Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds

We completely classify three-dimensional homogeneous Lorentzian manifolds, equipped with Einstein-like metrics. Similarly to the Riemannian case (E. Abbena et al., Simon Stevin Quart J Pure Appl Math 66:173–182, 1992), if (M, g) is a three-dimensional homogeneous Lorentzian manifold, the Ricci tensor of (M, g) being cyclic-parallel (respectively, a Codazzi tensor) is related to natural reductivity (respectively, symmetry) of (M, g). However, some exceptional examples arise. The author is supported by funds of MURST, GNSAGA and the University of Lecce.     

李奇曲率平行的黎曼流形的孤立现象

本文研究李奇曲率平行的封闭黎曼流形，证明了黎曼曲率平方的一个拚挤定理。     

Three-Dimensional Lorentz Metrics and Curvature Homogeneity of Order One

In this paper we investigate the existence of three-dimensional Lorentzian manifolds which are curvature homogeneous up to order one but which are not locally homogeneous, and we obtain a complete local classification of these spaces. As a corollary we determine, for each Segre type of the Ricci curvature tensor, the smallest k N for which curvature homogeneity up to order k guarantees local homogeneity of the three-dimensional manifold under consideration.     

M-Eigenvalues of the Riemann Curvature Tensor of Conformally Flat Manifolds

We investigate the M-eigenvalues of the Riemann curvature tensor in the higher dimensional conformally flat manifold. The expressions of Meigenvalues and M-eigenvectors are presented in this paper. As a special case,M-eigenvalues of conformal flat Einstein manifold have also been discussed,and the conformal the invariance of M-eigentriple has been found. We also reveal the relationship between M-eigenvalue and sectional curvature of a Riemannian manifold. We prove that the M-eigenvalue can determine the Riemann curvature tensor uniquely. We also give an example to compute the Meigentriple of de Sitter spacetime which is well-known in general relativity.     

满足$\varphi l=l\varphi$的3-维仿切触度量流形的分类

Let $M^{3}$ be a 3-dimensional paracontact metric manifold. Firstly, a classification of $M^{3}$ satisfying $\varphi Q=Q\varphi$ is given. Secondly, manifold $M^{3}$ satisfying $\varphi l=l\varphi$ and having $\eta$-parallel Ricci tensor or cyclic $\eta$-parallel Ricci tensor is studied.     

Complete Three-Dimensional Manifolds with Non-Negative Ricci Curvature and Quadratic Volume Growth

The paper proves that every three-manifold with non-negative Ricci curvature and quadratic volume growth in strict sense must be contractible provided that its universal covering is finite.AMS Subject Classification (2000) 53C20     

Logarithmic Gradient of the Heat Kernel on Riemannian Manifolds

Mathematical Notes -     

Classification of 4-dimensional homogeneous D’Atri spaces

The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold (M, g) satisfying the first odd Ledger condition is said to be of type . The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers by Podesta-Spiro and Bueken-Vanhecke (which are mutually complementary). The authors started with the corresponding classification of all spaces of type , but this classification was incomplete. Here we present the complete classification of all homogeneous spaces of type in a simple and explicit form and, as a consequence, we prove correctly that all homogeneous 4-dimensional D’Atri spaces are locally naturally reductive. The first author’s work has been partially supported by D.G.I. (Spain) and FEDER Project MTM 2004-06015-C02-01, by a grant AVCiTGRUPOS03/169 and by a Research Grant from Ministerio de Educación y Cultura. The second author’s work has been supported by the grant GA ČR 201/05/2707 and it is part of the research project MSM 0021620839 financed by the Ministry of Education (MŠMT).     

具有调和曲率与有限$L^p$曲率的完备流形

Let(M~n, g)(n ≥ 3) be an n-dimensional complete Riemannian manifold with harmonic curvature and positive Yamabe constant. Denote by R and R?m the scalar curvature and the trace-free Riemannian curvature tensor of M, respectively. The main result of this paper states that R?m goes to zero uniformly at infinity if for p ≥ n, the L~p-norm of R?m is finite.As applications, we prove that(M~n, g) is compact if the L~p-norm of R?m is finite and R is positive, and(M~n, g) is scalar flat if(M~n, g) is a complete noncompact manifold with nonnegative scalar curvature and finite L~p-norm of R?m. We prove that(M~n, g) is isometric to a spherical space form if for p ≥n/2, the L~p-norm of R?m is sufficiently small and R is positive.In particular, we prove that(M~n, g) is isometric to a spherical space form if for p ≥ n, R is positive and the L~p-norm of R?m is pinched in [0, C), where C is an explicit positive constant depending only on n, p, R and the Yamabe constant.     

~2Ric=0的Riemann流形的刚性定理(英文)

本文用Ｒｉｃ表示里奇曲率张量，研究了２Ｒｉｃ＝０的黎曼流形什么时候成为爱因斯坦流形或空间形式     

▽2Ric=0的Riemann流形的刚性定理

本文用Ｒｉｃ表示里奇曲率张量，研究了▽²Ｒｉｃ＝０的黎曼流形什么时候成为爱因斯坦流形或空间形式     

具有非负Ricci曲率的且满足大体积增长条件的开流形

In this paper, we prove that if M is an open manifold with nonnegative Ricci curvature and large volume growth, positive critical radius, then sup Cp=∞.p∈M As an application, we give a theorem which supports strongly Petersen‘s conjecture.     

On quasi-conformally flat weakly Ricci symmetric manifolds

The object of the present paper is to study quasi-conformally flat weakly Ricci symmetric manifolds.      

具有两个不同Ricci主曲率的局部共形平坦Riemann流形

得到了具有常m阶Schouten曲率与两个不同Schouten主曲率(或者等价地,两个不同Ricci主曲率)的完备局部共形平坦Riemann流形的分类结果.作为应用,得到了若干Schouten张量的pinching性质.     

Complete Open Manifolds with Nonnegative Ricci Curvature

In this paper, we study complete open manifolds with nonnegative Ricci curvature and injectivity radius bounded from below. We find that this kind of manifolds are diffeomorphic to a Euclidean space when certain distance functions satisfy a reasonable condition.     

A Class of Homogeneous Einstein Manifolds

A Riemannian manifold (M, g) is called Einstein manifold if its Ricci tensor satisfies r = c·g for some constant c. General existence results are hard to obtain, e.g., it is as yet unknown whether every compact manifold admits an Einstein metric. A natural approach is to impose additional homogeneous assumptions. M. Y. Wang and W. Ziller have got some results on compact homogeneous space G/H. They investigate standard homogeneous metrics, the metric induced by Killing form on G/H, and get some classification results. In this paper some more general homogeneous metrics on some homogeneous space G/H are studies, and a necessary and sufficient condition for this metric to be Einstein is given. The authors also give some examples of Einstein manifolds with non-standard homogeneous metrics.     

拟常曲率流形上的二阶平行张量

本文证明了如下结果：（1）在一个拟常曲率流形M上，[0,2]型平行张量是度量张量的常数倍。（2）在拟常曲率流形M上，不存在非零平行2－形式。除非对应于M的生成元的Ricci主曲率等于零。     

Homogeneity for Surfaces in Four-Dimensional Vector Space Geometry

We classify all surfaces in R ⁴ which are homogeneous in the sense of equi-centroaffine differential geometry. There result 21 group classes, some of them depending on one or two real parameters. The classification is cleared up, i.e. each copy is equivalent to exactly one representative. This applies as well to the corresponding groups as to the orbits (and also to the parameter cases). In particular, we can characterize the Clifford tori in a purely affine manner and determine all homogeneous centroaffine spheres. This answers a former question on the existence of centroaffine spheres which are not contained in a hyperplane. The classification and, in particular, the uniqueness is based on geometric insight and is essentially not computer dependent. The leading ideas are of a general nature and may also be applied to homogeneity for higher-dimensional cases and for related geometries.     

On the Existence of Homogeneous Geodesics in Homogeneous Riemannian Manifolds

It is proved that every homogeneous Riemannian manifold admits a geodesic which is an orbit of a one-parameter group of isometries.