scholarly journals Gray’s decomposition on doubly warped product manifolds and applications

Filomat ◽  
2020 ◽  
Vol 34 (11) ◽  
pp. 3767-3776
Author(s):  
Hoda El-Sayied ◽  
Carlo Mantica ◽  
Sameh Shenawy ◽  
Noha Syied
Keyword(s):  
Covariant Derivative ◽  
Ricci Tensor ◽  
Product Space ◽  
Warped Product ◽  
Necessary Conditions ◽  
Type A ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Sufficient And Necessary Conditions

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.

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 ON WARPED PRODUCT MANIFOLDS - CONFORMAL FLATNESS AND CONSTANT SCALAR CURVATURE PROBLEM

1998 ◽  
Vol 29 (3) ◽  
pp. 203-221
Author(s):  
KWANG-WU YANG
Keyword(s):  
Warped Product ◽  
Necessary Conditions ◽  
Constant Scalar Curvature ◽  
Topological Product ◽  
Product Manifold ◽  
Product Metrics ◽  
Conformal Flatness ◽  
Sufficient And Necessary Conditions ◽  
Linear Differential ◽  
Curvature Problem

In this paper, we study some geometric properties on doubly or singly warped­ product manifolds. In general, on a fixed topological product manifold, the problem for finding warped-product metrics satisfying certain curvature conditions are finally reduced to find positive solutions of linear or non-linear differential equations. Here, we are mainly interested in the following problems on essentially warped-product manifolds: one is the sufficient and necessary conditions for conformal flatness, and the other is to find warped-product metrics so that their scalar curvatures are contants.

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.

scholarly journals 2-Killing vector fields on warped product manifolds

2015 ◽  
Vol 26 (09) ◽  
pp. 1550065 ◽  
Author(s):  
Sameh Shenawy ◽  
Bülent Ünal
Keyword(s):  
Vector Field ◽  
Product Space ◽  
Warped Product ◽  
Vector Fields ◽  
Killing Vector ◽  
Killing Vector Field ◽  
Product Manifold ◽  
Killing Vector Fields ◽  
Warped Product Manifold

This paper provides a study of 2-Killing vector fields on warped product manifolds as well as characterization of this structure on standard static and generalized Robertson–Walker space-times. Some conditions for a 2-Killing vector field on a warped product manifold to be parallel are obtained. Moreover, some results on the curvature of a warped product manifolds in terms of 2-Killing vector fields are derived. Finally, we apply our results to describe 2-Killing vector fields of some well-known warped product space-time models.

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.

scholarly journals A note on Weingarten hypersurfaces in the warped product manifold

2014 ◽  
Vol 25 (14) ◽  
pp. 1450121 ◽  
Author(s):  
Haizhong Li ◽  
Yong Wei ◽  
Changwei Xiong
Keyword(s):  
Mean Curvature ◽  
Warped Product ◽  
Constant Mean Curvature ◽  
Higher Order ◽  
Product Manifold ◽  
Weingarten Surfaces ◽  
Warped Product Manifold ◽  
Higher Order Mean Curvature ◽  
Special Case ◽  
Differential Geom

In this paper, we consider the closed embedded hypersurface Σ in the warped product manifold [Formula: see text] equipped with the metric g = dr2 + λ(r)2 gN. We give some characterizations of slice {r} × N by the condition that Σ has constant weighted higher-order mean curvatures (λ′)αpk, or constant weighted higher-order mean curvature ratio (λ′)αpk/p1, which generalize Brendle's [Constant mean curvature surfaces in warped product manifolds, Publ. Math. Inst. Hautes Études Sci. 117 (2013) 247–269] and Brendle–Eichmair's [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] results. In particular, we show that the assumption convex of Brendle–Eichmair's result [Isoperimetric and Weingarten surfaces in the Schwarzschild manifold, J. Differential Geom. 94(3) (2013) 387–407] is unnecessary. Here pk is the kth normalized mean curvature of the hypersurface Σ. As a special case, we also give some characterizations of geodesic spheres in ℝn, ℍn and [Formula: see text], which generalize the classical Alexandrov-type results.

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.

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