H. K. El-Sayied, Sameh Shenawy, Noha Syied
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.
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In this paper, we study the doubly warped product manifolds with semisymmetric metric connection. We derive the curvatures formulas for doubly warped product manifold with semi-symmetric metric connection in terms of curvatures of components of doubly warped product manifolds. We also prove the necessary and sufficient condition for a doubly warped product manifold to be a warped product manifold. We obtain some results for Einstein doubly warped product manifold and Einstein-like doubly warped product manifold of class A with respect to a semi-symmetric metric connection.
On doubly warped product immersions
本文研究了双翘曲积浸入的基本几何性质，它是翘曲积浸入的推广。此外，我们还研究了等距浸入任意黎曼流形中的双曲积的几何不等式。
On Doubly Warped Product Finsler Manifolds
本文引入了水平和垂直翘积Finsler流形。证明了每一个C-可约或真Berwaldian双曲积Finsler流形都是黎曼流形。然后，我们找到了双曲积Finsler流形与其分支的Riemmanian曲率之间的关系，并考虑了这类流形是平坦的或具有标量旗曲率的情况。定义了翘曲积流形的双翘曲Sasaki-Matsumoto度量，并给出了水平和垂直切丛全测地的条件。同时，我们还得到了一个叶状流形约化为Reinhart流形的条件。最后，我们研究了双曲积Finsler流形的狭缝切丛上的一个殆复结构。
On doubly warped product submanifolds of Generalized $(κ,μ)$-Space Forms
本文建立了一个涉及黎曼流形中等距浸入的双弯曲积的弯曲函数的Laplacian与平方中曲率的一般不等式。此外，我们还得到了广义$(κ,μ)$ -空间型的C-全实双曲积子流形的一些几何不等式。
Gradient solitons on doubly warped product manifolds
该研究给出双重warped乘积流形的新刻画，分析其上多类梯度孤立子对因子流形的影响，并探究其共圆平坦与共谐平坦情形，丰富了相关微分几何研究。
Biharmonic maps between doubly warped product manifolds
本文研究了双曲积流形之间的双调和映射。证明了黎曼流形$B$和$F$到双曲积$_{f}B\times_{b}F$中的包含映射不可能是真的双调和映射.并分别分析了投影$_{f}B\times_{b}F\to B$和$_{f}B\times_{b}F\to F$的双调和性条件。利用调和映射与翘曲度量的乘积，给出了非调和双调和映射的一些刻划。特别地，在$f=1$的情况下，得到了\cite{Balmus-mont}中翘曲积的结果。
A Note on Doubly Warped Product Contact CR-Submanifolds in trans-Sasakian Manifolds
直到2001年，在B.Y.尘紧接着，另一条研究路线，类似于关于佐佐木几何的奇维版本的凯勒几何，被开发出来，即翘曲的产品接触CR-子流形在佐木流形。本文证明了trans-Sasakian流形中不存在真的双曲积切触CR-子流形。
Semi-Riemannian manifolds with a doubly warped structure
我们调查流形作为一个商的双翘曲产品。我们表明，他们总是覆盖的产品的两个合适的叶。这使我们能够证明，在正则性假设下，这些流形是一个双重翘曲产品的零措施由工会的叶子形成的子集。我们还得到了一个确保整个流形分解的充要条件，并利用它给出了几何性质的充分条件。最后，研究了非单连通情形下的直积分解的唯一性。
In this paper, we give an affirmative answer to Gromov's conjecture ([3, Conjecture E]) by establishing an optimal Lipschitz lower bound for a class of smooth functions on orientable open $3$-manifolds with uniformly positive sectional curvatures. For rigidity we show that the universal covering of the given manifold must be $\mathbf R^2\times (-c,c)$ with some doubly warped product metric if the optimal bound is attained. This gives a characterization for doubly warped product metrics with positive constant curvature. As a corollary, we also obtain a focal radius estimate for immersed toruses…
Doubly Warped Product Finsler Manifolds with Some Non-Riemannian Curvature Properties
本文考虑了具有某些非黎曼曲率性质的双曲积Finsler流形。首先，我们研究了Berwald流形和迷向平均Berwald DWP-Finsler流形。然后证明了每一个真Douglas DWP-Finsler流形都是黎曼流形。我们证明了一个适当的DWP-流形是Landsbergian当且仅当它是Berwaldian。接着，我们证明了每一个相对迷向的Landsberg DWP-流形都是Landsberg流形。证明了相对迷向平均Landsberg曲积流形是弱Landsberg流形。最后证明了不存在任何局部对偶平坦的真DWP-Finsler流形。
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.
Conformal Warped Product Submersion
本文将黎曼扭曲乘积淹没推广为共形扭曲乘积淹没，该淹没保持水平向量间角度，并推导了相应的基本张量。
Doubly warped product Einstein metrics on spheres
本文提出一种计算机辅助方法构造SO(d1+1)×SO(d2+1)-不变的协齐次一爱因斯坦度量，恢复了S^10和S^12上的已知度量，并在S^11、S^12、S^13和S^7×S^3上发现了新度量。
We give a characterization {\sl à la Obata} for certain families of K\''ahler manifolds. These results are in the same line as other extensions of the well-known Obata's rigidity theorem from \cite{Obata62}, like for instance the generalizations in \cite{RanjSant97} and \cite{Santhanam07}. Moreover, we give a complete description of the so-called K\''ahler doubly-warped product structures whose underlying metric is Einstein.
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.
Osserman and conformally Osserman manifolds with warped and twisted product structure
利用翘曲积的局部结构刻画了Osserman流形和共形Osserman黎曼流形.通过这种方法，我们分析了扭曲的产品结构，并获得，作为一个结果，唯一的Osserman流形，可以写为扭曲的产品是那些常曲率。
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.
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…
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.
Flat 3-manifolds with diagonal metrics and applications to warped products
本文研究了具有对角度量的3维子流形的平坦条件，并探讨了扭曲积类型流形的平坦性及其不存在结果。
A. Gray presented an interesting $O\left( n\right) $ 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 $M_{i},i=1,2$ of a doubly warped product manifold $M=_{f_{2}}M_{1}\times _{f_{1}}M_{2}$ 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…
Conformally-Kähler Ricci solitons and base metrics for warped product Ricci solitons
研究了与梯度Ricci孤子共形的Kähler度量，以及弯曲乘积梯度Ricci孤子的基度量.后者我们称之为准孤子。采用的一个主要假设是孤子势的函数相关，在第一种情况下，与保形因子，在第二种情况下与翘曲函数。第一种情况下的主要结果是在维度$n\geq 4$中的部分分类。在第二种情况下，满足上述主要假设的Kähler准孤子度量，在一个额外的一般性假设下，必然是黎曼积。另一个定理涉及满足上述主要假设的准孤子度量，它们也是共形的Kähler。在一些附加的假设下，证明了这些度量必然是爱因斯坦翘曲积的基度量，即准爱因斯坦度量。
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.
Spherically symmetric spacetimes are ambient spaces for models of stellar collapse and inhomogeneous cosmology. We obtain results for the Weyl tensor and the covariant form of the Ricci tensor on general doubly warped (DW) spacetimes. In a spherically symmetric metric, the Ricci and electric tensors become rank-2, built with a velocity vector field and its acceleration. Their structure dictates the general form of the energy-momentum tensor in the Einstein equations in DW spherical metrics. The anisotropic pressure and the heat current of an imperfect fluid descend from the gradient of the acc…
On the geometry of Riemannian warped product maps
本文引入Clairaut黎曼扭曲乘积映射，基于Ricci曲率建立正则曲线成为测地线的条件，利用曲率张量研究Ricci soliton结构，推导Bochner型公式与标量曲率积分公式，并扩展至conformal映射的几何性质，构造了相关实例。
Higher Fundamental Forms and Warped Product Hypersurfaces
本文利用高阶基本形式刻画乘积流形及共形流形中超曲面嵌入的几何，证明高阶共形基本形式在刻画中起关键作用。
Conformal Deformation of Warped Products and Scalar Curvature Functions on Open Manifolds
讨论了某些开流形上的共形形变和翘曲积。我们将讨论如何将这些可以应用到构建黎曼度量与特定的标量曲率函数。
On warped product generalized Roter type manifold
广义Roter型流形是共形平坦流形和Roter型流形的推广，它给出了曲率张量由基本度量张量和Ricci张量的代数组合到第二层的形式.本文的目的是研究翘曲积流形是广义Roter型的刻画。我们还给出了一个广义Roter型但不是Roter型的翘积流形的例子，以及一个Roter型但不是共形平坦的翘积流形的例子。这些例子确保了这些概念的正确存在。
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.
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…
Conformally weighted Einstein manifolds: the uniqueness problem
本文研究同一加权共形类中具有两个加权爱因斯坦代表的光滑度量测度空间，通过爱因斯坦与拟爱因斯坦扭曲乘积给出局部刻画和完备情形的全局分类，并证明紧致时必为加权球面。
Conformal de Rham decomposition of Riemannian manifolds
我们证明了黎曼流形作为黎曼流形或黎曼流形的翘曲积的de Rham和Hiepko局部分解定理的共形形式。也就是说，我们给出了黎曼流形局部共形于黎曼流形或翘曲积的充要条件。我们还得到了其他相关的de Rham型分解定理。作为应用，我们研究了处处具有两个不同特征值的Codazzi张量的黎曼流形。
A family of warped product semi-Riemannian Einstein metrics
研究了半黎曼爱因斯坦流形的曲积。我们考虑的情况下，该基地是共形的一个n维伪欧几里得空间和不变的作用下的一个平移群。我们提供了所有这样的解决方案的情况下，里奇平坦的基础是共形的n维伪欧几里得空间，不变的作用下的平移群和纤维F是里奇平坦的。特别是，我们得到明确的解决方案，在真空的情况下，爱因斯坦场方程。
The eta invariant in the doubly Kählerian conformally compact Einstein case
在紧4-流形的3-流形上，从一个共形紧化为Kähler度量的完备爱因斯坦度量导出一个共形结构。公式推导出这种共形结构的η不变的额外的假设下。一个这样的假设是，凯勒度规允许一个特殊的凯勒-里奇势在这个意义上定义的Derdzinski和Maschler。另一种观点是，凯勒度规是阿波斯托洛夫、卡尔德班克和高杜雄定义的任意结构的一部分，也是复曲面结构的一部分。公式推导使用Duistermaat-Heckman定理。这一结果是密切相关的早期工作希钦的爱因斯坦自对偶的情况下。
Non-collapsing of fibers in warped-product Poincare--Einstein manifolds
该研究针对扭曲乘积庞加莱-爱因斯坦流形，证明了共形无穷远具正 Yamabe 常数时纤维度量的非坍缩下界，推导了相关估计，给出显式界并验证假设的必要性。
Multiply warped products with harmonic Weyl tensor and zero radial Weyl curvature
本文研究具有调和Weyl张量和零径向Weyl曲率的多重扭曲积，证明其至多有三纤维，并利用共形变换和相容性方程刻画局部几何，所得度量源于两类构造，且界是精确的。
Curvature properties of some class of warped product manifolds
具有p维基（p= 1，2）的曲积流形满足某些伪对称型曲率条件.这些条件由所考虑流形的度量张量g、Riemann-Christoffel曲率张量R、Ricci张量S和Weyl共形曲率C构成。本文的主要结果是：若p=2，纤维是常曲率的半黎曼常曲率空间，若n ≥ 4，则这类翘曲积的（0，6）-张量R，R-Q（S，R）和C，C与（0，6）-张量Q（g，C），张量C由张量g和S的Kulkarni-Nomizu乘积的线性组合表示。因此，这些曲率条件满足非共形平坦的非爱因斯坦弯曲积时空（p=2，n=4）。我们还研究了拟爱因斯坦流形的伪对称型的曲率性质。特别地，我们得到了Goedel时空的一些曲率性质.
In this note we characterize 1+n doubly twisted spacetimes in terms of `doubly torqued' vector fields. They extend Bang-Yen Chen's characterization of twisted and generalized Robertson-Walker spacetimes with torqued and concircular vector fields. The result is a simple classification of 1+n doubly-twisted, doubly-warped, twisted and generalized Robertson-Walker spacetimes.
We study pseudo-Riemannian Einstein manifolds which are conformally equivalent with a metric product of two pseudo-Riemannian manifolds. Particularly interesting is the case where one of these manifolds is 1-dimensional and the case where the conformal factor depends on both manifolds simultaneously. If both factors are at least 3-dimensional then the latter case reduces to the product of two Einstein spaces, each of the special type admitting a non-trivial conformal gradient field. These are completely classified. If each factor is 2-dimensional, there is a special family of examples of non-c…
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 ($λ>0$).
Ricci solitons on singly warped product manifolds and applications
本文研究Ricci孤立子扭曲积流形对其底流形和纤维流形的蕴含，证明因子也是Ricci孤立子，并探讨共形或并发向量场及扭曲积时空上的Ricci孤立子。
Einstein metrics on conformal products
本文研究两个紧致共形流形共形乘积上的爱因斯坦度量，证明在自然几何假设下此类度量必为扭曲乘积度量。
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.
The simple structure of doubly torqued vectors allows for a natural characterization of doubly twisted down to warped spacetimes, as well as Kundt spacetimes down to PP waves. For the first ones the vectors are timelike, for the others they are null. We also discuss some properties, and their connection to hypersurface orthogonal conformal Killing vectors, and null Killing vectors.
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…
Curvature in Special Base Conformal Warped Products
引入了两个伪黎曼流形的基共形曲积的概念。我们还定义了一个子类的这种结构称为一个特殊的基础共形翘曲产品。之后，我们明确提到了许多相关领域，其中这些形式的度量以及对其曲率相关属性的考虑发挥了重要作用。除此之外，我们引用广义相对论，额外的维度，弦和超引力理论作为物理主题，也研究了拉普拉斯-贝尔特拉米算子在全局分析中的p-形式的频谱。然后，我们给出了基共形翘曲积的Ricci张量和数量曲率分别用基和纤维的Ricci张量和数量曲率表示的表达式。此外，我们引入特定的家庭，无论是标量或张量，非线性微分算子伪黎曼流形验证的具体身份。后者使我们能够获得新的有趣的表达式的Ricci张量和数量曲率的一个特殊的基础共形翘曲产品和事实证明，不仅是表达式，而且使用的分析方法是有趣的从物理，几何和分析的角度来看。最后，我们分析，调查和表征可能的解决方案的共形和翘曲因子的一个特殊的基础共形翘曲产品，这保证了相应的产品是爱因斯坦。此外，我们将这些结果应用到史瓦西度规的推广。
Harmonic morphisms are maps between Riemannian manifolds that pull back harmonic functions to harmonic functions. These maps are characterized as horizontally weakly conformal harmonic maps and they have many interesting links and applications to several areas in mathematics (see the book by Baird and Wood for details). In this paper, we study generalized harmonic morphisms which are defined to be maps between Riemannian manifolds that pull back harmonic functions to biharmonic functions. We obtain some characterizations of generalized harmonic morphisms into a Euclidean space and give two met…
Scalar-mean rigidity beyond warped product spaces
本文提出了一类具有边界的黎曼流形的标量-均值极值和刚性定理，扩展了传统扭曲积空间的范围，并引入了新的刚性流形家族。
Scalar curvature rigidity of domains in a warped product
本研究利用扭曲乘积度量与直积度量的共形性，构建扭旋量丛新联络及关联Dirac算子，得到扭曲乘积一般区域的Llarull型标量曲率刚性，并在角度匹配下研究双曲空间Gromov二面体刚性。
Conformally Kähler base metrics for Einstein warped products
一个具有Ricci曲率$\wht{\ri}$的黎曼度量$\wht{g}$称为非平凡拟Einstein，在Case，Shu和Wei的意义下，如果它满足$(-a/f)\wht{\nab} df+\wht{\ri}=λ\wht{g}$，对于一个光滑的非常数函数$f$和常数$λ$和$a>0$ .如果$a$是一个正整数，根据Kim和Kim的结果，这样的度量形成了某些扭曲的爱因斯坦度量的基础。在实维数至少为6的流形$M$上，设$(g,\t)$是由局部Kähler不可约的Kähler度量$g$和非常数Killing势$\t$组成的对。假设度量$\wht{g}=g/\t^2$是$M\setminus\t^{-1}(0)$上的非平凡的\bee，并且相关联的函数$f$是$\t$的局部函数。那么$(g,\t)$是一个\sk \ pair，一个由Derdzinski和Maschler定义的概念。这意味着$M$是双全纯的$CP^1$丛的全空间中的开集，其基流形允许Kähler-Einstein度量。如果$M$是额外紧的，它是这样一个丛或复射影空间的全空间。此外，函数$f$在具有非零常数的$\t^{-1}$中是仿射的。相反，在所有偶数维$n\geq 4$中，存在\sk对$(g,\t)$和相应的非零常数$K$和$L$，其中$g/\t^2$是具有$f=K\t^{-1}+L$的非平凡准爱因斯坦。此外，一个结果的情况…
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.
G2-structure deformations and warped products
我们概述了七维流形上G2-结构的非无穷小变形的性质，特别是，专注于G2的七维表示中的变形，从而由一个向量定义。然后，我们考虑从G2-结构的扭转类有一维和七维组件（所谓的共形近平行G2流形）的G2-结构只有一个一维的扭转组件（近平行G2流形）的变形。我们发现，这样的结构之间的变形存在当且仅当度量是一个特定的翘曲产品度量。
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…
On the classification of warped product Einstein metrics
本文采用Case-Shu-Wei提出的观点，通过底空间的里奇曲率方程来研究翘积Einstein度量。他们称这个方程为准爱因斯坦方程，但我们也称之为$m$ -爱因斯坦方程.本文推广了Case-Shu-Wei的工作和Kim-Kim的一些早期工作，使基具有非空边界。这是一个自然的情况，因为一个没有边界的流形经常出现作为一个扭曲的产品在一个带边流形的边界，在这种情况下，我们得到了一些有趣的新的典型的例子。我们还导出了一些新的公式，这些公式涉及类似于梯度Ricci孤子的曲率。作为应用，我们刻画了当基底局部共形平坦时的翘曲积爱因斯坦度量。
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