We introduce the concept of a base conformal warped product of two pseudo-Riemannian manifolds. We also define a subclass of this structure called as a special base conformal warped product. After, we explicitly mention many of the relevant fields where metrics of these forms and also considerations about their curvature related properties play important rolls. Among others, we cite general relativity, extra-dimension, string and super-gravity theories as physical subjects and also the study of the spectrum of Laplace-Beltrami operators on p-forms in global analysis. Then, we give expressions for the Ricci tensor and scalar curvature of a base conformal warped product in terms of Ricci tensors and scalar curvatures of its base and fiber, respectively. Furthermore, we introduce specific identities verified by particular families of, either scalar or tensorial, nonlinear differential operators on pseudo-Riemannian manifolds. The latter allow us to obtain new interesting expressions for the Ricci tensor and scalar curvature of a special base conformal warped product and it turns out that not only the expressions but also the analytical approach used are interesting from the physical, geometrical and analytical point of view. Finally, we analyze, investigate and characterize possible solutions for the conformal and warping factors of a special base conformal warped product, which guarantee that the corresponding product is Einstein. Besides, we apply these results to a generalization of the Schwarzschild metric.
Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. This paper deals with the study of conformal Ricci solitons within the framework of warped product manifolds which extends the notion of usual Riemannian product manifolds. First, we prove that if a warped product manifold admits conformal Ricci soliton then the base and the fiber also share the same property. In the next section the characterization of conformal Ricci solitons on warped product manifolds in terms of Killing and conformal vector fields has been studied. Next, we prove that a warped produc…
In this article, we present a complete study of two disjoint classes of conformal vector fields on doubly warped product manifolds as well as on doubly warped space-times. Then we study Ricci solitons on doubly warped product manifollds admitting these types of conformal vector fields.
In this paper, we investigate the condition on warped product manifold of a Kenmotsu manifold and the real line to be a conformal Kahler manifold. This result demonstrates the close relation of Kenmotsu and Kahler manifolds.
In this paper, we study the geometry of pointwise semi-slant warped products in a locally conformal Kaehler manifold. In particular, we obtain several results which extend Chen's inequality for CR-warped product submanifolds in Kaehler manifolds. Also, we study the corresponding equality cases. Several related results on pointwise semi-slant warped products are also proved in this paper.
In this paper we present not only some properties related to bi-warped product submanifolds of locally conformal almost cosymplectic manifolds, but also we show how the squared norm of the second fundamental form and the bi-warped product's warping functions are related when the bi-warped product submanifold has a proper slant submanifold as a base or fiber.
In this paper we study Inverse Mean Curvature Flow (IMCF) on manifolds that are conformal to a warped product manifold. To this end, we show how the gradient conformal vector field in warped product manifolds is related to the conformal vector field on the conformal metric and use this to gain control of the flow in order to show long time existence. Connections are made to recent results of the author on stability of the positive mass theorem (PMT) and the Riemannian Penrose inequality (RPI) where long time existence of IMCF is an important assumption.
n this paper, we obtain a geometric inequality between the length of the second fundamental form and the length of Lee form in terms of the warping function for a CR-warped product submanifold in a locally conformal Kaehler space form. The equality case is also investigated. Furthermore, the inequality is discussed for the important subclass of locally conformal Kaehler manifolds i.e., Vaisman manifold.
We investigate Kähler metrics conformal to gradient Ricci solitons, and base metrics of warped product gradient Ricci solitons. The latter we name quasi-solitons. A main assumption that is employed is functional dependence of the soliton potential, with the conformal factor in the first case, and with the warping function in the second. The main result in the first case is a partial classification in dimension $n\geq 4$. In the second case, Kähler quasi-soliton metrics satisfying the above main assumption are shown to be, under an additional genericity hypothesis, necessarily Riemannian produc…
We study Ricci-Bourguignon solitons on sequential warped products. The necessary conditions are obtained for a Ricci-Bourguignon soliton with the structure of a sequential warped product to be an Einstein manifold when we consider the potential field as a Killing or a conformal vector field.
Generalized Roter type manifold is a generalization of conformally flat manifold as well as Roter type manifold, which gives rise the form of the curvature tensor in terms of algebraic combinations of the fundamental metric tensor and Ricci tensors upto level 2. The object of the present paper is to investigate the characterizations of a warped product manifold to be generalized Roter-type. We also present an example of a warped product manifold which is generalized Roter type but not Roter type, and also an example of a warped product manifold which is Roter type but not conformally flat. The…
In this paper we prove that under certain conditions in a quasi Einstein semi Riemannian warped product the fiber is necessarily a Einstein manifold. We provide all the quasi Einstein manifolds when r Bakry Emery tensor is null, the base is conformal to an n-dimensional pseudo-Euclidean space invariant under the action of an n - 1 dimensional translation group and the fiber is Ricci flat. As an application, we have built a family of Ricci flat Einstein warped product whose base is not locally conformally flat.
We study warped products semi-Riemannian Einstein manifolds. We consider the case in that the base is conformal to an n-dimensional pseudo Euclidean space and invariant under the action of an translation group. We provide all such solutions in the case Ricci flat when the base is conformal to an n-dimensional pseudo-Euclidean space, invariant under the action of an translation group and the fiber F is Ricci flat. In particular, we obtain explicit solutions, in the case vacuum, for the Einstein field equation.
Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, then the (0,6)-tensors R.R - Q(S,R) and C.C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is expressed by a linear combina…
We derive the general formulas for a special configuration of the sequential warped product semi-Riemannian manifold to be Einstein, where the base-manifold is the product of two manifolds both equipped with a conformal metrics. Subsequently we study the case in which these two manifolds are conformal to a $n_1$-dimensional and $n_2$-dimensional pseudo-Euclidean space, respectively. For the latter case, we prove the existence of a family of solutions that are invariant under the action of a $(n_1-1)$-dimensional group of transformations to the case of positive constant Ricci curvature ($λ>0$).
In this note, we introduce a new type of warped products called as sequential warped products to cover a wider variety of exact solutions to Einstein's equation. First, we study the geometry of sequential warped products and obtain covariant derivatives, curvature tensor, Ricci curvature and scalar curvature formulas. Then some important consequences of these formulas are also stated. We provide characterizations of geodesics and two different types of conformal vector fields, namely, Killing vector fields and concircular vector fields on sequential warped product manifolds. Finally, we consid…
In this paper, we provide a necessary and sufficient conditions for the warped product $M=B\times_f F$ to be a gradient Yamabe soliton when the base is conformal to an n-dimensional pseudo-Euclidean space, which are invariant under the action of an (n-1)-dimensional translation group, and the fiber F is scalar-constant. As application, we obtain solutions in steady case with fiber scalar-flat. Besides, on the warped product we consider the potential function as separable variables and obtain some characterization of the base and the fiber.
In this paper we consider a class of Einstein warped product semi-Riemannian manifolds $\widehat{M} = M^{n}\times_{f}N^{m}$ with $n\geq 3$ and $m\geq 2$. For $\widehat{M}$ with compact base and Ricci-flat fiber, we prove that $\widehat{M}$ is simply a Riemannian product space. Then, when the base $M$ is conformal to a pseudo-Euclidean space which is invariant under the action of a $(n-1)$-dimensional translation group, we classify all such spaces. Furthermore, we get new examples of complete Einstein warped products Riemannian manifolds.
In this paper we consider $M = B\times_{f}F$ warped product gradient Ricci solitons. We proved that the potential function depends only on the base and the fiber $F$ is necessarily Einstein manifold. We provide all such solutions in the case of steady gradient Ricci solitons when the base is conformal to an $n$-dimensional pseudo-Euclidean space, invariant under the action of an $(n-1)$-dimensional translation group and the fiber $F$ is Ricci-flat.
The purpose of this article is to study gradient Yamabe soliton on warped product manifolds. First, we prove triviality results in the case of noncompact base with limited warping function, and for compact base. In order to provide nontrivial examples, we consider the base conformal to a semi-Euclidean space, which is invariant under the action of a translation group, and then we characterize steady solitons. We use this method to give infinitely many explicit examples of complete steady gradient Yamabe solitons.
In this paper we study gradient Ricci-Harmonic soliton with structure of warped product manifold. We obtain some triviality results for the potential function, warping function and the harmonic map which reaches maximum or minimum. In order to obtain nontrivial examples of warped product gradient Ricci-harmonic soliton, we consider the base and fiber conformal to a semi-Euclidean space which is invariant under the action of a translation group of co-dimension one. This approach provide infinitely many geodesically complete examples in the semi-Riemannian context, which is not contemplated in t…
The aim of this paper is to introduce a notion of mean curvature flow soliton general enough to encompass target spaces of constant sectional curvature, Riemannian products or, in increasing generality, warped product spaces.
We characterize Ricci almost solitons on semi-Riemannian warped products, considering the potential function to depend on the fiber or not. We show that the fiber is necessarily an Einstein manifold. As a consequence of our characterization we prove that when the potential function depends on the fiber, if the gradient of the warping function does not act by translations then the base and the warped product are also Einstein manifolds. Moreover, we show the existence of conformal vector fields on the base, the fiber and on the warped product. Assuming completeness of the warped product we prov…
In this paper, we mainly study gradient $ρ$-Einstein solitons on doubly warped product manifolds. More explicitly, we obtain necessary and sufficient conditions for a doubly warped product manifold to be a gradient $ρ$-Einstein soliton. We also apply our main result to warped product spacetime models such as generalized Robertson-Walker and standard static spacetimes as well as 3-dimensional Walker manifolds. We finally establish that there is no 3-dimensional esentially conformally symmetric gradient $ρ$-Einstein soliton.
We show that a complete gradient Ricci soliton $(M^n,\,g)$ with constant scalar curvature and a non-parallel closed conformal vector field is isometric to either the Euclidean space, or an Euclidean sphere, or negatively Einstein warped product of the real line with a complete non-positively Einstein manifold. Moreover, we show that a Kähler gradient Ricci soliton of real dimension $\ge 4$, with a non-parallel closed conformal real vector field is Ricci-flat (Calabi-Yau) and is flat in dimension 4. Finally, we show that a Ricci soliton whose associated 1-form is harmonic, has constant scalar c…
Any compact spacelike hypersurface immersed in a doubly warped product spacetime $I{}_{h} \times_ρ \mathbb{P}$ with nondecreasing warping factor $ρ$ must be a spacelike slice, provided that the mean curvature satisfies $H\geqρ'/hρ$ everywhere on the hypersurface. The conclusion also holds, under suitable assumptions on the immersion, when the hypersurface is complete and noncompact. A similar rigidity property is shown for compact hypersurfaces in spacetimes carrying a conformal, strictly expanding, timelike vector field.
Lyra Almost Ricci-Bourguignon Solitons on Twisted Warped Product Manifolds
Tanta University（坦塔大学）；Princess Nourah bint Abdlrahman University（公主诺拉·宾特·阿卜杜勒·拉赫曼大学）；University of Calcutta（加尔各答大学）；Arab Academy for Science, Technology and Maritime Transport（阿拉伯科学、技术与海运学院）；Ain-Shams University（艾因夏姆斯大学）
For a closed, connected direct product Riemannian manifold $(M, g)=(M_1\times\cdots\times M_l, g_1\oplus\cdots\oplus g_l)$, we define its multiconformal class $ [\![ g ]\!]$ as the totality $\{f_1^2g_1\oplus \cdots\oplus f_l^2g_l\}$ of all Riemannian metrics obtained from multiplying the metric $g_i$ of each factor $M_i$ by a function $f_i^2>0$ on the total space $M$. A multiconformal class $ [\![ g ]\!]$ contains not only all warped product type deformations of $g$ but also the whole conformal class $[\tilde{g}]$ of every $\tilde{g}\in [\![ g ]\!]$. In this article, we prove that $ [\![ g ]\!…
This paper aims to investigate the curvature restricted geometric properties admitted by Melvin magnetic spacetime metric, a warped product metric with $1$-dimensional fibre. For this, we have considered a Melvin type static, cylindrically symmetric spacetime metric in Weyl form and it is found that such metric, in general, is generalized Roter type, $Ein(3)$ and has pseudosymmetric Weyl conformal tensor satisfying the pseudosymmetric type condition $R\cdot R-Q(S,R)=\mathcal L' Q(g,C)$. The condition for which it satisfies the Roter type condition has been obtained. It is interesting to note t…
