射影Ricci平坦的Kropina度量（英文）

本文研究和刻画了射影Ricci平坦的Kropina度量.利用Kropina度量的S-曲率和Ricci曲率的公式,得到了Kropina度量的射影Ricci曲率公式.在此基础上得到了Kropina度量是射影Ricci平坦度量的充分必要条件.进一步,作为自然的应用,本文研究和刻画了由一个黎曼度量和一个具有常数长度的Killing 1-形式定义的射影Ricci平坦的Kropina度量,也刻画了具有迷向S-曲率的射影Ricci平坦的Kropina度量.在这种情形下,Kropina度量是Ricci平坦度量.     

关于一类弱Berwald的(α,β)-度量

本文研究了一类重要的形如F=α +εβ +β arctan(β/α) (ε为常数)的弱Berwald (α, β)-度量.利用S-曲率公式,获得了这类度量为弱Berwald度量的充要条件.并且还证明了F为具有标量旗曲率的弱Berwald度量当且仅当它们为Berwald度量且旗曲率消失.     

关于射影平坦Finsler空间

本文研究了射影平坦Finsler空间的几何量及其几何性质。证明了射影平坦Finsler空间的Ricci曲率可完全由射影因子简洁地刻画出来。同时还证明了，在射影平坦Finsler空间中，平均Berwald曲率S=0意味着Ricci曲率Ric是二次齐次的。此外，给出了一个射影平坦Finsler空间成为常曲率空间或局部Minkowski空间的充分条件。     

芬斯勒射影几何中的Ricci曲率

本文研究了保持Ricci曲率不变的Finsler射影变换。给定一个紧致无边的n维可微流形M，证明了：对于一个从M上的Berwald度量到Riemann度量的C-射影变换，如果Berwald度量的Ricci曲率关于Riemann度量的迹不超过Riemann度量的标量曲率，则该射影变换是平凡的。     

关于广义(α,β)-度量的若干Ricci曲率性质

本文研究了广义(α,β)-度量的Ricci曲率和Ricci曲率张量.首先,在一定条件下,本文给出了强Einstein广义(α,β)-度量的一个等价刻画.进一步,得到了广义(α,β)-度量是Ricci-齐次Finsler度量的一个充分必要条件.     

关于共形Berwald的Kropina度量(英文)

本文利用导航数据研究了共形Berwald的Kropina度量.首先利用导航数据刻画了Berwald Kropina度量.在此基础上,本文得到了Kropina度量是共形Berwald度量的一个充分必要条件.进一步,刻画了具有弱迷向旗曲率的共形Berwald Kropina度量的局部结构.     

一类射影平坦的Finsler度量

本文主要研究由两个Riemann度量和一个1-形式构成的Finsler度量.首先,本文给出这类度量局部射影平坦的等价条件;其次,给出这类度量局部射影平坦且具有常旗曲率的分类情形;最后,构造这类度量局部射影平坦且具有常旗曲率K=-1的例子.     

标量曲率Finsler空间与Finsler度量的射影变换

本文研究了与一个Ricci平坦Finsler空间或一个常曲率Finsler空间射影相关的标量曲率Finsler空间．我们给出了这种标量曲率Finsler空间成为常曲率空间的充分必要条件.特别地，我们给出了射影平坦Finsler空间具有常曲率的条件．     

一些球对称射影平坦的Finsler度量的构造

研究刻画球对称Finsler度量的射影平坦性质的偏微分方程,通过对射影平坦Finsler度量PDE的研究,构造了两类球对称射影平坦Finsler度量,得到了一些球对称的射影平坦Finsler度量,并进一步给出这些Finsler度量的射影因子和旗曲率.     

一些射影平坦Finsler度量的构造

通过研究刻画Finsler度量的射影平坦性质的偏微分方程组,得到了一些有用的解, 进一步证明了其中的一些度量还具有零旗曲率.     

General spherically symmetric Finsler metrics with constant Ricci and flag curvature

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.     

On a class of projectively Ricci-flat Finsler metrics

Every Finsler metric induces a spray on a manifold. With a volume form on a manifold, every spray can be deformed to a projective spray. The Ricci curvature of a projective spray is called the projective Ricci curvature. The projective Ricci curvature is an important projective invariant in Finsler geometry. In this paper, we study and characterize projectively Ricci-flat square metrics. Moreover, we construct some nontrivial examples on such Finsler metrics.     

On the Ricci curvature of a Randers metric of isotropic <Emphasis Type="Italic">S</Emphasis>-curvature

We derive the integral inequality of a Randers metric with isotropic S-curvature in terms of its navigation representation. Using the obtained inequality we give some rigidity results under the condition of Ricci curvature. In particular, we show the following result： Assume that an n-dimensional compact Randers manifold （M, F） has constant S-curvature c. Then （M, F） must be Riemannian if its Ricci curvature satisfies that Ric 〈 -（n - 1）c^2.     

Ricci-positive metrics on connected sums of projective spaces

It is a well known result of Gromov that all manifolds of a given dimension with positive sectional curvature are subject to a universal bound on the sum of their Betti numbers. On the other hand, there is no such bound for manifolds with positive Ricci curvature: indeed, Perelman constructed Ricci positive metrics on arbitrary connected sums of complex projective planes. In this paper, we revisit and extend Perelman's techniques to construct Ricci positive metrics on arbitrary connected sums of complex, quaternionic, and octonionic projective spaces in every dimension.     

Finsler metrics with special curvature properties

In this paper, we discuss the relationship between the flag curvature and some non-Riemannian quantities of Finsler metrics of scalar curvature. In particular, we characterize projectively flat Finsler metrics with isotropic S-curvature.     

Nonparallelism of the Ricci Tensor of Pseudo-cylindric Metrics

We describe examples of metrics in the conformal class [g] on some conformally flat Riemannian manifolds (M,g]. These metrics have a constant scalar curvature and an harmonic curvature with nonparallel Ricci tensor.     

Ricci flow on a 3-manifold with positive scalar curvature

In this paper we consider Hamilton's Ricci flow on a 3-manifold with a metric of positive scalar curvature. We establish several a priori estimates for the Ricci flow which we believe are important in understanding possible singularities of the Ricci flow. For Ricci flow with initial metric of positive scalar curvature, we obtain a sharp estimate on the norm of the Ricci curvature in terms of the scalar curvature (which is not trivial even if the initial metric has non-negative Ricci curvature, a fact which is essential in Hamilton's estimates [R.S. Hamilton, Three-manifolds with positive Ricci curvature, J. Differential Geom. 17 (1982) 255-306]), some L²-estimates for the gradients of the Ricci curvature, and finally the Harnack type estimates for the Ricci curvature. These results are established through careful (and rather complicated and lengthy) computations, integration by parts and the maximum principles for parabolic equations.