Warped Product Manifolds Research Articles

Starshaped compact hypersurfaces in warped product manifolds I: prescribed curvature equations

We derive global curvature estimates for closed, strictly star-shaped (n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2)$$\\end{document}-convex hypersurfaces in warped product manifolds, which satisfy the prescribed (n-2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(n-2)$$\\end{document}-curvature equation with a general right-hand side. The proof can be readily adapted to establish curvature estimates for semi-convex solutions to the general k-curvature equation. Furthermore, it can also be used to prove the same estimates for k-convex solutions to the prescribed curvature measure type equations.

Doubly warped product manifolds with the structure of hyperbolic Ricci solitons

In this paper we study hyperbolic Ricci solitons on doubly warped product manifolds. The necessary conditions are obtained for a hyperbolic Ricci soliton with the structure of a doubly warped product to be an Einstein manifold when we consider the potential field as a Killing or a conformal vector field. Also, we examined the behavior of hyperbolic Ricci solitons on doubly warped product space-times.

Geometry of Statistical Sequential Warped Product Manifolds.

The main purpose of the present study is to generalize the dualistic structures, specially the statistical structures on warped product manifolds to statistical structures on sequential warped product manifolds. On the one hand we proved that the statistical structure on sequential warped product manifold induces statistical on the factors and conversely. On the other hand we proved that if the statistical warped product is dually at space, then the base manifold is also dually at and ber is a space form, i.e. of constant sectional curvature.

On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form (M¯,g¯), where M¯=R×fM and g¯=ϵdt2+f2(t)gM. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where (M¯,g¯) is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ∂t and the so-called support function θ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below.

On Sequential Warped Products Whose Manifold Admits Gradient Schouten Harmonic Solitons

As part of our study, we investigate gradient Schouten harmonic solutions to sequential warped product manifolds. The main contribution of our work is an explanation of how it is possible to express gradient Schouten harmonic solitons on sequential warped product manifolds. Our analysis covers both sequential generalized Robertson–Walker spacetimes and sequential static spacetimes using gradient Schouten harmonic solitons. Studies conducted previously can be generalized from this study.

Characterization of warped product manifolds through the [formula omitted]-curvature tensor with applications to relativity

We first investigate how the flatness and the symmetry of the W2-tensor impact both the base manifold and the fiber manifold of a warped product manifold. In both cases, we determine the form of the W2-tensor on both the fiber and the base manifolds. Additionally, we establish that the fiber manifold is of constant curvature, while the base manifold is Einstein. It is also proved that a W2-curvature flat GRW space-time is perfect fluid and static. Furthermore, the space-time either represents dark matter era or the state equation has a specific form.

The Impact of Quasi-Conformal Curvature Tensor on Warped Product Manifolds

The Impact of Quasi-Conformal Curvature Tensor on Warped Product Manifolds

Warped Product Manifolds: Characterizations Through the Symmetry of the Semiconformal Curvature Tensor and Applications

Warped Product Manifolds: Characterizations Through the Symmetry of the Semiconformal Curvature Tensor and Applications

Doubly warped product manifolds: Investigations through the projective curvature tensor and relativistic applications

Doubly warped product manifolds: Investigations through the projective curvature tensor and relativistic applications

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Ricci Solitons on Spacelike Hypersurfaces of Generalized Robertson–Walker Spacetimes

Introduction to gradient h-almost η-Ricci soliton warped product

In this paper, we introduce the new concept of gradient h-almost η-Ricci soliton. We discuss here a steady or expanding gradient h-almost η-Ricci soliton warped product Bn ×f Fm, m 1. We show that the warping function f of this warped product attains minimum as well as maximum and it will definitely be a Riemannian product under certain conditions. We also describe some suitable restrictions to these constructions for the compact base of this warped product. Later, we study h-almost η-Ricci soliton and gradient h-almost η-Ricci soliton on warped product manifolds including a concurrent vector field.

Gradient almost Yamabe solitons immersed into a Riemannian warped productmanifold

Gradient almost Yamabe solitons immersed into a Riemannian warped productmanifold

Characterization of a special type of Ricci–Bourguignon soliton on sequential warped product manifold

In this paper, we aim to characterize the sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton. We derive applications of some vector fields like conformal vector field, torse-forming vector field, torqued vector field on [Formula: see text]-almost conformal Ricci–Bourguignon soliton. The inheritance properties of the Einstein-like sequential warped product [Formula: see text]-almost gradient conformal Ricci–Bourguignon solitons of class types [Formula: see text] are investigated in this paper. Later we show that if [Formula: see text] is a sequential warped product of a complete connected [Formula: see text]-dimensional Riemannian manifold [Formula: see text] and one-dimensional Riemannian manifolds [Formula: see text] and [Formula: see text] admitting [Formula: see text]-almost conformal Ricci–Bourguignon soliton, then [Formula: see text] becomes a [Formula: see text]-dimensional sphere of radius [Formula: see text]. We have discovered that for a [Formula: see text]-almost gradient conformal Ricci–Bourguignon soliton sequential warped product, the warping functions are constants under certain conditions.

Conharmonically flat and conharmonically symmetric warped product manifolds

This article presents characterizations of warped product manifolds based on the flatness and symmetry of the conharmonic curvature tensor. It is proved that when a warped product manifold is conharmonically flat, both the base and fiber manifolds exhibit constant sectional curvatures. In addition, the specific forms of the conharmonic curvature tensor are derived for both the base and fiber manifolds. It is demonstrated that in a conharmonically symmetric warped product manifold, the fiber manifold has a constant sectional curvature, while the base manifold is both Cartan-symmetric and conharmonically symmetric. In this scenario, the form of the conharmonic curvature tensor on the fiber manifold is determined. We characterize the generalized Robertson–Walker (GRW) space-time through the flatness and the symmetry of the conharmonic curvature tensor. It is shown that a conharmonically flat (symmetric) GRW space-time is a perfect fluid. Finally, a conharmonically flat standard static space-time is investigated.

2-Killing vector fields on multiply warped product manifolds

We characterize 2-Killing vector fields on multiply warped product manifolds. We find the necessary and sufficient conditions for the lift of a vector field on a factor manifold (Mi,gi), i=1,n¯, to be a 2-Killing vector field on the multiply warped product manifold, providing also conditions for the component of a Killing or a 2-Killing vector field on a multiply warped product to be a 2-Killing vector field on a factor manifold. Moreover, under certain assumptions, we prove that the component of a Killing or a 2-Killing vector field on a multiply warped product manifold is the potential vector field of a Ricci or a hyperbolic Ricci soliton factor manifold, respectively. As physical applications, we consider the spacetime case, constructing examples of 2-Killing vector fields on the generalized Robertson–Walker and on the generalized Kasner spacetimes.

Isometric Immersions of Lightlike Warped Product Manifolds

Isometric Immersions of Lightlike Warped Product Manifolds

Curvature analysis of concircular trajectories in doubly warped product manifolds

<p>The aim of this research paper was to explore the various characteristics of the doubly warped product manifold, focusing particularly on aspects such as the Hessian, Riemannian curvature, Ricci curvature, and concircular curvature tensor components. By examining the necessary conditions that would classify the manifold as Riemann-flat, Ricci-flat, and concircularly-flat, the study aimed to expand our understanding of these concepts. To achieve this, the research incorporated the application of these findings to a generalized Robertson-Walker doubly warped product manifold scenario. This approach allowed us to identify and analyze the specific circumstances under which the manifold displayed concircular flatness.</p>

On gradient normalized Ricci-harmonic solitons in sequential warped products

<p>Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in <sup>[<xref ref-type="bibr" rid="b26">26</xref>]</sup>.</p>

𝜁 (Ric)-vector fields on doubly warped product manifolds

𝜁 (Ric)-vector fields on doubly warped product manifolds

Killing and 2-Killing Vector Fields on Doubly Warped Products

We provide a condition for a 2-Killing vector field on a compact Riemannian manifold to be Killing and apply the result to doubly warped product manifolds. We establish a connection between the property of a vector field on a doubly warped product manifold and its components on the factor manifolds to be Killing or 2-Killing. We also prove that a Killing vector field on the doubly warped product gives rise to a Ricci soliton factor manifold if and only if it is an Einstein manifold. If a component of a Killing vector field on the doubly warped product is of a gradient type, then, under certain conditions, the corresponding factor manifold is isometric to the Euclidean space. Moreover, we provide necessary and sufficient conditions for a doubly warped product to reduce to a direct product. As applications, we characterize the 2-Killing vector fields on the doubly warped spacetimes, particularly on the standard static spacetime and on the generalized Robertson–Walker spacetime.