Abstract:
For a finite family of 3-dimensional almost contact metric manifolds with closed the structure form $\eta$ is described a construction of an almost contact metric manifold, where the members of the family are building blocks - cells. Obtained manifold share many properties of cells. One of the more important are nullity conditions. If cells satisfy nullity conditions - then - in the case of almost cosymplectic or almost $\alpha$-Kenmotsu manifolds - "sewed cells" also satisfies nullity condition - but generally with different constants. It is important that even in the case of the generalized nullity conditions - "sewed cells" are the manifolds which satisfy such conditions provided the cells satisfy the generalized nullity conditions.

Abstract:
We establish inequalities between the Ricci curvature and the squared mean curvature, and also between the k-Ricci curvature and the scalar curvature for a slant, semi-slant, and bi-slant submanifold in a locally conformal almost cosymplectic manifold with arbitrary codimension.

Abstract:
We obtain certain inequalities involving several intrinsic invariants namely scalar curvature, Ricci curvature and $k$-Ricci curvature, and main extrinsic invariant namely squared mean curvature for submanifolds in a locally conformal almost cosymplectic manifold with pointwise constant $% \phi $-sectional curvature. Applying these inequalities we obtain several inequalities for slant, invariant, anti-invariant and {\em CR}-submanifolds. The equality cases are also discussed.

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We study and obtain results on Ricci solitons in Kenmotsu manifolds satisfying , , , and , where and are C-Bochner and pseudo-projective curvature tensor. 1. Introduction A Ricci soliton is a natural generalization of an Einstein metric and is defined on a Riemannian manifold . A Ricci soliton is a triple with a Riemannian metric, a vector field, and a real scalar such that where is a Ricci tensor of and denotes the Lie derivative operator along the vector field . The Ricci soliton is said to be shrinking, steady, and expanding accordingly as is negative, zero, and positive, respectively [1]. In this paper, we prove conditions for Ricci solitons in Kenmotsu manifolds to be shrinking, steady, and expanding. In 1972, Kenmotsu [2] studied a class of contact Riemannian manifolds satisfying some special conditions and this manifold is known as Kenmotsu manifolds. Kenmotsu proved that a locally Kenmotsu manifold is a warped product of an interval and a Kaehler manifold with warping function , where is a nonzero constant. Kenmotsu proved that if in a Kenmotsu manifold the condition holds, then the manifold is of negative curvature , where is the curvature tensor of type and denotes the derivation of the tensor algebra at each point of the tangent space. The authors in [3–7] have studied Ricci solitons in contact and Lorentzian manifolds. The authors in [8] have obtained some results on Ricci solitons satisfying , , and and now we extend the work to , , , and . 2. Preliminaries An -dimensional differential manifold is said to be an almost contact metric manifold [9] if it admits an almost contact metric structure consisting of a tensor field of type , a vector field , a -form , and a Riemannian metric compatible with satisfying for all vector fields , on . An almost contact metric manifold is said to be Kenmotsu manifold [2] if From (3), we have where denotes the Riemannian connection of . In an -dimensional Kenmotsu manifold, we have where is the Riemannian curvature tensor. Let be a Ricci soliton in an -dimensional Kenmotsu manifold . From (4) we have From (1) and (9) we get The above equation yields that where is the Ricci tensor, is the Ricci operator, and is the scalar curvature on . 2.1. Example for 3-Dimensional Kenmotsu Manifolds We consider -dimensional manifold , where are the standard coordinates in . Let be linearly independent given by Let be the Riemannian metric defined by , , where is given by The structure is given by The linearity property of and yields that , , , for any vector fields on . By definition of Lie bracket, we have Let be the

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It is proved that if an almost Hermitian manifold of dimension greater than 4 has vanishing (classical) Bochner curvature tensor and is not Kaehlerian at a point, then it is flat in a neighbourhood of this point.

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We mainly investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. It is proved that a tangential half-lightlike submanifold of an indefinite Kenmotsu space form with semiparallel second fundamental form either satisfies or is -mixed geodesic. 1. Introduction As the intersection of normal bundle and tangent bundle of a submanifold of a semi-Riemannian manifold may not be trivial, it is more difficult and interesting to study the geometry of lightlike submanifolds than nondegenerate submanifolds. The two standard methods to deal with the above difficulties were developed by Kupeli [1], Duggal and Bejancu [2], Duggal and Jin [3], and Duggal and Sahin [4], respectively. Let be a lightlike submanifold immersed in a semi-Riemannian manifold, it is obvious to see that there are two cases of codimension 2 lightlike submanifolds, since for this type the dimension of their radical distributions is either 1 or 2. A codimension 2 lightlike submanifold of a semi-Riemannian manifold is called a half-lightlike submanifold [5] if , where denotes the degenerate radical distribution of . For more results about half-lightlike submanifolds, we refer the reader to [4, 6, 7]. In the theory of submanifolds of Riemannian manifolds, the parallel and semiparallel immersions were studied by Ferus [8] and Deprez [9], respectively. Recently, Massamba [10–13] and Upadhyay and Gupta [14] studied the parallel and semiparallel lightlike hypersurfaces of an indefinite Sasakian, Kenmotsu, and cosymplectic manifolds, respectively. However, the parallel and semiparallel half-lightlike submanifolds of an indefinite Kenmotsu manifolds have not yet been considered. The aim of this paper is to investigate the parallelism of half-lightlike submanifolds of indefinite Kenmotsu manifolds. This paper is organized in the following way. In Section 2, we provide some well-known basic formulas and properties of indefinite Kenmotsu manifolds and half-lightlike submanifolds. Section 3 is devoted to presenting some main results on semiparallel half-lightlike submanifolds of indefinite Kenmotsu space form. Finally, in Section 4, some properties of parallel half-lightlike submanifolds of indefinite Kenmotsu manifolds are investigated. 2. Preliminaries In this section, we follow Duggal and Sahin [4] for the notation and fundamental equations for half-lightlike submanifolds of indefinite Kenmotsu manifolds. A -dimensional semi-Riemannian is said to be an indefinite Kenmotsu manifold if it admits a normal almost contact metric structure , where is a tensor field of type

Abstract:
We study $\mathcal D$-homothetic deformations of almost $\alpha$-Kenmotsu structures. We characterize almost contact metric manifolds which are $CR$-integrable almost $\alpha$-Kenmotsu manifolds, through the existence of a canonical linear connection, invariant under $\mathcal D$-homothetic deformations. If the canonical connection associated to the structure $(\varphi,\xi,\eta,g)$ has parallel torsion and curvature, then the local geometry is completely determined by the dimension of the manifold and the spectrum of the operator $h'$ defined by $2\alpha h'=({\mathcal L}_\xi\varphi)\circ\varphi$. In particular, the manifold is locally equivalent to a Lie group endowed with a left invariant almost $\alpha$-Kenmotsu structure. In the case of almost $\alpha$-Kenmotsu $(\kappa,\mu)'$-spaces, this classification gives rise to a scalar invariant depending on the real numbers $\kappa$ and $\alpha$.

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The object of the present paper is to study some curvature conditions on -contact metric manifolds. 1. Introduction The notion of the odd dimensional manifolds with contact and almost contact structures was initiated by Boothby and Wong in 1958 rather from topological point of view. Sasaki and Hatakeyama reinvestigated them using tensor calculus in 1961. Tanno [1] classified the connected almost contact metric manifolds whose automorphism groups possess the maximum dimension. For such a manifold, the sectional curvature of plain sections containing is a constant, say . He showed that they can be divided into three classes: (i) homogeneous normal contact Riemannian manifolds with , (ii) global Riemannian products of line or a circle with a K？hler manifold of constant holomorphic sectional curvature if , and (iii) a warped product space if . It is known that the manifolds of class (i) are characterized by admitting a Sasakian structure. Kenmotsu [2] characterized the differential geometric properties of the manifolds of class (iii); so the structure obtained is now known as Kenmotsu structure. In general, these structures are not Sasakian [2]. On the other hand in Pokhariyal and Mishra [3] defined a tensor field on a Riemannian manifold as where and . Such a tensor field is known as m-projective curvature tensor. Later, Ojha [4] defined and studied the properties of m-projective curvature tensor in Sasakian and K hler manifolds. He also showed that it bridges the gap between the conformal curvature tensor, conharmonic curvature tensor, and concircular curvature tensor on one side and H-projective curvature tensor on the other. Recently m-projective curvature tensor has been studied by Chaubey and Ojha [5], Singh et al. [6], Singh [7], and many others. Motivated by the above studies, in the present paper, we study flatness and symmetry property of -contact metric manifolds regarding m-projective curvature tensor. The present paper is organized as follows. In this paper, we study the m-projective curvature tensor of -contact metric manifolds. In Section 2, some preliminary results are recalled. In Section 3, we study m-projectively semisymmetric -contact metric manifolds. Section 4 deals with m-projectively flat -contact metric manifolds. -m-projectively flat -contact metric manifolds are studied in Section 5 and obtained necessary and sufficient condition for an -contact metric manifold to be -m-projectively flat. In Section 6, m-projectively recurrent -contact metric manifolds are studied. Section 7 is devoted to the study of -contact metric manifolds

Abstract:
Main interest of the present paper is to investigate the almost {\alpha}-cosymplectic manifolds for which the characteristic vector field of the almost {\alpha}-cosymplectic structure satisfies a specific ({\kappa},{\mu},{\nu})-nullity condition. This condition is invariant under D-homothetic deformation of the almost cosymplectic ({\kappa},{\mu},{\nu})-spaces in all dimensions. Also, we prove that for dimensions greater than three, {\kappa},{\mu},{\nu} are not necessary constant smooth functions such that df^{\eta}=0. Then the existence of the three-dimensional case of almost cosymplectic ({\kappa},{\mu},{\nu})-spaces are studied. Finally, we construct an appropriate example of such manifolds.

Abstract:
In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient condition is given for an almost $s-$contact metric manifold to be a generalized Kenmotsu manifold.We show that a generalized Kenmotsu manifold is a locally warped product space. In addition, we study some curvature properties of generalized Kenmotsu manifolds. Moreover, we show that the $\varphi $% -sectional curvature of any semi-symmetric and projective semi-symmetric $% (2n+s)$-dimensional generalized Kenmotsu manifold is $-s$.