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

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Uday Chand De ◽  
Abdallah Abdelhameed Syied ◽  
Nasser Bin Turki ◽  
Suliman Alsaeed
Keyword(s):  
Scalar Curvature ◽  
Curvature Tensor ◽  
Warped Product ◽  
Einstein Manifold ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Divergence Free

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.

scholarly journals On warped product gradient η-Ricci solitons

Filomat ◽  
2017 ◽  
Vol 31 (18) ◽  
pp. 5791-5801 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Warped Product ◽  
Constant Scalar Curvature ◽  
Ricci Soliton ◽  
Product Manifold ◽  
Potential Vector ◽  
Warped Product Manifold ◽  
Gradient Type ◽  
Potential Vector Field

If the potential vector field of an ?-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by f . We give a way to construct a gradient ?-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

scholarly journals The W2-curvature tensor on warped product manifolds and applications

2016 ◽  
Vol 13 (07) ◽  
pp. 1650099 ◽  
Author(s):  
Sameh Shenawy ◽  
Bülent Ünal
Keyword(s):  
Curvature Tensor ◽  
Product Space ◽  
Warped Product ◽  
Space Time ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Static Space

The purpose of this paper is to study the [Formula: see text]-curvature tensor on (singly) warped product manifolds as well as on generalized Robertson–Walker and standard static space-times. Some different expressions of the [Formula: see text]-curvature tensor on a warped product manifold in terms of its relation with [Formula: see text]-curvature tensor on the base and fiber manifolds are obtained. Furthermore, we investigate [Formula: see text]-curvature flat warped product manifolds. Many interesting results describing the geometry of the base and fiber manifolds of a [Formula: see text]-curvature flat warped product manifold are derived. Finally, we study the [Formula: see text]-curvature tensor on generalized Robertson–Walker and standard static space-times; we explore the geometry of the fiber of these warped product space-time models that are [Formula: see text]-curvature flat.

scholarly journals A note on warped product almost quasi-Yamabe solitons

Filomat ◽  
2019 ◽  
Vol 33 (7) ◽  
pp. 2009-2016 ◽  
Author(s):  
Adara Blaga
Keyword(s):  
Riemannian Manifolds ◽  
Warped Product ◽  
Sufficient Conditions ◽  
Constant Scalar Curvature ◽  
Product Manifold ◽  
Base Manifold ◽  
Warped Product Manifold ◽  
Necessary And Sufficient ◽  
Yamabe Solitons ◽  
Yamabe Soliton

We consider almost quasi-Yamabe solitons in Riemannian manifolds, derive a Bochner-type formula in the gradient case and prove that under certain assumptions, the manifold is of constant scalar curvature. We also provide necessary and sufficient conditions for a gradient almost quasi-Yamabe soliton on the base manifold to induce a gradient almost quasi-Yamabe soliton on the warped product manifold.

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

2019 ◽  
Vol 11 (2) ◽  
pp. 332-349
Author(s):  
Buddhadev Pal ◽  
Santu Dey ◽  
Sampa Pahan
Keyword(s):  
Warped Product ◽  
Einstein Manifold ◽  
Warped Products ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Symmetric Connection

Abstract 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.

scholarly journals Gradient almost Ricci solitons on multiply warped product manifolds

2021 ◽  
Vol 13 (2) ◽  
pp. 386-394
Author(s):  
S. Günsen ◽  
L. Onat
Keyword(s):  
Potential Field ◽  
Warped Product ◽  
Einstein Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Product Manifold ◽  
Rigidity Result ◽  
Warped Product Manifold ◽  
Almost Ricci Soliton

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.

On generalized quasi-Einstein manifolds

2019 ◽  
Vol 10 (3) ◽  
pp. 193-202 ◽  
Author(s):  
Mir Ahmad Mirshafeazadeh ◽  
Behroz Bidabad
Keyword(s):  
Fourth Order ◽  
Weyl Tensor ◽  
Warped Product ◽  
Regular Point ◽  
Product Manifold ◽  
Einstein Manifolds ◽  
Warped Product Manifold ◽  
Divergence Free

Abstract 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.

scholarly journals Harmonicity of horizontally conformal maps and spectrum of the Laplacian

2002 ◽  
Vol 30 (12) ◽  
pp. 709-715 ◽  
Author(s):  
Gabjin Yun
Keyword(s):  
Warped Product ◽  
Warping Function ◽  
Conformal Map ◽  
Product Manifold ◽  
Conformal Maps ◽  
Warped Product Manifold ◽  
Spectrum Of The Laplacian ◽  
Divergence Free

We discuss the harmonicity of horizontally conformal maps and their relations with the spectrum of the Laplacian. We prove that ifΦ:M→Nis a horizontally conformal map such that the tension field is divergence free, thenΦis harmonic. Furthermore, ifNis noncompact, thenΦmust be constant. Also we show that the projection of a warped product manifold onto the first component is harmonic if and only if the warping function is constant. Finally, we describe a characterization for a horizontally conformal map with a constant dilation preserving an eigenfunction.

scholarly journals On warped product manifolds satisfying Ricci-Hessian class type equations

2018 ◽  
Vol 103 (117) ◽  
pp. 69-75 ◽  
Author(s):  
Sinem Güler ◽  
Altay Demirbağ
Keyword(s):  
Smooth Function ◽  
Ricci Tensor ◽  
Warped Product ◽  
Einstein Manifold ◽  
Warped Products ◽  
Product Manifold ◽  
Warped Product Manifold

We deal with a study of warped product manifold which is also a generalized quasi Einstein manifold. Then, we investigate the relationships between such warped products and certain manifolds that provide some Ricci-Hessian type equations, such as Ricmf = ?g for some smooth function ?, where Ricmf denotes the m-Bakery-Emery Ricci tensor. Finally, we obtain some rigidity conditions for such manifolds.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

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