A generalization of 4-dimensional locally conformal flat Walker manifolds

A 4-dimensional Walker metrics with c = 0 on a semi-Riemannian manifold M have been investigated by E. García-Río and Y.Matsushita. The case c=constant has been studied in [1]. In this paper we generalize these notions to the case of non-constant c. We find the form of the defining functions that makes this manifold similar to locally conformal flat 4-dimensional Walker manifold.     

共形平坦黎曼流形上的Schouten张量

M是一个紧致的局部共形平坦黎曼流形,其上定义的Schouten张量是一个Codazzi张量.本文借助这个Codazzi张量引入Cheng和Yau的自伴算子,从而获得了局部共形平坦流形上的一些性质,改进了已有的结论.     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

本文研究局部对称共形平坦黎曼流形N^n p(p≥2）中具有平行平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了最佳拼挤常数。     

On conformally flat almost Hermitian manifolds

We study some of 2n-dimensional conformally flat almost Hermitian manifolds with J-(anti)-invariant Ricci tensor. Received 13 May 2000; revised 15 February 2001.     

A fully nonlinear version of the Yamabe problem on locally conformally flat manifolds with umbilic boundary

We prove existence and compactness of solutions to a fully nonlinear Yamabe problem on locally conformally flat Riemannian manifolds with umbilic boundary.     

4-dimensional anti-Kähler manifolds

We show that every 4-dimensional anti-Kähler manifold is Einstein and locally symmetric. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.     

共形平坦的Randers度量

本文研究共形平坦的Randers 度量的性质. 证明了具有数量旗曲率的共形平坦的Randers 度量都是局部射影平坦的, 并且给出了这类度量的分类结果. 本文还证明了不存在非平凡的共形平坦且具有近迷向S 曲率的Randers 度量.     

4-dimensional anti-Kähler manifolds and Weyl curvature

On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.     

On quasi-conformally flat weakly Ricci symmetric manifolds

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

Yang-mills equations in 4-dimensional conformally connected manifolds

In this paper on the simplest examples of compact 4-dimensional conformally connected manifolds (real quadrics in a 5-dimensional projective space) we show that the only invariant, which is quadratic with respect to the curvature Φ of the connectivity, is the Yang-Mills functional ε |tr (*Φ Λ Φ)|. We do not know whether the 4-form |tr (*Φ Λ Φ)| is invariant in any 4-dimensional conformally connected manifolds.     

The Huber theorem for non-compact conformally flat manifolds

It was proved in 1957 by Huber that any complete surface with integrable Gauss curvature is conformally equivalent to a compact surface with a finite number of points removed. Counterexamples show that the curvature assumption must necessarily be strengthened in order to get an analogous conclusion in higher dimensions. We show in this paper that any non compact Riemannian manifold with finite -norm of the Ricci curvature satisfies Huber-type conclusions if either it is a conformal domain with volume growth controlled from above in a compact Riemannian manifold or if it is conformally flat of dimension 4 and a natural Sobolev inequality together with a mild scalar curvature decay assumption hold. We also get partial results in other dimensions. Received: April 14, 2000; revised version: March 20, 2001     

A sphere theorem on locally conformally flat even-dimensional manifolds

In this paper, we prove that a closed even-dimensional manifold which is locally conformally flat with positive scalar curvature, positive Euler characteristic and which satisfies some additional condition on its curvature is diffeomorphic to the sphere or projective space.     

Convex-cocompactness of Kleinian groups and conformally flat manifolds with positive scalar curvature

We give a sufficient condition for a higher dimensional Kleinian group to be convex cocompact in terms of the critical exponent of . As a consequence, we see that the fundamental group of a compact conformally flat manifold with positive scalar curvature is hyperbolic in the sense of Gromov. We give some other applications to geometry and topology of conformally flat manifolds with positive scalar curvature.

     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

研究局部对称共形平坦黎曼流形N^n＋p（p≥2）中具有平等平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了量佳拼挤常数．     

Schouten curvature functions on locally conformally flat Riemannian manifolds

Consider a compact Riemannian manifold (M, g) with metric g and dimension n ≥ 3. The Schouten tensor A _g associated with g is a symmetric (0, 2)-tensor field describing the non-conformally-invariant part of the curvature tensor of g. In this paper, we consider the elementary symmetric functions {σ _k (A _g ), 1 ≤ k ≤ n} of the eigenvalues of A _g with respect to g; we call σ _k (A _g ) the k-th Schouten curvature function. We give an isometric classification for compact locally conformally flat manifolds which satisfy the conditions: A _g is semi-positive definite and σ _k (A _g ) is a nonzero constant for some k ∈ {2, ... , n}. If k = 2, we obtain a classification result under the weaker conditions that σ₂(A _g ) is a non-negative constant and (M ⁿ , g) has nonnegative Ricci curvature. The corresponding result for the case k = 1 is well known. We also give an isometric classification for complete locally conformally flat manifolds with constant scalar curvature and non-negative Ricci curvature. Udo Simon: Partially supported by Chinese-German cooperation projects, DFG PI 158/4-4 and PI 158/4-5, and NSFC.     

局部对称共形平坦黎曼流形中的紧致子流形

本文研究局部对称共形平坦黎曼流形中具有平行平均曲率向量的紧致子流形的性质，通过一个代数不等式的证明，改进了已有的结果．     

Projective Blaschke manifolds

In this paper we define the concept of projective Blaschke manifolds and extend the theory of equiaffine differential geometry to the projective Blaschke manifolds. partially supported by NSFC 10771146 and RFDP(20060610004)     

A note on complete manifolds with finite volume

In this article, we concern on complete manifolds with finite volume. We prove that under some assumptions about scalar curvature and the Yamabe constant, the manifolds must be compact, and we also give the diameter estimates in terms of the scalar curvature and the Yamabe constant.     

Harmonic Riemannian maps on locally conformal Kaehler manifolds

We study harmonic Riemannian maps on locally conformal Kaehler manifolds (lcK manifolds). We show that if a Riemannian holomorphic map between lcK manifolds is harmonic, then the Lee vector field of the domain belongs to the kernel of the Riemannian map under a condition. When the domain is Kaehler, we prove that a Riemannian holomorphic map is harmonic if and only if the lcK manifold is Kaehler. Then we find similar results for Riemannian maps between lcK manifolds and Sasakian manifolds. Finally, we check the constancy of some maps between almost complex (or almost contact) manifolds and almost product manifolds.