In this paper, we study Ricci-flat and Einstein Lorentzian multiply warped products. We also consider the case of having constant scalar curvatures for this class of warped products. Finally, after we introduce a new class of spacetimes called as generalized Kasner space-times, we apply our results to this kind of space-times as well as other relativistic space-times, i.e., Reissner-Nordstrom, Kasner space-times, Banados-Teitelboim-Zanelli and de Sitter Black hole solutions.
In this paper, we define a semi-symmetric metric Killing vector field, then study semi-symmetric metric Killing vector fields on warped and multiply warped products with a semi-symmetric metric connection. We also study Killing and 2-Killing vector fields on multiply warped products.
In this paper, we study the Einstein warped products and multiply warped products with a quarter-symmetric connection. We also study warped products and multiply warped products with a quarter-symmetric connection with constant scalar curvature. Then apply our results to generalized Robertson-Walker spacetimes with a quarter-symmetric connection and generalized Kasner space-times with a quarter-symmetric connection.
In this paper, we generalize the results in [Y. Wang: Affine connections on singular warped products. Int. J. Geom. Methods Mod. Phys. 18(5), 2150076, (2021).] to singular multiply warped products and singular twisted products. We study singular multiply warped products and singular twisted products and their curvature with the semi-symmetric metric connection and the semi-symmetric non-metric connection. We also discuss Koszul forms associated with the almost product structure and their curvature of singular multiply warped products and singular twisted products. Finally, several examples are…
In this paper, we study the Einstein multiply warped products with a semi-symmetric non-metric connection and the multiply warped products with a semi-symmetric non-metric connection with constant scalar curvature, we apply our results to generalized Robertson-Walker spacetimes with a semi-symmetric non-metric connection and generalized Kasner spacetimes with a semi-symmetric non-metric connection and find some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature.ß We also consider the multiply warped products with an affine connection with a zero torsi…
By applying an average method in PDE, we obtain a dichotomy between "constancy" and "infinity" of the warping functions on complete noncompact Riemannian manifolds for an appropriate isometric immersion of a multiply warped product manifold $N_1\times_{f_2} N_2 \times \cdots \times _{f_k} N_k\, $ into a Riemannian manifold. Generalizing the earlier work of the authors in [{Glasg. Math. J. 51 (2009) 579-592], we establish sharp inequalities between the mean curvature of the immersion and the sectional curvatures of the ambient manifold under the influence of quantities of a purely analytic natu…
In this paper, we compute the index form of the multiply twisted products. We study the Killing vector fields on the multiply twisted product manifolds and determine the Killing vector fields in some cases. We compute the curvature of the multiply twisted products with a semi-symmetric metric connection and show that the mixed Ricci-flat multiply twisted products with a semi-symmetric metric connection can be expressed as multiply warped products. We also study the Einstein multiply warped products with a semi-symmetric metric connection and the multiply warped products with a semi-symmetric m…
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 …
This study aims mainly at investigating the effects of concircular flatness and concircular symmetry of a warped product manifold on its fibre and base manifolds. Concircularly flat and concircularly symmetric warped product manifolds are investigated. The divergence free concircular curvature tensor on warped product manifolds is considered. Finally, we apply some of these results to generalized Robertson-Walker and standard static space-times.
We prove some isoperimetric type inequalities in warped product manifolds, or more generally, multiply warped product manifolds. We then relate them to inequalities involving the higher order mean-curvature integrals. We also apply our results to obtain sharp eigenvalue estimates and some sharp geometric inequalities in space forms.
We consider the curvature of a family of warped products of two pseduo-Riemannian manifolds $(B,g_B)$ and $(F,g_F)$ furnished with metrics of the form $c^{2}g_B \oplus w^2 g_F$ and, in particular, of the type $w^{2 μ}g_B \oplus w^2 g_F$, where $c, w \colon B \to (0,\infty)$ are smooth functions and $μ$ is a real parameter. We obtain suitable expressions for the Ricci tensor and scalar curvature of such products that allow us to establish results about the existence of Einstein or constant scalar curvature structures in these categories. If $(B,g_B)$ is Riemannian, the latter question involves …
In this paper, we investigate the null (light-like) sectional curvatures of Lorentzian warped product manifolds. We derive the formulas for the null sectional curvature of many well-known warped product space-time models such as multiply generalized Robertson-Walker space-times, generalized Kasner spacetimes and standard static space-times.
In this paper we review $G_2$ and $Spin(7)$ geometries in relation with a special type of metric structure which we call warped-like product metric. We present a general ansatz of warped-like product metric as a definition of warped-like product. Considering fiber-base decomposition, the definition of warped-like product is regarded as a generalization of multiply-warped product manifolds, by allowing the fiber metric to be non block diagonal. For some special cases, we present explicit example of $(3+3+2)$ warped-like product manifolds with $Spin(7)$ holonomy of the form $M=F\times B$, where …
We consider surfaces with constant mean curvature in certain warped product manifolds. We show that any such surface is umbilic, provided that the warping factor satisfies certain structure conditions. This theorem can be viewed as a generalization of the classical Alexandrov theorem in Euclidean space. In particular, our results apply to the deSitter-Schwarzschild and Reissner-Nordstrom manifolds.
In this paper we study warped product Finsler metrics and show that the notion of isotropic $E$-curvature and isotropic $S$-curvature are equivalent for this class of metrics.
In this paper we provide several uniqueness and non-existence results for complete parabolic constant mean curvature spacelike hypersurfaces in Lorentzian warped products under appropriate geometric assumptions. As a consequence of this parametric study, we obtain very general uniqueness and non-existence results for a large family of uniformly elliptic EDP's, so solving the Calabi-Bernstein problem in a wide family of spacetimes.
We prove local and global upper estimates for the infimum of the mean curvature, the scalar curvature and the norm of the shape operator of graphs in a warped product space. Using these estimates, we obtain some results on pseudo-hyperbolic spaces and space forms.
We discuss conformal deformation and warped products on some open manifolds. We discuss how these can be applied to construct Riemannian metrics with specific scalar curvature functions.
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…
We give estimates on the intrinsic and the extrinsic curvature of manifolds that are isometrically immersed as cylindrically bounded submanifolds of warped products. We also address extensions of the results in the case of submanifolds of the total space of a Riemannian submersion.
Let $(M^m,g_M)$ be a closed, connected manifold with positive scalar curvature and $(T^k,g)$ some flat $k$-Torus of unit volume. By a result of F. Dobarro and E. Lami Dozo, there exists a unique $f: M \rightarrow \mathbf{R}_{>0}$ such that the warped product $M\times_f T^k$ has constant scalar curvature and unit volume. We study the Yamabe equation on these spaces. We use techniques from bifurcation theory, along with spectral theory for warped products, to prove multiplicity results for the Yamabe problem.
We obtain necessary conditions for the existence of complete vertical graphs of constant mean curvature in the Hyperbolic and Steady State spaces. In the two-dimensional case we prove Bernstein-type results in each of these ambient spaces.
In this paper, we introduce semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms on singular semi-Riemannian manifolds. Semi-symmetric metric Koszul forms and semi-symmetric non-metric Koszul forms and their curvature of semi-regular warped products are expressed in terms of those of the factor manifolds. We also introduce Koszul forms associated to the almost product structure on singular almost product semi-Riemannian manifolds. Koszul forms associated to the almost product structure and their curvature of semi-regular almost product warped products are expressed in …
In this paper we obtain a sharp height estimate concerning compact hypersurfaces immersed into warped product spaces with some constant higher order mean curvature, and whose boundary is contained into a slice. We apply these results to draw topological conclusions at the end of the paper.
We introduce a notion of doubly warped product of weighted graphs that is consistent with the doubly warped product in the Riemannian setting. We establish various discrete Bakry-Émery Ricci curvature-dimension bounds for such warped products in terms of the curvature of the constituent graphs. This requires deliberate analysis of the quadratic forms involved, prompting the introduction of some crucial notions such as curvature saturation at a vertex. In the spirit of being thorough and to provide a frame of reference, we also introduce the $\left(R_1,R_2\right)$-doubly warped products of smoo…
In this article, we introduce the sectional curvature in contravariant warped product space $(M= M_{1}\times_{f_{1}}M_{2},Π,g^{f_{1}})$, where $Π=Π_1+ν_{1}Π_2$). After that we find the sectional curvature of $M$ for which $M_{1}$ and $M_{2}$ are Poisson manifolds of positive sectional curvatures. In dual space of $M$, we introduce the notion of null, spacelike, timelike $1 -$ forms and then by using these forms, qualar curvature is defined. Finally, as an examples we obtain the sectional curvatures for $M_{1} = H_{1}^2$, $M_{2} = S_{0}^2 , E_{2}^2$ and qualar curvature for $M$.
The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers $p\leq 1$ of a curvature function satisfying few common properties.
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…
In this paper, we first investigate several rigidity problems for hypersurfaces in the warped product manifolds with constant linear combinations of higher order mean curvatures as well as "weighted'' mean curvatures, which extend the work \cite{Mon, Brendle,BE} considering constant mean curvature functions. Secondly, we obtain the rigidity results for hypersurfaces in the space forms with constant linear combinations of intrinsic Gauss-Bonnet curvatures $L_k$. To achieve this, we develop some new kind of Newton-Maclaurin type inequalities on $L_k$ which may have independent interest.
In this paper we study self-similar solutions in warped products satisfying $F-\mathcal{F}=\bar{g}(λ(r)\partial_{r},ν)$, where $\mathcal{F}$ is a nonnegative constant and $F$ is in a class of general curvature functions including powers of mean curvature and Gauss curvature. We show that slices are the only closed strictly convex self-similar solutions in the hemisphere for such $F$. We also obtain a similar uniqueness result in hyperbolic space $\mathbb{H}^{3}$ for Gauss curvature $F$ and $\mathcal{F}\geq 1$.
We study the $(2+2)$-Einstein warped product manifolds, where the scalar curvature of the Base is a multiple of the warping function, and we called this condition (inside a warped product manifold) $f$-curvature-Base ($R_{f_B}$).The aim of this paper is to check if there are Base-manifolds with non-flat metrics that satisfy this condition, and this check was done in cases where $M$ and Fiber-manifold are not both non-Ricci-flat. As a results of the our cases we find that the "$f$-curvature-Base" is equivalent to requesting a flat metric on the Base-manifold.
In this paper we characterize compact and complete hypersurfaces with some constant higher order mean curvature into warped product spaces. Our approach is based on the use of a new trace operator version of the Omori-Yau maximum principle which seems to be interesting in its own.
We investigate the mean curvature flows in a class of warped product manifolds with closed hypersurfaces fibering over $\mathbb{R}$. In particular, we prove that under natural conditions on the warping function and Ricci curvature bound for the ambient space, there exists a large class of closed initial hypersurfaces, as geodesic graphs over the totally geodesic hypersurface $Σ$, such that the mean curvature flow starting from $S_0$ exists for all time and converges to $Σ$.
We present several rigidity results for Riemannian manifolds $(M^n,g)$ with scalar curvature $S \ge -n(n-1)$ (or $S\ge 0$), and having compact boundary $N$ satisfying a related mean curvature inequality. The proofs make use of results on marginally outer trapped surfaces applied to appropriate initial data sets. One of the results involves an analysis of Obata's equation on manifolds with boundary. This result is relevant to recent work of Lan-Hsuan Huang and the second author concerning the rigidity of asymptotically locally hyperbolic manifolds with zero mass.
In this paper, under suitable settings, we can obtain the existence of solutions to a class of prescribed Weingarten curvature equations in warped product manifolds of special type by the standard degree theory based on the a priori estimates for the solutions. This is to say that the existence of closed hypersurface (which is graphic with respect to the base manifold and whose $k$-th Weingarten curvature satisfies some constraint) in a given warped product manifold of special type can be assured.
We consider the warped product manifold, $\mathbb{R}_+ \times_{\bf{Id}} M^n$, with Riemannian metric $γ\equiv \mathrm{d} r^2 \oplus r^2 σ$, where $(M^n, σ)$ is a smooth closed Riemannian $n$-manifold. We investigate what sufficient curvature condition is required of $σ$ to ensure that a solution to the inverse mean curvature flow - commencing with a star-shaped surface - exists for all times $t>0$.
In this short survey, we show how two (classes of) known examples of inhomogeneous, curvature homogeneous Riemannian manifolds with nontrivial $κ$-nullity can be seen as deformations of homogeneous metrics along the vertical distribution of an integrable Riemannian submersion. We also pose two open questions.
We study inverse mean curvature flows of starshaped, mean convex hypersurfaces in warped product manifolds with a positive warping factor $φ(r)$. If $φ'(r)>0$ and $φ''(r)\geq 0$, we show that these flows exist for all times, remain starshaped and mean convex. Plus the positivity of $φ''(r)$ and a curvature condition we obtain a lower positive bound of mean curvature along these flows independent of the initial mean curvature. We also give a sufficient condition to extend the asymptotic behavior of these flows in Euclidean spaces into some more general warped product manifolds.
In this paper, we study rigidity problems for hypersurfaces with constant curvature quotients $\frac{\mathcal{H}_{2k+1}}{\mathcal{H}_{2k}}$ in the warped product manifolds. Here $\mathcal{H}_{2k}$ is the $k$-th Gauss-Bonnet curvature and $\mathcal{H}_{2k+1}$ arises from the first variation of the total integration of $\mathcal{H}_{2k}$. Hence the quotients considered here are in general different from $\frac{σ_{2k+1}}{σ_{2k}}$, where $σ_k$ are the usual mean curvatures. We prove several rigidity and Bernstein type results for compact or non-compact hypersurfaces corresponding to such quotients…
In this paper, we consider doubly warped product (DWP) Finsler manifolds with some non-Riemannian curvature properties. First, we study Berwald and isotropic mean Berwald DWP-Finsler manifolds. Then we prove that every proper Douglas DWP-Finsler manifold is Riemannian. We show that a proper DWP-manifold is Landsbergian if and only it is Berwaldian. In continue, we prove that every relatively isotropic Landsberg DWP-manifold is a Landsberg manifold. We show that a relatively isotropic mean Landsberg warped product manifold is a weakly Landsberg manifold. Finally, we show that there is no any lo…
We study a multiply warped products manifold associated with the Reissner-Nordstrom metric to investigate the physical properties inside the black hole event horizons. It is shown that, different from the uncharged Schwarzschild metric, the Ricci curvature components inside the Reissner-Nordstrom black hole horizons are nonvanishing, while the Einstein scalar curvature vanishes even in the interior of the charged metric. Introducing a perfect fluid inside the Reissor-Nordstrom black hole, it is also shown that the charge plays effective roles of decreasing the mass-energy density and the press…
In this paper we study warped product Einstein metrics over spaces with constant scalar curvature. We call such a manifold rigid if the universal cover of the base is Einstein or is isometric to a product of Einstein manifolds. When the base is three dimensional and the dimension of the fiber is greater than one we show that the space is always rigid. We also exhibit examples of solvable four dimensional Lie groups that can be used as the base space of non-rigid warped product Einstein metrics showing that the result is not true in dimension greater than three. We also give some further natura…
We prove generalized lower Ricci curvature bounds for warped products over complete Finsler manifolds. On the one hand our result covers a theorem of Bacher and Sturm concerning euclidean and spherical cones. On the other hand it can be seen in analogy to a result of Bishop and Alexander in the setting of Alexandrov spaces with curvature bounded from below. For the proof we combine techniques developed in these papers. Because the Finsler product metric can degenerate we regard a warped product as metric measure space that is in general neither a Finsler manifold nor an Alexandrov space again …
Given a compact Riemannian manifold $M$, we consider a warped product $\bar M = I \times_h M$ where $I$ is an open interval in $\Bbb R$. For a positive function $ψ$ defined on $\bar M$, we generalized the arguments in \cite{GRW2015} and \cite{RW16}, to obtain the curvature estimates for Hessian equations $σ_k(κ)=ψ(V,ν(V))$. We also obtain some existence results for the starshaped compact hypersurface $Σ$ satisfying the above equation with various assumptions.
Let $M$ be a complete Riemannian manifold which either is compact or has a pole, and let $φ$ be a positive smooth function on $M$. In the warped product $M\times_φ\mathbb R$, we study the flow by the mean curvature of a locally Lipschitz continuous graph on $M$ and prove that the flow exists for all time and that the evolving hypersurface is $C^\infty$ for $t>0$ and is a graph for all $t$. Moreover, under certain conditions, the flow has a well defined limit.
For any manifold $N^p$ admitting an Einstein metric with positive Einstein constant, we study the behavior of the Ricci flow on high-dimensional products $M = N^p \times S^{q+1}$ with doubly-warped product metrics. In particular, we provide a rigorous construction of local, type II, conical singularity formation on such spaces. It is shown that for any $k > 1$ there exists a solution with curvature blow-up rate $\| Rm \|_{\infty} (t) \gtrsim (T-t)^{-k}$ with singularity modeled on a Ricci-flat cone at parabolic scales.
Alexandrov's theorem asserts that spheres are the only closed embedded constant mean curvature hypersurfaces in space forms. In this paper, we consider Alexandrov's theorem in warped product manifolds and prove a rigidity result in the spirit of Alexandrov's theorem. Our approach generalizes the proofs of Reilly and Ros and, under more restrictive assumptions, it provides an alternative proof of a recent theorem of Brendle.
We will obtain the warped product decompositions of spaces of constant curvature (with arbitrary signature) in their natural models as subsets of pseudo-Euclidean space. This generalizes the corresponding result by S. Nolker to arbitrary signatures, and has a similar level of detail. Although our derivation is complete in some sense, none is proven. Motivated by applications, we will give more information for the spaces with Euclidean and Lorentzian signatures. This is an expository article which is intended to be used as a reference. So we also give a review of the theory of circles and spher…
