Volodymyr Kiosak, Vladimir S. Matveev
We prove an important partial case of the pseudo-Riemannian version of the projective Lichnerowicz conjecture stating that a complete manifold admitting an essential group of projective transformations is the round sphere (up to a finite cover).
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We construct the first known examples of compact pseudo-Riemannian manifolds having an essential group of conformal transformations, and which are not conformally flat. Our examples cover all types $(p,q)$, with $2 \leq p \leq q$.
The original version of the paper claimed to disprove the pseudo-Riemannian Lichnerowicz conjecture of D'Ambra and Gromov. However, the argument contains a crucial sign error in the lines following equation (8).
Projective Lichnerowicz-Obata conjecture
我们解决了两个经典的代数证明，如果一个连通的李群的作用在一个完整的黎曼流形保持测地线（被认为是非参数化曲线），那么度量有常数正截面曲率，或集团的行为通过仿射变换。
We prove a lower bound for the first eigenvalue of the sub-Laplacian on sub-Riemannian manifolds with transverse symmetries. When the manifold is of H-type, we obtain a corresponding rigidity result: If the optimal lower bound for the first eigenvalue is reached, then the manifold is equivalent to a 1 or a 3-Sasakian sphere.
A variational analysis of Einstein-scalar field Lichnerowicz equations on compact Riemannian manifolds
建立了紧流形上Einstein-标量场Lichnerowicz方程正解的存在性和不存在性的新结果。该方程源于广义相对论中爱因斯坦-标量场系统的哈密顿约束方程。我们的分析介绍变分技术，在山路引理的形式，分析的哈密顿约束方程，这是以前研究过的其他方法。
We report on some aspects and recent progress in certain problems in the sub-Riemannian CR and quaternionic contact (QC) geometries. The focus are the corresponding Yamabe problems on the round spheres, the Lichnerowicz-Obata first eigenvalue estimates, and the relation between these two problems. A motivation from the Riemannian case highlights new and old ideas which are then developed in the settings of Iwasawa sub-Riemannian geometries.
For a second order operator on a compact manifold satisfying the strong Hörmander condition, we give a bound for the spectral gap analogous to the Lichnerowicz estimate for the Laplacian of a Riemannian manifold. We consider a wide class of such operators which includes horizontal lifts of the Laplacian on Riemannian submersions with minimal leaves.
Relationship between sectional curvature and null spaces of Lichnerowicz-type Laplacians and their smallest eigenvalues
该研究修正了首版错误，证明截面曲率严格正且Ricci曲率适当夹逼的黎曼流形第二类曲率算子严格正，还得到三类拉普拉斯算子零空间的消没定理及最小特征值估计。
The title is self-explanatory. We aim to give an easy to read and self-contained introduction to the field of harmonic manifolds. Only basic knowledge of Riemannian geometry is required. After we gave the definition of harmonicity and derived some properties, we concentrate on Z. I. Szabó's proof of Lichnerowicz's conjecture in the class of compact simply connected manifolds.
Finite-energy solutions to Einstein-scalar field Lichnerowicz equations on complete Riemannian manifolds
本文研究了完备黎曼流形上爱因斯坦-标量场利钦诺维奇方程的有限能量解的存在性，通过正则化问题、山峰点论证和Harnack不等式，证明了在特定条件下非负解和正解的存在性。
Finsler conformal Lichnerowicz-Obata conjecture
我们证明了共形Lichnerowicz-Obata猜想的Finsler类比，证明了非黎曼Finsler流形上的完备本质共形向量场是Minkowski度量的位似向量场.
Supersymmetry and the generalized Lichnerowicz formula
Lichnerowicz [8]的一个经典结果涉及黎曼流形$M$上Clifford模${\cal E}$ \上Clifford联络定义的Dirac算子的平方的分解.最近，这个公式被推广到任意Dirac算子[2]。在本文中，我们证明了一个超对称版本的广义Lichnerowicz公式，动机的事实，有一个一对一的对应关系Clifford超连接和Dirac算子。我们推广这个结果，得到一个简单的公式的超曲率的广义Bismut超连接。这可以被看作是证明任意狄拉克算子族的局部指数定理的第一步。
Conformal quotients of plane waves, and Lichnerowicz conjecture in a locally homogeneous setting
本文研究平面波上的共形群作用，证明存在非等距的真余紧相似作用，并在局部齐性背景下证明了洛伦兹共形 Lichnerowicz 猜想。
On the normal holonomy representation of spacelike submanifolds in pseudo-Riemannian space forms
本文研究了伪黎曼空间中类空子流形的正规联络的弱不可约完整表示。我们将屏蔽表示与弱不可约正规和乐群联系起来，并对具有Borel-Lichnérowicz性质的屏蔽表示进行了分类.特别是，我们得到一个分类的洛伦兹正常holonomy表示。
Pseudo-Conformal actions of semisimple Lie groups
本文研究半单李群在伪黎曼流形上的共形作用，证明此类流形在特定条件下为共形平坦。
In the present work, we introduce a linear connection (preserving the almost product structure and the Riemannian metric) on Riemannian almost product manifolds. This connection, called P-connection, is an analogue of the first canonical connection of Lichnerowicz in the Hermitian geometry and the B-connection in the geometry of the almost complex manifolds with Norden metric. Particularly, we consider the P-connection on a the class of manifolds with nonintegrable almost product structure.
On the Stability of Riemannian Manifold with Parallel Spinors
受物理学家Hertog-Horowitz-Maeda最近工作的启发，我们证明了具有非零平行旋量的紧致黎曼流形的两个稳定性结果。我们的第一个结果说，Ricci平坦的度量，也承认非零平行旋量是稳定的（在共形结构的变化方向）作为临界点的总数量曲率功能。事实上，我们表明，Lichnerowicz拉普拉斯算子，它支配的二阶变分，是扭曲狄拉克算子的平方。我们的第二个结果，这是第一个的本地版本，表明任何度量的正数量曲率不能躺在太接近一个度量与非零平行旋量。我们还证明了一个刚性的特殊holonomy度量的结果。在$SU(m)$完整性情形下，刚性结果意味着Calabi-Yau度规的标量平坦形变必定是Calabi-Yau的.最后，我们探讨了与正质量定理的联系，这提供了另一种方法来证明这些稳定性和刚性的结果。
Contact Riemannian manifolds, with not necessarily integrable complex structures, are the generalization of pseudohermitian manifolds in CR geometry. The Tanaka-Webster-Tanno connection on such a manifold plays the role of Tanaka-Webster connection in the pseudohermitian case. We prove the contact Riemannian version of the pseudohermitian Bochner-type formula, and generalize the CR Lichnerowicz theorem about the sharp lower bound for the first nonzero eigenvalue of the sub-Laplacian to the contact Riemannnian case.
Inspired by the Lichnerowicz-Obata theorem for the first eigenvalue of the Laplacian, we define a new family of invariants $\{Ω_k(g)\}$ for closed Riemannian manifolds. The value of $Ω_k(g)$ delicately reflects the spherical part of the manifold. Indeed, $Ω_1(g)$ and $Ω_2(g)$ characterize the standard sphere.
Remark on a conjecture of conformal transformations of Riemannian manifolds
Ejiri否定了Lichnerowicz关于数量曲率为常数的黎曼流形的一个猜想，并给出了一个无穷小的非等距保角变换。有了这个目标，他建造了一个扭曲的产品圈的长度$T$和一个紧凑的流形。但他在分析中忽略了$T$必须足够大的条件。在这里，我们给出了一个明确的明确的界限$T_0 < T$，这将使证明完整。我们的演示是独立的，主要使用分叉技术。此外，我们还证明了还有其他这样的例子，并对这些流形的分类做出了一些贡献。
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in dimension three. The proof has parallels with both the Schoen-Yau minimal hypersurface technique and Witten's spinorial approach. In particular, the role of harmonic spinors and the Lichnerowicz formula in Witten's argument is replaced by that of harmonic functions and a formula introduced by the fourth named author in recent work, while the level sets of harmoni…
Two pseudo-Riemannian metrics are called projectively equivalent if their unparametrized geodesics coincide. The degree of mobility of a metric is the dimension of the space of metrics that are projectively equivalent to it. We give a complete list of possible values for the degree of mobility of Riemannian and Lorentzian Einstein metrics on simply connected manifolds, and describe all possible dimensions of the space of essential projective vector fields.
About curvature, conformal metrics and warped products
我们考虑两个伪黎曼流形$(B,g_B)$和$(F,g_F)$的翘积族的曲率，这两个流形具有$c^{2}g_B \oplus w^2 g_F$形式的度量，特别是$w^{2 μ}g_B \oplus w^2 g_F$类型的度量，其中$c, w \colon B \to (0,\infty)$是光滑函数，$μ$是实参数.我们得到合适的表达式的Ricci张量和数量曲率的产品，使我们能够建立爱因斯坦或常数数量曲率结构的存在，在这些类别的结果。如果$(B,g_B)$是黎曼的，则后一个问题涉及具有凹凸非线性的非线性椭圆偏微分方程和Lichnerowicz-York型的奇异偏微分方程等。
The degree of mobility of a (pseudo-Riemannian) Kähler metric is the dimension of the space of metrics h-projectively equivalent to it. We prove that a metric on a closed connected manifold can not have the degree of mobility $\ge 3$ unless it is essentially the Fubini-Study metric, or the h-projective equivalence is actually the affine equivalence. As the main application we prove an important special case of the classical conjecture attributed to Obata and Yano, stating that a closed manifold admitting an essential group of h-projective transformations is $(CP(n), g_{Fubini-Study})$ (up to a…
In this paper we prove local existence of a Ricci de Turck flow starting at a space with incomplete edge singularities and flowing for a short time within a class of incomplete edge manifolds. We derive regularity properties for the corresponding family of Riemannian metrics and discuss boundedness of the Ricci curvature along the flow. For Riemannian metrics that are sufficiently close to a flat incomplete edge metric, we prove long time existence of the Ricci de Turck flow. Under certain conditions, our results yield existence of Ricci flow on spaces with incomplete edge singularities. The p…
Degree of mobility of a (pseudo-Riemannian) metric is the dimension of the space of metrics geodesically equivalent to it. We describe all possible values of the degree of mobility on a simply connected n-dimensional manifold of lorentz signature. As an application we calculate all possible differences between the dimension of the projective and the isometry groups. One of the main new technical results in the proof is the description of all parallel symmetric (0,2)-tensor fields on cone manifolds of signature $(n-1,2).
For a closed, spin, odd dimensional Riemannian manifold $(Y,g)$, we define the rho invariant $ρ_{spin}(Y,E,H, g)$ for the twisted Dirac operator $D^E_H$ on $Y$, acting on sections of a flat hermitian vector bundle $E$ over $Y$, where $H = \sum i^{j+1} H_{2j+1}$ is an odd-degree closed differential form on $Y$ and $H_{2j+1}$ is a real-valued differential form of degree ${2j+1}$. We prove that it only depends on the conformal class of the pair $[H,g]$. In the special case when $H$ is a closed 3-form, we use a Lichnerowicz-Weitzenbock formula for the square of the twisted Dirac operator, to show …
Given a probability measure $μ$ supported on a convex subset $Ω$ of Euclidean space $(\mathbb{R}^d,g_0)$, we are interested in obtaining Poincaré and log-Sobolev type inequalities on $(Ω,g_0,μ)$. To this end, we change the metric $g_0$ to a more general Riemannian one $g$, adapted in a certain sense to $μ$, and perform our analysis on $(Ω,g,μ)$. The types of metrics we consider are Hessian metrics (intimately related to associated optimal-transport problems), product metrics (which are very useful when $μ$ is unconditional, i.e. invariant under reflection with respect to the principle hyperpla…
On the eigenvalues of the Laplacian on fibred manifolds
本文研究了纤维流形上拉普拉斯算子特征值的比较定理，推广了李钦罗维奇和法伯-克拉恩不等式，并证明了在里奇曲率下界条件下陈定理的对应结果。
Two Kaehler metrics on one complex manifold are said to be c-projectively equivalent if their J-planar curves, i.e., curves defined by the property that their acceleration is complex proportional to their velocity, coincide. The degree of mobility of a Kaehler metric is the dimension of the space of metrics that are c-projectively equivalent to it. We give the list of all possible values of the degree of mobility of simply connected 2n-dimensional Riemannian Kaehler manifolds. We also describe all such values under the additional assumption that the metric is Einstein. As an application, we de…
Marked Poincar{é} rigidity near hyperbolic metrics
本文研究了双曲度量附近带标记的庞加莱刚性问题，并证明了在三维中测地流的不稳定体积膨胀与利钦尼奇拉普拉斯算子的单射性之间的关系。
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