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In this paper, we consider the tangent bundle TM of a Riemannian manifold (M, g) with the Sasaki metric G and using the Cauchy-Kowalevski Theorem, we answer the question of how many analytic statistical structures are there on (TM,G). Also, we study the Ricci tensor of linear affine connections on the tangent bundle TM. In addition, we answer the question of how many Ricci flat affine connections with and without torsion are there on the tangent bundle. [ABSTRACT FROM AUTHOR]

An almost Yamabe soliton is a generalization of the Yamabe soliton. In this article, we deduce some results regarding almost gradient Yamabe solitons. More specifically, we show that a compact almost gradient Yamabe soliton having non-negative Ricci curvature is trivial. Again, we prove that an almost gradient Yamabe soliton with a non-negative potential function and scalar curvature bound admitting an integral condition is trivial. Moreover, we give a characterization of a compact almost gradient Yamabe solitons. [ABSTRACT FROM AUTHOR]

The object of the present paper is to investigate biharmonic and sub-biharmonic functions on Riemannian manifolds. [ABSTRACT FROM AUTHOR]

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric ρ-connection if its Riemannian metrics are Ricci and gradient Ricci solitons, respectively. It is proved that if a three-dimensional Riemannian manifold equipped with a semi-symmetric metric ρ-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature −1 and the soliton is expanding with λ = −2. Next, we study the gradient Ricci solitons in such a manifold. Finally, we construct a non-trivial example of a three-dimensional Riemannian manifold endowed with a semi-symmetric metric ρ-connection admitting a Ricci soliton and validate our some results. [ABSTRACT FROM AUTHOR]

In this paper, we study almost coKähler manifolds admitting k -almost Yamabe solitons (g , V , k , λ). First, we obtain a classification of almost coKähler (κ , μ) -manifolds admitting nontrivial closed k -almost Yamabe solitons. Next, we consider an almost α -coKähler manifold admitting a nontrivial k -almost Yamabe soliton and prove that it is locally the Riemannian product of an almost Kähler manifold with the real line if the potential vector field V is collinear with the Reeb vector field. For the potential vector field V being orthogonal to the Reeb vector field, we also obtain two results. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to study Ricci semisymmetric paracontact metric manifolds satisfying ∇ ξ ⁡ h = 0 {\nabla_{\xi}h=0} and such that the sectional curvature of the plane section containing ξ equals a non-zero constant c. Also, we study paracontact metric manifolds satisfying the curvature condition Q ⋅ R = 0 {Q\cdot R=0} , where Q and R are the Ricci operator and the Riemannian curvature tensor, respectively, and second order symmetric parallel tensors in paracontact metric manifolds under the same conditions. Several consequences of these results are discussed. [ABSTRACT FROM AUTHOR]

We study certain curvature properties of Kenmotsu manifolds with respect to the quarter-symmetric metric connection. First we consider Ricci semisymmetric Kenmotsu manifolds with respect to a quarter-symmetric metric connection. Next, we study ξ-conformally flat and ξ-concircularly flat Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Moreover, we study Kenmotsu manifolds satisfying the condition Z(ξ,Y),S = 0, where Z and S are the concircular curvature tensor and Ricci tensor respectively with respect to the quarter-symmetric metric connection. Then, we prove the non-existence of ξ-projectively flat and pseudo-Ricci symmetric Kenmotsu manifolds with respect to the quarter-symmetric metric connection. Finally, we construct an example of a 5-dimensional Kenmotsu manifold admitting a quarter-symmetric metric connection for illustration. [ABSTRACT FROM AUTHOR]

In this paper, we introduce a new tensor named -tensor which generalizes the -tensor introduced by Mantica and Suh [Pseudo symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. Then, we study pseudo--symmetric manifolds which generalize some known structures on pseudo-Riemannian manifolds. We provide several interesting results which generalize the results of Mantica and Suh [Pseudo symmetric Riemannian manifolds with harmonic curvature tensors, Int. J. Geom. Methods Mod. Phys. 9(1) (2012) 1250004]. At first, we prove the existence of a . Next, we prove that a pseudo-Riemannian manifold is -semisymmetric if and only if it is Ricci-semisymmetric. After this, we obtain a sufficient condition for a to be pseudo-Ricci symmetric in the sense of Deszcz. Also, we obtain the explicit form of the Ricci tensor in a if the -tensor is of Codazzi type. Finally, we consider conformally flat pseudo--symmetric manifolds and prove that a spacetime is a -wave under certain conditions. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study N(k)-quasi-Einstein manifolds. study We an N(k)-quasi-Einstein manifold satisfying the curvature conditions R(ξ,X)· Z = 0, Z(X, ·) · R = 0, and P(·,X) · Z = 0, where R, P and Z denote the Riemannian curvature tensor, the projective curvature tensor and Z tensor respectively. Next we prove that the curvature condition C · Z = 0 holds in an N(k)-quasi-Einstein manifold, where C is the conformal curvature tensor. We also study Z-recurrent N(k)-quasi-Einstein manifolds. Finally, we construct an example of an N(k)-quasi-Einstein manifold and mention some physical examples. [ABSTRACT FROM AUTHOR]

We study a type of semisymmetric metric connection on a Riemannian manifold whose torsion tensor is almost pseudo symmetric and the associated 1-form of almost pseudo symmetric manifold is equal to the associated 1-form of the semisymmetric metric connection. [ABSTRACT FROM AUTHOR]

In this paper, we investigate pseudo-projectively symmetric P-Sasakian manifolds and P-Sasakian manifolds satisfying the condition R(X; ξ) . P = P(X; ξ).R: Next we study P-Sasakian manifolds satisfying the curvature condition P . S = 0 and pseudoprojectively at P-Sasakian manifolds. Further, we discuss about ϕ-projectively semisymmetric and locally ϕ-projectively symmetric P-Sasakian manifolds. Finally, we construct an example of a 5-dimensional P-Sasakian manifold to verify some results. [ABSTRACT FROM AUTHOR]

The object of the present paper is to study a type of Riemannian manifold called generalized quasi-Einstein manifold. The existence of a generalized quasi-Einstein manifold have been proved by non-trivial examples. [ABSTRACT FROM AUTHOR]

We characterize the three-dimensional Riemannian manifolds equipped with a semi-symmetric metric ρ -connection under the assumption that the Riemannian metric is a Yamabe soliton. It is shown that a three-dimensional Riemannian manifold endowed with a semi-symmetric ρ -connection, whose metric is Yamabe soliton, is a manifold of constant sectional curvature − 1 and the Yamabe soliton is expanding with soliton constant − 6. Finally, we give an example of a three-dimensional Riemannian manifold and validate our findings. [ABSTRACT FROM AUTHOR]