds of quasi-constant holomorphic sectional curvature, providing conditions under which each of these structures induces the other one. ...

In this work, we first derive a generalized Wintgen type inequality for a Lagrangian submanifold in a generalized complex space form. Further, we extend this inequality to the case of bi-slant submanifolds in generalized complex and generalized Sasakian space forms and derive some applications in various slant cases. Finally, we obtain obstructions to the existence of non-flat generalized complex space forms and non-flat generalized Sasakian space forms in terms of dimension of the vector space of solutions to the first fundamental equation on such spaces.

... If we take λ1 = − 3λ Example 4.2. [14] Let N (c) be a complex space form, and by the warped product M = (− π 2 , π 2 ) × f N endowed with the almost contact metric structure (φ, ξ, η, g f ), Sasakian space form M (f1, f2, f3) is generalized with functions ...

The present paper deals the study of generalised Sasakian-space-forms with the conditions Cq(ξ,X).S = 0, Cq(ξ,X).R = 0 and Cq(ξ,X).Cq = 0, where R, S and Cq denote Riemannian curvature tensor, Ricci tensor and quasi-conformal curvature tensor of the space-form, respectively and at last, we have given some examples to improve our results.

... where σ is positive function depending only on the direction of ξ. By virtue of Lemma 2.2 of ( [2]) it is easy to see that the resulting manifoldM(ϕ, ξ, η,ḡ) isβ-Kenmotsu, whereβ = β + ξσ 2σ . Next, we chooseβ in such a way that the manifoldM under consideration will be a Kenmotsu. ...

First, we prove that if the Reeb vector field ξ of a Kenmotsu manifold M leaves the Ricci operator
Q invariant, then M is Einstein. Next, we study Kenmotsu manifold whose metric represents a Ricci
soliton and prove that it is expanding. Moreover, the soliton is trivial (Einstein) if either (i) V is a
contact vector field, or (ii) the Reeb vector field ξ leaves the scalar curvature invariant. Finally, it is
shown that if the metric of a Kenmotsu manifold represents a gradient Ricci almost soliton, then it
is η-Einstein and the soliton is expanding. We also exhibited some examples of Kenmotsu manifold
that admit Ricci almost solitons.

... where, σ is a positive function which depends only on ξ = ∂ ∂t . Using Lemma 4.1 in the paper [2], first we derive a β-Kenmotsu manifold (M * , φ, ξ, η, g * ) with ...

In this paper the notion of Ricci $\rho$-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3- dimensional almost Kenmotsu Einstein manifold $M$ be a $\rho$-soliton, then $M$ is a Kenmotsu manifold of constant sectional curvature $-1$ and the $\rho$-soliton is expanding, with $\lambda =2$.

... In particular, if f 1 = c+3 4 , f 2 = f 3 = c−3 4 , then the generalized Sasakianspace-form reduces to the notion of Sasakian-space-form. The generalized Sasakian-spaceforms have also been studied in [2,3,4,12,13,14,20,21,22,23,25] and many other instances. ...

The object of this paper is to study the invariant submanifolds of generalized Sasakian-space-forms. Here, we obtain some equivalent conditions for an invariant submanifold of a generalized Sasakian-space-forms to be totally geodesic.

... for arbitrary vector fields X, Y and Z and smooth functions f 1 , f 2 and f 3 on M. We denote such class of manifolds by M 2n+1 (f 1 , f 2 , f 3 ). In particular, if we choose f 1 = c+3 4 and f 2 = f 3 = c−1 4 , where c is the constant ϕ-sectional curvature of M, then the generalized Sasakian-space-forms convert into Sasakian-space-forms. Indeed, Sasakian-space-forms seem to be natural examples of the generalized Sasakian-space-forms. ...

We set a definition of a {(0,2)} -type tensor on the generalized Sasakian-space-forms. The necessary and sufficient conditions for W -semisymmetric generalized Sasakian-space forms are studied. Certain results of the Ricci solitons, the Killing vector fields and the closed 1-form on the generalized Sasakian-space-forms are derived. We also verify our results by taking non-trivial examples of the generalized Sasakian-space-forms.

... The properties of generalized Sasakian space form was studied by many geometers such as [2,3,4,5,6,7,8,9,10,11]. The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. ...

In this paper, we studied generalized Sasakian space forms admitting Sasakian structure with respect to the quarter symmetric metric connection and the locally ϕ-symmetric, η-recurrent, ϕ-recurrent and flatness of projective curvature tensor on generalized Sasakian space forms. We establish the relation between the Riemannian connection and the quarter symmetric metric connection.

... The properties of generalized Sasakian space form was studied by many geometers such as [2,3,4,5,6,7,8,9,10,11]. The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. ...

In this paper, we studied generalized Sasakian space forms admitting Sasakian structure with respect to the quarter symmetric metric connection and the locally ϕ-symmetric, η-recurrent, ϕ-recurrent and flatness of projective curvature tensor on generalized Sasakian space forms. We establish the relation between the Riemannian connection and the quarter symmetric metric connection.

... The properties of generalized Sasakian space form was studied by many geometers such as [2,3,4,5,6,7,8,9,10,11]. The notion of local symmetry of a Riemannian manifold has been weakened by many authors in several ways to a different extent. ...

... where c denotes constant φ−sectional curvature. The generalized Sasakian-space-forms have been extensively studied by [2,3,14,15,16,21] and many others. ...

The main purpose of the present paper is to introduce the notion of
generalized φ−recurrency of generalized Sasakian-space-forms. We studied
generalized φ−recurrent generalized Sasakian-space-forms, generalized
concircular φ−recurrent generalized Sasakian-space-forms and obtained
a number of results. We also proved generalized Sasakian-spaceforms
satisfying the condition S(X, ξ).R = 0 is reduced to η−Einstein.

... Moreover, cosymplectic space-form and Kenmotsu space-form are also considered as particular types of generalized Sasakian-space-form. The generalized Sasakianspace-forms have also been studied in ( [2][3][4][5][6][7][8][9][10][11][12][13][14]) and many others. ...

The object of this present paper is to study generalized ϕ-recurrent generalized Sasakian-space-forms and its various geometric properties. Among the results established here, it is shown that a generalized ϕ-recurrent generalized Sasakian-space-form is an Einstein manifold. Further, we study generalized concircular ϕ-recurrent generalized Sasakian-space-forms. © 2017 by the Mathematical Association of Thailand. All rights reserved.

... The generalized Sasakian-space-forms have been studied by several authors such as Alegre and Carriazo ([2], [3], [4]), Belkhelfa et al. [9], Carriazo [11], Al-Ghefari et al. [5], Gherib et al. [20], Hui et al. ([30], [30]), Kim [33] and many others. ...

The present paper deals with the study of invariant submanifolds of generalized Sasakian-space-forms with respect to Levi-Civita connection as well as semi-symmetric metric connection. We provide some examples of such submanifolds and obtain many new results including, the necessary and sufficient conditions under which the submanifolds are totally geodesic. The Ricci solitons of such submanifolds are also studied.

... Their conformal deformations as well as their so-called D-homothetic deformations are also generalized Sasakian space forms (see [1]). Other examples can be found in [2]. Now, let (M, g) be a submanifold of an almost contact metric manifold ( M , g, ξ, η, φ). ...

We study f-biharmonic and bi-f-harmonic submanifolds in both generalized complex and Sasakian space forms. We prove necessary and sufficient condition for f-biharmonicity and bi-f-harmonicity in the general case and many particular cases. Some non-existence results are also obtained.

... The generalized Sasakian-space-forms have been studied by several authors such as Alegre and Carriazo ([2], [3], [4]), Belkhelfa, Deszcz and Verstraelen [9], Carriazo [12], Ghefari, Al-Solamy and Shahid [16], Gherib et. al ( [17], [18]), Hui et. ...

The present paper deals with the study of generalized Sasakian-space-forms whose
metric is Ricci soliton with potential vector field is conformal killing and obtain the conditions of such
type of Ricci solitons to be expanding, steady and shrinking respectively.

... The generalized Sasakian-space-forms have been studied by several authors such as Alegre and Carriazo ([2], [3], [4]), Belkhelfa, Deszcz and Verstraelen [5], Carriazo [7], Cîrnu [8], De and Sarkar ([10], [11]), Ghefari, Solamy and Shahid [12], Gherib, Gorine and Belkhelfa [13], Kim [14], Narain, Yadav and Dwivedi [16], Olteanu ([17], [18]), Shukla and Chaubey [27], Sular andÖzgür [28], Yadav, Suthar and Srivastava [32] and many others. In [14] Kim studied conformally flat generalized Sasakian-space-forms. Also in [10] De and Sarkar studied the Weyl projective curvature tensor of generalized Sasakian-space-forms. ...

The aim of this article is to study the h-almost Ricci solitons and h-almost gradient Ricci solitons on generalized Sasakian-space-forms. First, we consider h-almost Ricci soliton with the potential vector field V as a contact vector field on generalized * Corresponding Author. Sasakian-space-form of dimension greater than three. Next, we study h-almost gradient Ricci solitons on a three-dimensional quasi-Sasakian generalized Sasakian-space-form. In both the cases, several interesting results are obtained.

In this paper we introduced the new notion called block-line forest signed graph
of a signed graph and its properties are studied. Also, we obtained the structural characterization
of this new notion and presented some switching equivalent characterizations.

في هذا الكتاب سنتناول البنية المترية التلامسية تقريبا و البنى التلامسية على المنوّعات الريمانية ذات البُعد الفردي، ندرس أهم الخواص مع براهين مفصّلة و أمثلة توضيحية تمهيدا للفصل الأساسي الموالي و الذي يحوي دراسة مفصّلة عن البُنى المترية التلامسية تقريباً ثلاثية الأبعاد مع استعراض لأصنافها الخمسة حسب تصنيف تشينيا-غونزليز. نشير أنه من بين الأصناف الخمسة، يوجد صنفين غير ناظميين لم يحظيا باهتمام من قبل الباحثين في هذا الميدان. هنا، أدرجنا دراسة مستفيظة عن هاذين الصنفين مدعمين بأبحاث أصلية نُشرت لنا مؤخرا في مجلات دولية محكّمة ثم أدرجنا تعميما يشمل أربعة أصناف من البُنى المترية التلامسية تقريبا مع دراسة مفصّلة و أمثلة ملموسة. في الأخير، قدّمنا مجموعة من التحويلات القديمة و الحديثة بما في ذلك الأصلية للتنقل بين أصناف البُنى المترية التلامسية تقريبا ثلاثية الأبعاد على نفس المنوّعة الريمانية معتمدين على التعميم المُدرج في الباب السابق مع تعزيز هذه الدراسة بأمثلة مفصّلة

In this paper, we obtain a generalized Wintgen-type inequality for submanifolds in generalized (κ,µ)-space forms. Further, we discuss the inequality for bi-slant submanifolds, semi-slant submanifolds, hemi-slant submanifolds, CR-submanifolds and slant submanifolds in various ambient spaces. In particular, we generalize some recent results of Mihai, Tohoku Math. J. 69 (2017), and Aquib et al., Tamkang J. Math. 50 (2019) and Mathematics 7(12) (2019).

The purpose of the present paper is to study the notion of generalized recurrent and generalized Ricci recurrent generalized Sasakian space forms and to study various properties related with the existence of such notion.

In the present frame work, we studied the semi generalized recurrent, semi generalized ϕ -recurrent, extended generalized ϕ -recurrent and concircularly locally ϕ -symmetric on generalized Sasakian space forms.

The object of present paper is to study (epsilon, delta)-trans-Sasakian manifold with eta
parallel Ricci tensor admitting a non null concircular vector �field

In this paper, we study Pseudo-projective curvature tensor in K-contact mani-
fold.

The object of the present paper is to study η-recurrent *-Ricci tensor, *-Ricci semisymmetric and globally ϕ-*-Ricci symmetric contact metric generalized (κ, µ)-space form. Besides these, *-Ricci soli-ton and gradient *-Ricci soliton in contact metric generalized (κ, µ)-space form have been studied.

The object of the present paper is to study the properties of generalized Sasakian-space-forms. We prove the results related to Ricci symmetric, Ricci recurrent, cyclic parallel and Codazzi type Ricci tensors. Results on Ricci soliton and gradient Ricci soliton are proved. Also, we provide the examples of generalized Sasakian-space-forms which are verified our results.

In this paper, we study W2-pseudosymmetric, W2-locally symmetric, W2-locally φ-symmetric and W2-φ-recurrent generalized Sasakian space form. Further, illustrative examples are given.

In this work, the complete (k, 2)- arcs with 6≤k≤10 in the left near field projective plane of order 9 were determined and classified by using a computer program.

The object of this paper is to study of pseudo projective curvature tensor and C-Bochner curvature tensor on generalized Sasakian-space-forms. We show that pseudo projective generalized Sasakian-space-forms are Einstein manifolds. Additionally, generalized Sasakian-space-forms under some conditions related to pseudo projective curvature tensor are studied. In next, we consider the conditions \(B(X,Y)Z=0\) and \(R.B=0\) on a generalized Sasakian-space-form, where B is the C-Bochner curvature tensor.

The present paper deals with a study of contact CR-warped product submanifolds of generalized Sasakian-space-forms and contact CR-warped product semi-slant submanifolds of generalized Sasakian-space-forms. It is shown that there exists no proper contact CR-warped product submanifolds of generalized Sasakian-space-forms. However, we obtain some results for the existence or non-existence of contact CR-warped product semi-slant submanifolds of generalized Sasakian-space-forms.

The object of the present paper is to study Ricci semisymmetric, locally ϕ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi $$\end{document}-symmetric and η\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\eta $$\end{document}-parallel Ricci tensor in a three dimensional generalized Sasakian-space-form. The three dimensional quasi-Sasakian generalized Sasakian-space-forms have been studied. Besides these, Ricci solitons and gradient Ricci solitons have been studied. Finally, illustrative examples are given.

In this paper, we introduce two new classes of almost contact structures, called trans-Sasakian and almost trans-Sasakian structures, which are obtained from certain classes of almost Hermitian manifolds closely related to locally conformal Kahler or almost Kahler manifolds, respectively. In particular, although transSasakian structures are normal almost contact metric structures containing both cosymplectic and Sasakian structures, they are different from quasi-Sasakian structures, as it is shown constructing explicit examples, and in fact no inclusion relation between these classes exists.

The purpose of the present paper is to characterize pseudoprojectively flat and pseudoprojective semisymmetric generalized Sasakian-space-forms.

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 a 3-dimensional trans-Sasakian manifold, explicit formulae for Ricci operator, Ricci tensor and curvature tensor are obtained. In particular, expressions for Ricci tensor are obtained in a 3-dimensional trans-Sasakian manifold in cases of the manifold being η-Einstein or satisfying R (X, Y) · S = 0.

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.

In this paper, we completely characterize the local structure of trans-Sasakian manifolds of dimension 5 by giving suitable examples.

The topology of contact Riemannian manifolds. Ill Spain e-mail: carriazo@us.es Received

- S Tanno

Tanno, S.: The topology of contact Riemannian manifolds. Ill. J. Math. 12, 700– 717 (1968) Spain e-mail: carriazo@us.es Received: November 9, 2010.