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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.

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... since M d is a warped product bi-slant submanifold. Moreover, it is easy to verify that M d is D-totally geodesic pointwise semi-slant warped product submanifold inM 2p+1 (f 1 , f 2 , f 3 ) using the second condition in (3.22), and (3.23), while the mixed totally geodesy derives from the first condition in (3.22), proving (a) in assertion (3). ...
... Since (a) of statement (3) implies that M d is D-totally geodesic and (b) of statement ...
... If (3.25) is true, M d is a D θ -totally geodesic warped product submanifold iñ M 2p+1 (f 1 , f 2 , f 3 ), based on the second condition in(3.24). This is the first situation in part (b) of the theorem's statement(3). ...
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We find Ricci curvature bounds for pointwise semi-slant warped products submanifolds in non-Sasakian generalized Sasakian space forms in this work, and analyze the equality case of the inequality. The derived inequality is also used to develop a number of applications.
... Later, D. E. Blair and A. Carriazo [3] established an analogue inequality for anti-invariant submanifolds in R 2m+1 with its standard Sasakian structure and characterized the equality case with a specific expression of the second fundamental form, similar to Equation (1). In a previous paper [4], we studied the corresponding inequality for slant submanifolds of generalized Sasakian space forms; ...
... Finally, a slant submanifold of an (α, β) trans-Sasakian generalized Sasakian space form [4], if its second fundamental form σ is given by the following expression: ...
... They are specially interesting because it was proven in [4] that this expression of the second fundamental form characterizes the equality case of the following inequality involving the squared mean curvature H 2 and the scalar curvature τ: ...
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The Maslov form is a closed form for a Lagrangian submanifold of C m , and it is a conformal form if and only if M satisfies the equality case of a natural inequality between the norm of the mean curvature and the scalar curvature, and it happens if and only if the second fundamental form satisfies a certain relation. In a previous paper we presented a natural inequality between the norm of the mean curvature and the scalar curvature of slant submanifolds of generalized Sasakian space forms, characterizing the equality case by certain expression of the second fundamental form. In this paper, first, we present an adapted form for slant submanifolds of a generalized Sasakian space form, similar to the Maslov form, that is always closed. And, in the equality case, we studied under which circumstances the given closed form is also conformal.
... The submanifolds ofM 2n+1 (f1, f2, f3) are * E-mail: skishormath@gmail.com 285 Submanifolds of Generalized Sasakian-space-forms with Respect to Certain Connections studied in [3,10,16]. In [3], Alegre and Carriazo studied submanifolds ofM 2n+1 (f1, f2, f3) with respect to Levi-Civita connection∇. ...
... 285 Submanifolds of Generalized Sasakian-space-forms with Respect to Certain Connections studied in [3,10,16]. In [3], Alegre and Carriazo studied submanifolds ofM 2n+1 (f1, f2, f3) with respect to Levi-Civita connection∇. The present paper deals with study of such submanifolds ofM 2n+1 (f1, f2, f3) with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection respectively. ...
... Note thatR (X, Y )X should be tangent if 3f2(p)g(Y, φX)φX is tangent. Since f2(p) = 0 for each p ∈ M , as similar as proof of Lemma 3.2 of [3], we may conclude that either M is invariant or anti-invariant. This proves the Lemma. ...
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The present paper deals with some results of submanifolds of generalized Sasakian-space-forms in \cite{ALEGRE3} with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-webster connection.
... It has been noted that the class of bi-warped product submanifolds is a generalization of several classes, such as CR-warped products, warped product semi-slant submanifolds, and warped product pseudo-slant submanifolds. On the other hand, as a generalization of nearly cosymplectic, nearly Sasakian [13], nearly Kenmotsu [8,14], nearly α-Sasakian, and nearly β-Kenmotsu manifolds, nearly trans-Sasakian manifolds have been studied on a large scale; see [15][16][17][18][19]. Therefore, our objective was to remove the gap in the nearly trans-Sasakian manifold literature, as they are an interesting structure of the almost contact manifolds that have generalized many others structures. ...
... Again, interchanging X 4 with TX 4 in (46),Lemma 3,and (16), we obtain 2g(B(X 2 , X 3 ), FTX4 ) = g(B(TX 4 , X 2 ), ψX 3 ) + g(B(TX 4 , X 3 ), ψX 2 ) − 2 cos 2 φ − η(X 4 ) + (X 4 ln f 2 ) g(X 2 , X 3 ). (54) Lemma 4. Let M = M φ × f 1 M T × f 2 M ⊥be a bi-warped product submanifold of a nearly trans-Sasakian manifold M. ...
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In the present work, we consider two types of bi-warped product submanifolds, M=MT×f1M⊥×f2Mϕ and M=Mϕ×f1MT×f2M⊥, in nearly trans-Sasakian manifolds and construct inequalities for the squared norm of the second fundamental form. The main results here are a generalization of several previous results. We also design some applications, in view of mathematical physics, and obtain relations between the second fundamental form and the Dirichlet energy. The relationship between the eigenvalues and the second fundamental form is also established.
... Recently, we have proved in [4] that, for a proper θ-slant submanifold, M m+1 , of an (α, β) trans-Sasakian generalized Sasakian space form M 2m+1 (f 1 , f 2 , f 3 ), the squared mean curvature H 2 and the scalar curvature τ verify at each point the following inequality: ...
... We call * -slant submanifold such a submanifold verifying the equality case, [4]. ...
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This book contains an up-to-date survey and self-contained chapters on complex slant submanifolds and geometry, authored by internationally renowned researchers. The book discusses a wide range of topics, including slant surfaces, slant submersions, nearly Kaehler, locally conformal Kaehler, and quaternion Kaehler manifolds. It provides several classification results of minimal slant surfaces, quasi-minimal slant surfaces, slant surfaces with parallel mean curvature vector, pseudo-umbilical slant surfaces, and biharmonic and quasi biharmonic slant surfaces in Lorentzian complex space forms. Furthermore, this book includes new results on slant submanifolds of para-Hermitian manifolds.
... Recently, we have proved in [4] that, for a proper θ-slant submanifold, M m+1 , of an (α, β) trans-Sasakian generalized Sasakian space form M 2m+1 (f 1 , f 2 , f 3 ), the squared mean curvature H 2 and the scalar curvature τ verify at each point the following inequality: ...
... We call * -slant submanifold such a submanifold satisfying the equality case, [4]. ...
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The book gathers a wide range of topics such as warped product semi-slant submanifolds, slant submersions, semi-slant ξ^⊥-, hemi-slant ξ^⊥-Riemannian submersions, quasi hemi-slant submanifolds, slant submanifolds of metric f-manifolds, slant lightlike submanifolds, geometric inequalities for slant submanifolds, 3-slant submanifolds, and semi-slant submanifolds of almost paracontact manifolds. The book also includes interesting results on slant curves and magnetic curves, where the latter represents trajectories moving on a Riemannian manifold under the action of magnetic field. It presents detailed information on the most recent advances in the area, making it of much value to scientists, educators and graduate students.
... 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. ...
... Now, we prove the following. (18) and the equality in Equation (18) holds at a point p ∈ N if and only if the shape operator takes similar forms as in Lemma 1 with respect to some suitable tangent and normal orthonormal bases. ...
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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.
... 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. ...
... Yildiz and Murathan [28] studied invariant submanifolds of Sasakian-space-forms. In [3], Alegre and Carriazo studied some submanifolds of generalized Sasakian-space-forms. Recently, Hui et. al. [16] studied parallel, semiparallel and 2-semiparallel invariant submanifolds of generalized Sasakian-space-forms. ...
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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.
... Aiming to extend and unify the notions of Sasakian space forms, Kenmotsu space forms, etc. (from almost contact metric geometry), Alegre et al. [1], introduced the concept of generalized Sasakian space forms. There are many results on their geometric properties, several examples and some studies of their submanifolds, [1,2,8]. ...
... Related to Proposition 1, we have the following: 2 , it follows that c 0 is constant and c 1 only depends on ξ . ...
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In the present paper, two geometric notions, namely K\"ahler manifolds of quasi-constant holomorphic sectional curvature and generalized Sasakian space forms, are related to each other, for the first time. Some conditions under which each of these structures induces the other one, are provided here. Several results are obtained on direct products (which are special cases of Naveira's classification), warped products or hypersurfaces of manifolds and relevant examples are included. A result of Niebergall and Ryan is generalized here. Some necessary and sufficient conditions for Einsteinian hypersurfaces are given at the end.
... In [3], Alegre and Carriazo studied almost semi-invariant submanifolds of generalized Sasakian-space-form with respect to Levi-Civita connection. In this paper, we have studied the results of [3] with respect to certain connections, namely semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection, Tanaka-Webster connection. ...
... In [3], Alegre and Carriazo studied almost semi-invariant submanifolds of generalized Sasakian-space-form with respect to Levi-Civita connection. In this paper, we have studied the results of [3] with respect to certain connections, namely semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection, Tanaka-Webster connection. ...
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The present paper deals with some results of almsot semi-invariant submanifolds of generalized Sasakian-space-forms in \cite{ALEGRE3} with respect to semisymmetric metric connection, semisymmetric non-metric connection, Schouten-van Kampen connection and Tanaka-Webster connection.
... Note that a generalized Sasakian-space-form can be considered a natural generalization of metric contact structures relevant to classical ones, known to be contained in the classical space forms. This generalization has attracted considerable attention from researchers worldwide, as evidenced by numerous contributions [2,3,4,5,12,14,15,16,18,20,22]. This is a more involved way of describing things in terms of curvature tensors. ...
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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.
... 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. ...
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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.
... In [18], Kim studied conformally flat generalized Sasakian-space-forms under the assumption that the characteristic vector field ξ is Killing and he classified locally symmetric generalized Sasakian-space-forms. Also he proved some geometric properties of generalized Sasakian-space-forms which depend on the nature of the functions f 1 , f 2 and f 3 . Generalized Sasakian-space-forms have also been studied in ([2]- [7], [11], [15], [16], [17], [19]- [24], [27], [28]) and many others. ...
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In this paper, we study Yamabe solitons on generalized Sasakian-space-forms. We prove that the scalar curvature of such man-ifolds is constant as well as harmonic and the flow vector field is Killing. Moreover, we prove that either £V φ is orthogonal to the structure vector field ξ or V is an infinitesimal automorphism of the contact metric structure of (M, φ, ξ, η, g). Finally, we give a valuable remark.
... Recently, many geometers such as [1,2,14,16] and many others have made an attempt to weakened the notion of generalized Sasakian space forms with different extent. ...
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This paper deals with the certain survey of generalized Sasakian space forms admitting pseudo projective curvature tensor. \linebreak Firstly, we have considered ϕ\phi-pseudo projectively flat and quasi-pseudo projectively flat generalized Sasakian space form. Later we characterize pseudo projective pseudo-symmetric generalized Sasakian space form. Finally, we have investigated generalized Sasakian space form endowed with P~P~=0\tilde{P}\cdot\tilde{P}=0.
... 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. ...
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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]. ...
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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]. ...
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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.
... Alegre et al. [2] introduced the notion of generalized Sasakian space forms and gave many examples of it. Throughout the years, many geometers [3,4,13,15,16,17] focused on generalized Sasakian space forms under different geometric conditions. Blair et al. [5] introduced the notion of (k, µ)-contact metric manifolds. ...
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In this paper, the geometric structures of generalized (k, µ)-space forms and their quasi-umbilical hypersurface are analyzed. First ξ-Q and conformally flat generalized (k, µ)-space form are investigated and shown that a conformally flat generalized (k, µ)-space form is Sasakian. Next, we prove that a generalized (k, µ)-space form satisfying Ricci pseudosymmetry and Q-Ricci pseudosymmetry conditions is η-Einstein. We obtain the condition under which a quasi-umbilical hypersurface of a generalized (k, µ)-space form is a generalized quasi Einstein hypersurface. Also ξ-sectional curvature of a quasi-umbilical hypersurface of generalized (k, µ)-space form is obtained. Finally, the results obtained are verified by constructing an example of 3-dimensional generalized (k, µ)-space form.
... 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]. ...
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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.
... In [1], Alegre, Blair and Carriazo defined the notion of a generalized Sasakian space form. In [2], Alegre and Carriazo studied submanifolds of generalized Sasakian space forms. For some recent study of generalized Sasakian space forms see [7], [8], [15], [23], [24], [25]. ...
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We study f-biharmonic integral submanifolds and integral C-parallel submanifolds in generalized Sasakian space forms. As an application, we find the f-biharmonicity conditions for the integral and integral C-parallel submanifolds in Sasakian, λ-Sasakian, Kenmotsu and cosymplectic space forms. Finally, we give also some examples of f-biharmonic integral submanifolds in Sasakian space forms.
... This idea was introduced by P. Alegre, D. Blair and A. Carriazo [13] in 2004. P. Alegre and Carriazo [15], A. Sarkar, S. K. Hui, etc. [19,21,22] studied generalized Sasakian-space-forms by considering the cosymplectic space of Kenmotsu space form as particular types of generalized Sasakian-spaceforms. In 2006, U. Kim [22] studied conformally flat generalized Sasakian-spaceform and locally symmetric generalized Sasakian-space-form. ...
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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.
... thenM is a Sasakian space form [5]. IfM is a Kenmotsu space form, a cosympletic space form and a λ-Sasakian space form respectively [14], [11]). Generalized Sasakian-space-forms have also been studied in ( [2], [8], [9]) and many others. ...
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We study anti-invariant minimal pseudo-parallel and Ricci generalized pseudoparallel submanifolds of generalized Sasakian space forms.
... Let M be an almost contact metric manifold of dimension (2n + 1). Then M is said to be a generalized Sasakian-space-form if there exists a non-vanishing Riemannian curvature tensor R on M of type (1,3) that satisfies the relation R(X, Y)Z = f 1 {g(Y, Z)X − g(X, Z)Y} + f 2 {g(X, ϕZ)ϕY − g(Y, ϕZ)ϕX + 2g(X, ϕY)ϕZ} ...
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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. ...
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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. ...
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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. ...
... The generalized Sasakian-space-forms have been studied by several authors such as Alegre and Carriazo ( Alegre and Carriazo, 2008, 2009), Belkhelfa, Deszcz andVerstraelen (2005), Carriazo (2005), C ˆ irnu, Ghefari, Solamy and Shahid (2006), Gherib, Gorine andBelkhelfa (2008), Sarkar (2011), Kim (2006), Narain, Yadav and Dwivedi (2011), Olteanu (2009, 2010), Shukla and Chaubey (2010), Yadav, Suthar and Srivastava (2011) and many others. ...
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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.
... 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. ...
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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. ...
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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.
... However, we can find generalized Sasakian-space-forms with non-constant functions and arbitrary dimensions. 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. ...
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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.
... Hence we can state the following: 2 1−2n or the manifold under consideration is a special type of η-Einstein manifold. Remark. ...
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The aim of the paper is to study *-Ricci tensor in generalized Sasakian space form. We study the generalized Sasakian space form admitting *-conformal η-Ricci soliton and analyse the behaviour of the soliton. Also, we prove *-Ricci semi-symmetric and Pseudo *-Ricci semisymmetric generalized Sasakian space forms are *-Ricci flat.
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The Volume 3 of International Journal of Mathematical Combinatorics
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This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {\Phi}-holomorphic sectional (G{\Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{\Phi}SH-curvature. In addition, the notion of {\Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {\eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.
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This paper determined the components of the generalized curvature tensor for the class of Kenmotsu type and established the mentioned class is {\eta}-Einstein manifold when the generalized curvature tensor is flat; the converse holds true under suitable conditions. It also introduced the notion of generalized {\Phi}-holomorphic sectional (G{\Phi}SH-) curvature tensor and thus found the necessary and sufficient conditions for the class of Kenmotsu type to be of constant G{\Phi}SH-curvature. In addition, the notion of {\Phi}-generalized semi-symmetric was introduced and its relationship with the class of Kenmotsu type and {\eta}-Einstein manifold established. Furthermore, this paper generalized the notion of the manifold of constant curvature and deduced its relationship with the aforementioned ideas. It finally showed that the class of Kenmotsu type exists as a hypersurface of the Hermitian manifold and derived a relation between the components of the Riemannian curvature tensors of the almost Hermitian manifold and its hypersurfaces.
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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.
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The Mathematical Combinatorics (International Book Series) is a fully refereed international book series with ISBN number on each issue, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 110-160 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences.
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  • D E Blair
D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Birkhäuser, Boston, 2002.