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In this paper, we extend the notion of generalized Sasakian space form to the semi-Riemannian setting. We consider several interesting cases and we give examples of them all. We also study their structures.

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... In [12], Alegre and Carriazo described a generic indefinite Sasakian-space-form as an indefinite almost contact metric manifold, obeying a comparable equation with differentiable functions in place of constant values. As a result, we have ...

... Afterwards, Riemannian manifolds and Lorentzian manifolds form the most important subclass of pseudo-Riemannian manifolds, and they are especially crucial in general relativity and cosmology applications. In [12], authors were focused on Lorentzian case, along with certain examples of them. In this case: = −1 and the index of the metric is one. ...

... Putting V 5 = ζ in (24) then using (12) and (13), we lead to ...

In this note, the generalized Lorentzian Sasakian-space-form M12n+1(f1,f2,f3) satisfying certain constraints on the M-projective curvature tensor W* is considered. Here, we characterize the structure M12n+1(f1,f2,f3) when it is, respectively, M-projectively flat, M-projectively semisymmetric, M-projectively pseudosymmetric, and φ−M-projectively semisymmetric. Moreover, M12n+1(f1,f2,f3) satisfies the conditions W*(ζ,V1)·S=0, W*(ζ,V1)·R=0 and W*(ζ,V1)·W*=0 are also examined. Finally, illustrative examples are given for obtained results.

... 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. ...

... Then, using Equation (4), we arrive at the first part (17). The second part, (18), can be obtained through a similar process as the first part. ...

... Using terms (i) and (ii), which were not considered in (17), and (42), we can derive B(D, D)⊥ ψD ⊥ and B(D, D)⊥FD φ . It is obtained that B(D, D) = 0. ...

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.

... In [5], the authors defined the generalized indefinite Sasakian-space-form. It is the generalized Sasakian-spaceform with a semi-Riemannian metric. ...

... In [5], the author replaced the constants with three smooth functions defining the manifold. For an ε-almost contact metric manifold M, if the curvature tensor is given by ...

... We can use warped product to construct LGSSF (see [5]). Let h > 0 be a function on ℝ and (N 2n , J, and G) be an almost complex manifold. ...

In this paper, we investigate the Lorentzian generalized Sasakian-space-form. We give the necessary and sufficient conditions for the Lorentzian generalized Sasakian-space-form to be projectively flat, conformally flat, conharmonically flat, and Ricci semisymmetric and their relationship between each other. As the application of our theorems, we study the Ricci almost soliton on conformally flat Lorentzian generalized Sasakian-space-form.

... In di¤erential geometry, the Weyl conformal curvature tensor vanishes on a 3dimensional pseudo-Riemannian manifold and hence one can consider an another type of conformal invariant, which is the Cotton tensor. Cotton tensor C is a tensor of type (1,2), de…ned by ...

... where S is the Ricci tensor. Alegre et al. [1] have studied semi-Riemannian generalized Sasakian space-forms. The class of Ricci semisymmetric manifolds includes the set of Ricci symmetric manifolds (rS = 0) as a proper subset. ...

... The generalized Sasakian-space-form have been studied by many authors such as Sarkar and De ([17,10,11]), Singh ([18, 19]), De and Majhi ([8,9,12]), Kishor et al. [15], Alegre and Carriazo [3,4], Akbar and Sarkar [1], Sular and Ozgur [20,21] and many others. ...

The object of this paper is to study symmetric properties of Sasakian generalized Sasakian-space-form with respect to generalized Tanaka–Webster connection. We studied semisymmetry and Ricci semisymmetry of Sasakian generalized Sasakian-space-form with respect to generalized Tanaka–Webster connection. Further we obtain results for Ricci pseudosymmetric and Ricci-generalized pseudosymmetric Sasakian generalized Sasakian-space-form.

... The case of contact Lorentzian structures (η, g), where η is a contact 1-form and g a Lorentzian metric associated to it, has a particular relevance for physics and was considered in [12] and [4]. A systematic study of almost contact pseudo-metric manifolds was undertaken by Calvaruso and Perrone [7] in 2010, introducing all the technical apparatus which is needed for further investigations, and such manifolds have been extensively studied under several points of view in [1][2][3]6,9,10,[15][16][17]24], and references cited therein. ...

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields \(h:=\frac{1}{2}\pounds _\xi \varphi \) and \(\ell := R(\cdot ,\xi )\xi \), emphasizing analogies and differences with respect to the contact metric case. Certain identities involving \(\xi \)-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure \((\mathcal {H}(M), J, \theta )\) corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold \((M, \varphi ,\xi ,\eta ,g)\) is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure \((\mathcal {H}(M), J)\) is integrable and J is parallel along \(\xi \) with respect to the Bott partial connection.

... The case of contact Lorentzian structures (η, g), where η is a contact one-form and g a Lorentzian metric associated to it, has a particular relevance for physics and was considered in [19] and [4]. A systematic study of almost contact semi-Riemannian manifolds was undertaken by Calvaruso and Perrone [8] in 2010, introducing all the technical apparatus which is needed for further investigations, and such manifolds have been extensively studied under several points of view in [1,7,2,10,26,27,28,32,3,11,31], and references therein. ...

(Miskolc Mathematical Notes, Vol. 20, No. 2, pp. 1083-1099, (2019))
In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant $\varphi$-sectional curvature, and prove the structure theorem for $\xi$-conformally flat and $\varphi$-conformally flat Kenmotsu pseudo-metric manifolds. Next, we consider Ricci solitons on this manifolds. In particular, we prove that an $\eta$-Einstein Kenmotsu pseudo-metric manifold of dimension higher than 3 admitting a Ricci soliton is Einstein, and a Kenmotsu pseudo-metric 3-manifold admitting a Ricci soliton is of constant curvature $-\varepsilon$.

... The case of contact Lorentzian structures (η, g), where η is a contact 1-form and g a Lorentzian metric associated to it, has a particular relevance for physics and was considered in [12] and [4]. A systematic study of almost contact pseudo-metric manifolds was undertaken by Calvaruso and Perrone [7] in 2010, introducing all the technical apparatus which is needed for further investigations, and such manifolds have been extensively studied under several points of view in [1,6,2,9,14,15,16,22,3,10], and references cited therein. ...

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}\pounds _\xi \varphi$ and $\ell := R(\cdot,\xi)\xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $\xiξ-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost $CR$ structure $(\mathcal{H}(M), J, \theta)$ corresponding to almost contact pseudo-metric manifold $M$ to be $CR$ manifold. Finally, we prove that a contact pseudo-metric manifold $(M, \varphi, \xi, \eta, g)$ is Sasakian if and only if the corresponding nondegenerate almost $CR$ structure $(\mathcal{H}(M), J)$ is integrable and $J$ is parallel along $\xi$ with respect to
the Bott partial connection. Mathematics Subject Classification (2010). 53C15; 53C25; 53D10.

In this paper, the projective curvature tensor field of a generalized indefinite Sasakian space form is investigated. Some results dealing with projectively semi-symmetric and [Formula: see text]-projectively semi-symmetric generalized indefinite Sasakian space forms are obtained. Furthermore, biharmonic Legendre Frenet curves are discussed on these manifolds.

The purpose of this paper is to study the almost contact pseudo-metric manifolds of dimension three which are normal. We derive certain necessary and sufficient conditions for an almost contact pseudo-metric manifold to be normal. We prove that in a normal almost contact pseudo-metric 3-manifold M of constant curvature k the function \(\beta \) is harmonic, and if \(\beta =0\) on M, then the function \(\alpha \) is harmonic. Furthermore we give the necessary and sufficient condition for normal almost contact pseudo-metric 3-manifold to be \(\beta \)-Sasakian pseudo-metric manifold. Finally, we study the class of symmetric parallel (0, 2)-tensor field and its consequences.

In this paper we investigate (ε, δ)-trans-Sasakian manifolds which generalize the notion of (ε)-Sasakian and (ε)-Kenmotsu manifolds. We prove the existence of such a structure by an example and we consider φ -recurrent, pseudo-projectively flat and pseudo-projective semi-symmetric (ε, δ)-trans-Sasakian manifolds.

We obtain some basic results for Riemannian curvature tensor of (ε)-Sasakian manifoldsand then establish equivalent relations among φ-sectional curvature, totally real sectionalcurvature, and totally real bisectional curvature for (ε)-Sasakian manifolds

In the present paper we investigate CR-submanifolds of a nearly trans-hyperbolic Sasakian manifold. We also study parallel distribution relating to ξ-vertical CR-submanifold of a nearly trans-hyperbolic Sasakian manifold.

We introduce the concept of ()-almost paracontact manifolds,
and in particular, of ()-para-Sasakian manifolds. Several examples are presented. Some
typical identities for curvature tensor and Ricci tensor of ()-para Sasakian manifolds are
obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent
or proper Ricci-recurrent, then it cannot admit an ()-para Sasakian structure. We
show that, for an ()-para Sasakian manifold, the conditions of being symmetric, semi-symmetric,
or of constant sectional curvature are all identical. It is shown that a symmetric
spacelike (resp., timelike) ()-para Sasakian manifold is locally isometric to a pseudohyperbolic
space (resp., pseudosphere ). At last, it is proved that for an ()-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric,
and Einstein are all identical.

Bonome et al., 1997, provided an algebraic characterization for an indefinite Sasakian manifold to reduce to a space of constant ϕ -holomorphic sectional curvature. In this present paper, we generalize the same characterization for indefinite g · f · f -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.

We first prove some results on invariant lightlike
submanifolds of indefinite Sasakian manifolds. Then, we introduce a general notion of contact Cauchy-Riemann (CR) lightlike submanifolds and study the geometry of leaves of their distributions. We also study a class, namely, contact screen Cauchy-Riemann (SCR) lightlike submanifolds which include invariant and screen real subcases. Finally, we prove characterization theorems on the existence of contact SCR, screen real, invariant, and contact CR minimal lightlike submanifolds.

A new class of contact manifolds (carring a global non-vanishing timelike vector field) is introduced to establish a relation between spacetime manifolds and contact structures. We show that odd dimensional strongly causal (in particular, globally hyperbolic) spacetimes can carry a regular contact structure. As examples, we present a causal spacetime with a non regular contact structure and a physical model [GÃƒÂ¶del Universe] of Homogeneous contact manifold. Finally, we construct a model of 4-dimensional spacetime of general relativity as a contact CR-submanifold.

If the scalar normal curvature of a spacelike maximal surface in a 5-dimensional normal contact Lorentzian manifold with constant φ-sectional curvature is constant, then the surface is totally geodesic or nonpositively curved.

A complete decomposition of the space of curvature tensors over a Hermitian vector space into irreducible factors under the action of the unitary group is given. The dimensions of the factors, the projections, their norms and the quadratic invariants of a curvature tensor are determined. Several applications for almost Hermitian manifolds are given. Conformal invariants are considered and a general Bochner curvature tensor is introduced and shown to be a conformal invariant. Finally curvature tensors on four-dimensional manifolds are studied in detail.

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.

We introduce a systematic study of contact structures with pseudo-Riemannian associated metrics, emphasizing analogies and differences with respect to the Riemannian case. In particular, we classify contact pseudo-metric manifolds of constant sectional curvature, three-dimensional locally symmetric contact pseudo-metric manifolds and three-dimensional homogeneous contact Lorentzian manifolds.

In this article, we introduce ()-trans-Sasakian manifolds and give an example of such manifolds. Some basic results regarding ()-trans-Sasakian manifolds have been obtained in this context. Conformally flat and Weyl-semi symmetric ()-trans-Sasakian manifolds are also studied.

On sectional curvatures of

- R Kumar
- R Rani
- R K Nagaich