On the Existence of Certain Metrics with Non Parallel Ricci Tensor
A. Raouf Chouikha
A. Derdzinki [D]给出了紧致流形$(M,g]$上具有调和曲率和非平行Ricci张量的黎曼度量的例子.我们研究它们的存在以及它们的数量自然取决于几何形状的流形。
A. Derdzinki [D] gave examples of Riemannian metrics with harmonic curvature and non parallel Ricci tensor on some compact manifolds $(M,g]$ . We examine their existence as well as their number wich naturally depends on the geometry of the manifolds.
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Non Parallelism of Ricci tensor of Pseudo-cylindric Metrics
我们描述了完备共形平坦黎曼流形$(M,g].$上共形类$[g]$中度量的例子。这些度量具有常数数量曲率和具有非平行Ricci张量的调和曲率。
Algebraic Classification of the Ricci Curvature Tensor and Spinor for Neutral Signature in Four Dimensions
我申请的代数分类的自伴自同态${\bf R}^{2,2}$提供的若尔当典范形的里奇曲率张量的四维中性流形和相关的代数分类的里奇曲率旋量。这些结果平行类似的结果众所周知，在四维洛伦兹几何。该分类总结在本文末尾的表2中。
Ricci Tensors with Rotational Symmetry on R^n
本文考虑微分方程Ric（g）=T，其中Ric（g）是度量g的Ricci张量，T是R^n上的旋转对称张量.本文基于隐式微分方程的定性理论，给出了该方程光滑解存在性的一个新的几何证明。这个结果是DeTurck和Cao在1994年得到的。
We define the notion of the Ricci tensor for NQ symplectic manifolds of degree 2 and show that it corresponds to the standard generalized Ricci tensor on Courant algebroids. We use an appropriate notion of connections compatible with the generalized metric on the graded manifold.
Ricci flow for homogeneous compact models of the universe
利用四元数，我们给出了具有三维球面拓扑的齐性空间的Ricci张量的一个简明的推导。我们得到明确的和数值解的Ricci流偏微分方程，并讨论其性质。在这些模型的坍缩（或膨胀）中，研究了Ricci张量的各个分量的相互作用。我们将这篇论文献给Josh Goldberg的工作。
We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmonic if and only if it is annihilated by Chen's vector, and any of the two conditions is necessary and sufficient for the GRW space-time to be a quasi-Einstein (perfect fluid) manifold. Finally, the general structure of the Riemann tensor for Robertson-Walker space-times is given, i…
We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete classification in dimension three.
Parallel *-Ricci tensor of real hypersurfaces in CP^2 and CH^2
证明了在非平坦复空间型中不存在具有平行 *-Ricci张量的三维实超曲面，最后对 *-Ricci张量的进一步研究提出了一些设想.
The Long-Time Behavior of the Ricci Tensor Under The Ricci Flow
我们证明了，给定曲率和直径一致有界的闭流形上的里奇流的不朽解，当t趋于无穷大时，里奇张量趋于零。我们还证明了，如果在一个闭的三维流形上存在一个不朽解，使得直径的平方与曲率张量的范数的乘积是一致有界的，那么这个解一定是III型的.
Improved Chen-Ricci inequality for curvature-like tensors and its applications
给出了类曲率张量的Chen-Ricci不等式和改进的Chen-Ricci不等式。应用我们改进的Chen-Ricci不等式研究了复空间型的Lagrange和Kaehler斜子流形以及Sasakian空间型的C-全实子流形。
Curvature tensor under the Ricci-Harmonic flow
证明了若里奇曲率在Ricci-Harmonic流下对所有时间$t$ \in [0，T）都是一致有界的，则曲率张量也必须是一致有界的.
The purpose of this paper is to study *-Ricci tensor on Sasakian manifold. Here, ϕ-confomally flat and confomally flat *-η-Einstein Sasakian manifold are studied. Next, we consider *-Ricci symmetric conditon on Sasakian manifold. Finally, we study a special type of metric called *-Ricci soliton on Sasakian manifold.
Estimating the trace-free Ricci tensor in Ricci flow
在四维及四维以上空间中分析里奇流奇异性形成时，一个重要而自然的问题是：控制黎曼曲率张量范数的最弱条件是什么？在这个简短的说明中，我们表明，在一个紧凑的流形上，无迹的里奇张量是控制在一个精确的方式由其他组件的不可约分解的曲率张量，没有任何假设的初始数据。
In this paper, we prove some compactness theorems of Myers, Ambrose, and Galloway for complete Riemannian manifold in the concept of $h$-almost Ricci tensors and generalized quasi-Einstein tensors. Also, we extend the previous theorems when $h$ has at most linear growth in the distance function.
Some Geometric Properties of the Bakry-Emery-Ricci Tensor
Bakry-Emery张量给出了具有光滑测度的黎曼流形的Ricci张量的类似物。我们表明，具有正或非负的Ricci张量的拓扑后果的一些也是有效的Bakry-Emery张量。我们表明，Bakry-Emery张量是非递减的黎曼淹没下，其纤维运输保持措施常数。给出了Bakry-Emery张量与Gromov-Hausdorff极限之间的关系。
Almost Kähler 4-dimensional Lie groups with $J$-invariant Ricci tensor
本文的目的是确定4维李群$(g, J, Ω)$上严格左不变的几乎Kähler结构，使得Ricci张量是$J$不变的。
Real hypersurfaces in the complex quadric with commuting Ricci tensor
在复二次曲面$Q^m = SO_{m+2}/SO_mSO_2$中引入实超曲面的交换Ricci张量的概念。证明了交换Ricci张量使单位法向量场$N$成为$\frak A$ -主场或$\frak A$ -各向同性场.然后根据每种情况，给出了具有交换Ricci张量的$Q^m = SO_{m+2}/SO_mSO_2$中实超曲面的完全分类.
Generalized Tanaka-Webster Parallel Ricci tensor in complex two-plane Grassmannians
证明了复两平面Grassmannian中Ricci张量平行于广义Tanaka-Webster联络的Hopf实超曲面的不存在性。
The classification of simple Jacobi--Ricci commuting algebraic curvature tensors
我们对代数曲率张量进行分类，使得Ricci算子是简单的（即Ricci算子是复对角化的，并且复谱由单个实特征值组成，或者复谱由一对相互复共轭的特征值组成），并且是Jacobi-Ricci交换的（即Ricci算子与任何向量的Jacobi算子交换）。
On rigidity of gradient Kähler-Ricci solitons with harmonic Bochner tensor
本文证明了具有调和Bochner张量的完全梯度稳定Kähler-Ricci孤子必然是Kähler-Ricci平坦的，即Calabi-Yau，以及具有调和Bochner张量的完全梯度收缩（或扩张）Kähler-Ricci孤子必须等距于$N^k\times \mathbb{C}^{n-k}$的商，其中$N$是具有正（或负）数量曲率的Kähler-Einstein流形.
In this article, we introduce the notion of star-Ricci tensors in the real hypersurfaces of complex quadric $Q^m$. It is proved that there exist no Hopf hypersurfaces in $Q^m,m\geq3$, with commuting star-Ricci tensor or parallel star-Ricci tensor. As a generalization of star-Einstein metric, star-Ricci solitons on $M$ are considered. In this case we show that $M$ is an open part of a tube around a totally geodesic $\mathbb{C}P^\frac{m}{2}\subset Q^{m},m\geq4$.
This paper focuses on the study of three dimensional real hypersurfaces in non-flat complex space forms whose $^{*}$-Ricci tensor satisfies conditions of parallelism. More precisely, extension of existing results concerning real hypersurfaces with vanishing, semi-parallel and pseudo-parallel $^{*}$-Ricci tensor in case of ambient space being the complex hyperbolic space are provided. Furthermore, new results concerning $ξ$-parallelism of $^{*}$-Ricci tensor of real hypersurfaces in non-flat complex space forms are presented.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified, respectively, with different integral-differential formulas and semigroup inequalities. Moreover, by using derivative and Hessian formulas for the heat semigroup $P_t$ developed from stochastic analysis, explicit Hessian estimates are derived on Einstein and Ricci parallel manifolds.
We examine the moduli spaces of Type~A connections on oriented and unoriented surfaces both with and without torsion in relation to the signature of the associated symmetric Ricci tensor. If the signature of the symmetric Ricci tensor is (1,1) or (0,2), the moduli spaces are smooth. If the signature is (2,0), there is an orbifold singularity.
The Cotton tensor and the Ricci flow
本文计算了黎曼流形中里奇流下Cotton张量和Bach张量的发展方程，特别是三维情形，并讨论了一些应用.
Cotton tensor and conformal deformations of three-dimensional Ricci flow
研究了三维共形结构在里奇流作用下的变形。我们在里奇流下导出了Cotton-York张量的发展方程及其L1模。特别地，我们研究了三维单连通齐性流形上Cotton-York张量在里奇流下的L1-范数的性质。对于非齐次情形，我们还研究了两球面上里奇流的Rosenau解与圆的标准度量的乘积度量的Cotton-York张量的L1-范数的行为.
Evolution of the Weyl Tensor under the Ricci Flow
计算了黎曼流形中里奇流下的Weyl张量的发展方程，并讨论了局部共形平坦Ricci孤子分类的一些结果.
On Ricci-Tensor of Randers Metrics
本文研究了Randers度量，并给出了Randers度量的Ricci张量是Berwaldian的一个条件。这推广了沈的定理：在完备Randers度量下的每个R-°是局部Minkowski的。在此基础上，我们得到了Ricci张量的一个充要条件，在此条件下，标量°ag曲率的Randers度量为零°ag曲率.
Connections with skew-symmetric Ricci tensor on surfaces
将曲面上具有反对称Ricci张量的无挠联络的一些已知结果推广到有挠联络，并简化了这类联络的Wong正则坐标形式。
About the Injectivity Radius and the Ricci Tensor of a Complete Riemannian Manifold
本文得到了内射半径i（M）仅依赖于i（M）的完备黎曼流形的里奇曲率下确界的一个简单上界。在刚性的情况下，黎曼流形必须是一个欧氏球面（欧几里得空间），且内射半径是有限（无限）的。此外，在Ricci张量的二阶导数为零的假设下，证明了里奇曲率的下确界的上界也是上确界，并且$M^n$实际上具有平行的Ricci张量.
Eigenvalue estimates of the Dirac operator depending on the Ricci tensor
利用精化Weitzenböck技巧证明了紧致黎曼自旋流形上Dirac算子第一特征值的一个新的下界.它适用于具有调和曲率张量的流形，并依赖于Ricci张量。示例显示了它与其他已知边界相比的行为。
Almost Kahler 4-manifolds with J-invariant Ricci tensor and special Weyl tensor
研究了紧致4-辛流形上相容的殆复结构的可积性问题，在对相应的殆Kahler度量曲率的各种自然假设下.
Spectral comparison results for the $N$-Bakry-Emery Ricci tensor
本文研究了 $N$-Bakry-Emery Ricci 张量在谱意义下具有正下界时的几何比较问题，建立了直径和全局加权体积比较结果。
Stable weighted minimal surfaces in manifolds with nonnegative Bakry-Emery Ricci tensor
本文研究了流形中具有非负Bakry-Emery里奇曲率的稳定加权极小超曲面.我们将给出一些几何和拓扑应用。特别地，在$f$有界的假设下，给出了具有非负Bakry-Emery里奇曲率的完备三维流形的部分分类.
In this paper, we first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing the results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f'(t)$.
A relation between the Ricci tensor and the spectrum of the Dirac operator
使用Weitzenböck技术在任何紧黎曼自旋流形，我们得到一个一般不等式依赖于一个实参数，并加入狄拉克算子的频谱与依赖于Ricci张量和它的第一共变导数的条款。这个不等式的讨论产生消灭定理的狄拉克操作$D$的核和新的下界的$D^2$的频谱，如果Ricci张量满足一定的条件。
Evaluation of the ADM mass and center of mass via the Ricci tensor
我们不用密度定理直接证明了：（i）在渐近平坦流形上按通常方法定义的ADM质量等于用Ricci张量内在定义的质量;（ii）质心的哈密顿公式与用Ricci张量内在定义的质心是相同的.
Gradient Shrinking Sasaki-Ricci Solitons with Harmonic Weyl Tensor
本文研究完备梯度收缩Sasaki-Ricci孤立子，建立其积分曲率估计，并证明具有调和Weyl张量的此类孤立子必为球面的有限商，推广了Munteanu和Sesum关于Ricci孤立子的结果。
Let (M,g) be a complete noncompact riemannian manifold with bounded geometry and parallel Ricci curvature. We show that some operators, "affine" relatively to the Ricci curvature, are locally invertible, in some classical Sobolev spaces, near the metric g.
We show that any left invariant metric with harmonic curvature on a solvable Lie group is Ricci-parallel. We show the same result for any Lie group of dimension $\leq$ 6.
Let $(M,g)$ be a compact riemannian manifold without boundary., with parallel Rici curvature. We show that some operators, affine relatively to the Ricci curvature,are locally invertible, near the metric $g$
Real Hypersurfaces in complex two-plane Grassmannians with recurrent Ricci tensor
本文在复二平面Grassmannian算子$G_2({\mathbb C}^{m+2})$中引入了广义Tanaka-Webster Reeb递归Ricci张量的新概念.其次，在这样的条件下，我们给出了实超曲面$M$在$G_2({\mathbb C}^{m+2})$中的不存在性。
On DeTurck uniqueness theorems for Ricci tensor
黎曼几何中的里奇曲率问题是：给定一个光滑流形$M$和一个对称的2-张量$r$，在$M$上构造一个度量，其Ricci张量等于$r$。特别地，DeTurck和Koiso证明了以下著名的结果：里奇曲率唯一地决定任何具有非负截面曲率的紧致Einstein流形上的Levi-Civita联络。本文推广了DeTurck和Koiso关于具有非负截面曲率的黎曼流形的结果。此外，我们将我们的结果推广到具有非负截面曲率和有限总数量曲率的完全非紧黎曼流形。
Curvature tensor under the Ricci flow
考虑$t\in [0,T)$的未归一化里奇流$(g_{ij})_t = -2R_{ij}$，其中$T < \infty$。理查德汉密尔顿表明，如果曲率算子是一致有界的流动下的所有时间$t\in [0,T)$然后解决方案可以扩展到超越$T$。我们证明了，如果里奇曲率是一致有界的流动下的所有时间$t\in [0,T)$，那么曲率张量必须一致有界。
We show that a basis of a semisimple Lie algebra of compact type, for which any diagonal left-invariant metric has a diagonal Ricci tensor, is characterized by the Lie algebraic condition of being "nice". Namely, the bracket of any two basis elements is a multiple of another basis element. This extends the work of Lauret and Will on nilpotent Lie algebras. The result follows from a more general characterization for diagonalizing the Ricci tensor for homogeneous spaces. Finally, we also study the Ricci flow behavior of diagonal metrics on cohomogeneity one manifolds.
Curvature tensor under the complete non-compact Ricci Flow
证明了完备非紧黎曼流形上曲率有界的里奇流的解$(M^n,g(t))$，$t\in[0,T)$，其中$T<\infty$的里奇曲率张量被$M^n\times [0,T)$上的某个常数$C$一致有界，曲率张量在$M^n\times [0,T)$上保持一致有界.还介绍了其他一些结果。
In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci tensor and the potential function or its gradient are bounded. As applications, we prove diameter estimates and eigenvalue estimates on smooth metric measure spaces. These results not only give a supplement of the author's previous results under integral Bakry-Émery Ricci tensor bounds, but also are generalizations of the Wei-Wylie's pointwise results.
On the global structure of conformal gradient solitons with nonnegative Ricci tensor
本文证明了任何具有非负Ricci张量的完备共形梯度孤子要么等距于直积$\mathbb{R}\times N^{n-1}$，要么整体共形等价于欧几里得空间$\mathbb{R}^{n}$或球面$\mathbb{S}^{n}$ .特别地，我们证明了任何完整的，非紧的，梯度Yamabe型孤子与正Ricci张量是旋转对称的，只要势函数是非常数。
On Einstein-type manifold with cyclic parallel Ricci tensor
本文推导了涉及张量D_{ijk}的积分公式，用于分类满足循环平行瑞奇张量条件的三维紧致爱因斯坦型流形，扩展并统一了现有文献中的研究成果。
The Ricci tensor of a gradient Ricci soliton with harmonic Weyl tensor
本文为具有调和 Weyl 张量且维度 $n \geq 4$ 的梯度 Ricci 孤立子的 Ricci 张量至多有三个特征值这一定理提供了更简短且无需活动标架的新证明。
The rigity of spaces with cyclic parallel Ricci tensor
本文对满足循环平行Ricci张量的特殊黎曼流形分类，涵盖梯度收缩Ricci孤立子等结构，通过构造积分恒等式结合曲率条件证明其Ricci张量平行，明确这类空间的刚性性质。
Scalar-flat Kähler surfaces whose Weyl tensor annihilates the Ricci form
研究标量平坦凯勒表面中威利张量作用于2形式 annihilates 里奇形式的性质，证明其为里奇平坦或局部等距于两个曲率相反的实表面乘积。
We prove mean curvature and volume comparison estimates on smooth metric measure spaces when their integral Bakry-Émery Ricci tensor bounds, extending Wei-Wylie's comparison results to the integral case. We also apply comparison results to get diameter estimates, eigenvalue estimates and volume growth estimates on smooth metric measure spaces with their normalized integral smallness for Bakry-Émery Ricci tensor. These give generalizations of some work of Petersen-Wei, Aubry, Petersen-Sprouse, Yau and more.
Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators
我们研究自然产生的仿射和黎曼几何的代数结构的几何可实现性的问题。假设给定流形M的点P处的代数曲率算子R，并假设给定实解析（分别。C-k对于定义在P附近的M上的有限k（至少2）伪黎曼度量，我们构造了一个无挠的无挠的实解析（分别为. C-k）联络D，它定义在M的切丛上的P附近，其曲率算子是P处的给定算子R，使得D具有常数数量曲率.证明了若R是Ricci对称的，则D可被选为Ricci对称的;若R有迹自由Ricci张量，则D可被选为迹自由Ricci张量;若R是Ricci交错的，则D可被选为Ricci交错的.
We prove in a simple and coordinate-free way the equivalence bteween the classical definitions of the mass or the center of mass of an asymptotically flat manifold and their alternative definitions depending on the Ricci tensor and conformal Killing fields. This enables us to prove an analogous statement in the asymptotically hyperbolic case.
Direct proofs for spaces with cyclic parallel Ricci tensor
该研究为梯度里奇孤子、带边界紧致拟爱因斯坦流形及具循环平行里奇张量的临界空间的刚性结果提供直接证明，通过推导张量恒等式等方法得到相关刚性结论与里奇平行性结果。
Real hypersurfaces in complex two-plane Grassmannians with Reeb parallel Ricci tensor in generalized Tanaka-Webster connection
复二平面Grassmannian $G_2({\mathbb C}^{m+2})$中的实超曲面有几类分类问题。其中，Suh在Levi-Civita联络下用Reeb平行Ricci张量将Hopf超曲面$M$分类为$G_2({\mathbb C}^{m+2})$.本文在$G_2({\mathbb C}^{m+2})$中引入了$M$的广义Tanaka-Webster Reeb平行Ricci张量的概念.利用这种平行条件，我们给出了$G_2({\mathbb C}^{m+2})$中Hopf超曲面的完全分类.
Weyl structures with positive Ricci tensor
本文证明了当Ricci张量满足一定的正性条件时，紧流形上的第一贝蒂数为零，从而得到了Weyl几何中的一个Bochner型消灭定理.我们还研究了正则Weyl联络的Ricci张量具有非负对称部分的紧致Hermitian-Weyl流形，并证明了每一个这样的流形都有第一贝蒂数$b_1 =1$和Hodge数$h^{p,0} =0$（对于$p>0$），$h^{0,1} =1$，$h^{0,q} =0$（对于$q>1$）.
The Weyl Tensor of Gradient Ricci Solitons
本文导出了梯度Ricci孤子的Weyl张量的新恒等式，特别是在四维情形。首先，我们证明了自对偶Weyl张量的范数的Bochner-Weitzenböck型公式，并讨论了它的应用，包括几何和拓扑之间的联系。在第二部分中，我们考虑了黎曼曲率的不同分量与孤子势函数（梯度和Hessian）的相互作用。外尔张量自然出现在这些调查。这里的应用是刚性结果。
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