复射影空间中具有常数量曲率的实超曲面

将球面上常数量曲率超曲面推广到复射影空间中,得到此类实超曲面的某些积分不等式.     

Yamabe不变量与在非紧流形上预定纯量曲率

我们得到关于Ｙａｍａｂｅ不变量的某些条件,使得非紧流形通过共形形变后,纯量曲率为常数．     

具有非负Ricci曲率和大体积增长的开流形

In this paper,we prove that a complete n-dimensional Riemannian manifold with n0nnegative kth-Ricci curvature,large volume growth has finite topological type provided that lim{((vol[B(p,r))]/(ω_nr~n)-αM)r(k(n-1))/(k 1)(1-α/2)}＜=εfor some constantε＞0.We also prove that a complete Riemannian manifold with nonnegative kth-Ricci curvature and under some pinching conditions is diffeomorphic to R~n.     

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     

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。     

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.     

Convexity of Domains of Riemannian Manifolds

In this paper the problem of the geodesic connectedness and convexity ofincomplete Riemannian manifolds is analyzed. To this aim, a detailedstudy of the notion of convexity for the associated Cauchy boundary iscarried out. In particular, under widely discussed hypotheses,we prove the convexity of open domains (whose boundaries may benondifferentiable) of a complete Riemannian manifold. Variationalmethods are mainly used. Examples and applications are provided,including a result for dynamical systems on the existence oftrajectories with fixed energy.     

Volume Comparison Theorems for Lorentzian Manifolds

Using Riccati equation techniques and the Raychaudhuri equation from General Relativity, volume comparison results are obtained for compact geodesic wedges in the chronological future of some point in a globally hyperbolic space-time and corresponding wedges in a Lorentzian space-form.     

Geodesic $\gamma$-Pre-$E$-Convex Functions on Riemannian Manifolds

In this paper, we generalize geodesic $E$-convex function and define geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions on Riemannian manifolds. The sufficient condition of equivalence class of geodesic $\gamma$-pre-$E$-convexity and geodesic $\gamma$-$E$-convexity for differentiable function on Riemannian manifolds is studied. We discuss the sufficient condition for $E$-epigraph to be geodesic $E$-convex set. At the end, we establish some optimality results with the aid of geodesic $\gamma$-pre-$E$-convex and geodesic $\gamma$-$E$-convex functions and discuss the mean value inequality for geodesic $\gamma$-pre-$E$-convex function.     

Projective Curvature and Integral Invariants

In this paper, an extension of all Lie group actions on R ²to coordinates defined by potentials is given. This provides a new solution to the equivalence problems of curves under the projective group and two of its subgroups. The potentials correspond to integrals of higher and higher-order producing an infinite number of independent integral invariants. Applications to computer vision are discussed.     

Ricci曲率平行的一类Riemann流形的C∞紧性

本文证明,在Gromov-Hausdorff拓扑下,Ricci曲率平行,截面曲率和单一半径有下界,体积有上界的Riemann流形的集合是c^∞紧的.作为应用,我们证明一个pinching结果,即在某些条件下,Ricci平坦的流形必定平坦.     

VOLUME GROWTH ESTIMATES OF MANIFOLDS WITH NONNEGATIVE CURVATURE OUTSIDE A COMPACT SET

In this article, using the properties of Busemann functions, the authors prove that the order of volume growth of Kahler manifolds with certain nonnegative holomorphic bisectional curvature and sectional curvature is at least half of the real dimension. The authors also give a brief proof of a generalized Yau＇s theorem.     

Best Sobolev Constants and Manifolds with Positive Scalar Curvature Metrics

We study the Yamabe invariant of manifolds which admit metrics of positive scalar curvature. Analysing `best Sobolev constants'we give a technique to find positive lower bounds for the invariant.We apply these ideas to show that for any compact Riemannian manifold (N ⁿ ,g) of positive scalarcurvature there is a positive constant K =K(N, g), which depends only on (N, g), such that for any compact manifold M ^m , the Yamabe invariantof M ^m × N ⁿ is no less than K times the invariant ofS ^{n + m} . We will find some estimates for the constant K in the case N =S ⁿ .     

On Saddle Submanifolds of Riemannian Manifolds

Saddle submanifolds are considered. A characterization of such submanifolds of Euclidean space is given in terms of sectional curvature. Extending results of T. Frankel, K. Kenmotsu and C. Xia, we determine under what conditions two complete saddle submanifolds of a complete connected Riemannian manifold M, with nonnegative k-Ricci curvature, must intersect. Moreover, if M has positive k -Ricci curvature and the dimension of a compact saddle submanifold satisfies a certain inequality then we show that the homomorphism of the fundamental groups _1(M) and ₁(M) is surjective.     

On Einstein Manifolds of Positive Sectional Curvature

Let (M, g) be a compact oriented four-dimensional Einstein manifold. If M has positive intersection form and g has non-negative sectional curvature, we show that, up to rescaling and isometry, (M, g) is ₂, with its standard Fubini–Study metric.     

On Projective Maps of Volume-Decreasing

ＯｎＰｒｏｊｅｃｔｉｖｅＭａｐｓｏｆＶｏｌｕｍｅ－Ｄｅｃｒｅａｓｉｎｇ￥ＣａｉＫａｉｒｅｎ（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＨａｎｇｚｈｏｕＴｅａｃｈｅｒ＇ｓＣｏｌｌｅｇｅ）Ａｂｓｔｒａｃｔ：ＧｉｖｅｎａｐｒｏｊｅｃｔｉｖｅｍａｐＦ：...     

关于复射影空间中具有平行平均曲率向量的全实叶的注记(英文)

We discussed a totally real Riemannian foliations with parallel mean curvature on a complex projective space.We carried out the divergence of a vector field on it and obtained a formula of Simons’type.     

f-Harmonic Morphisms Between Riemannian Manifolds

f-Harmonic maps were first introduced and studied by Lichnerowicz in 1970. In this paper, the author studies a subclass of f-harmonic maps called f-harmonic morphisms which pull back local harmonic functions to local f-harmonic functions. The author proves that a map between Riemannian manifolds is an f-harmonic morphism if and only if it is a horizontally weakly conformal f-harmonic map. This generalizes the well-known characterization for harmonic morphisms. Some properties and many examples as well as some non-existence of f-harmonic morphisms are given. The author also studies the f-harmonicity of conformal immersions.     

Asymptotic Formulae for Brillouin Index on Riemannian Manifolds

We establish several asymptotic formulae for Brillouin index on fiat tori. As an application of these formulae it is proved that the topological entropy of a geodesic flow on a fiat torus is zero.     

非负曲率流形的体积增长估计

本文利用非负曲率流形上的Busemann函数和穷竭函数的性质,得出了在某紧致子集外满足一定非负曲率条件的完备非紧的(复) n维K■hler流形的体积增长至少是n次的．推广了陈兵龙和朱熹平教授新近的一个结果．