A Study of Generalized Projective P − Curvature Tensor on Warped Product Manifolds

The main aim of this study is to investigate the effects of the P − curvature flatness, P − divergence-free characteristic, and P − symmetry of a warped product manifold on its base and fiber (factor) manifolds. It is proved that the base and the fiber manifolds of the P − curvature flat warped manifold are Einstein manifold. Besides that, the forms of the P − curvature tensor on the base and the fiber manifolds are obtained. The warped product manifold with P − divergence-free characteristic is investigated, and amongst many results, it is proved that the factor manifolds are of constant scalar curvature. Finally, P − symmetric warped product manifold is considered.

Gradient almost Ricci solitons on multiply warped product manifolds

In this paper, we investigate multiply warped product manifold \[M =B\times_{b_1} F_1\times_{b_2} F_2\times_{b_3} \ldots \times_{b_m} F_m\] as a gradient almost Ricci soliton. Taking $b_i=b$ for $1\leq i \leq m$ lets us to deduce that potential field depends on $B$. With this idea we also get a rigidity result and show that base is a generalized quasi-Einstein manifold if $\nabla b$ is conformal.

Geometry of Chen invariants in statistical warped product manifolds

In this paper, we derive Chen inequality for statistical submanifold of statistical warped product manifolds [Formula: see text]. Further, we derive Chen inequality for Legendrian statistical submanifold in statistical warped product manifolds [Formula: see text]. We also provide some applications of derived inequalities in a statistical warped product manifold which is equivalent to a hyperbolic space. Moreover, we construct new examples of statistical warped product manifolds to support results.

Einstein doubly warped product manifolds with semi-symmetric metric connection

In this paper, we study the doubly warped product manifolds with semi-symmetric metric connection. We derive the curvature formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for an Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.

Gray’s decomposition on doubly warped product manifolds and applications

A. Gray presented an interesting O(n) invariant decomposition of the covariant derivative of the Ricci tensor. Manifolds whose Ricci tensor satisfies the defining property of each orthogonal class are called Einstein-like manifolds. In the present paper, we answered the following question: Under what condition(s), does a factor manifold Mi,i = 1,2 of a doubly warped product manifold M =f2 M1 x f1 M2 lie in the same Einstein- like class of M? By imposing sufficient and necessary conditions on the warping functions, an inheritance property of each class is proved. As an application, Einstein-like doubly warped product space-times of type A,B or P are considered.

On a non flat Riemannian warped product manifold with respect to quarter-symmetric connection

In this paper, we study generalized quasi-Einstein warped products with respect to quarter symmetric connection for dimension n ≥ 3 and Ricci-symmetric generalized quasi-Einstein manifold with quarter symmetric connection. We also investigate that in what conditions the generalized quasi-Einstein manifold to be nearly Einstein manifold with respect to quarter symmetric connection. Example of warped product on generalized quasi-Einstein manifold with respect to quarter symmetric connection are also discussed.

On generalized quasi-Einstein manifolds

We study generalized quasi-Einstein manifolds, or briefly, GQE manifolds. Here, we present relations between the Bach, Cotton and D tensors on GQE manifolds. Next, a 3-tensor E which measures the deviation of m-quasi-Einstein manifolds from GQE manifolds is introduced. Among others in dimension 3, it is shown that Bach-flatness implies locally conformally flatness. Furthermore, it is proved that, around a regular point of the fourth-order divergence free Weyl tensor, a GQE manifold is a locally warped product manifold with {(n-1)} -dimensional Einstein fibers in suitable cases.