COMPLETE KAHLER METRICS WITH POSITIVE HOLOMORPHIC SECTIONAL CURVATURES ON CERTAIN LINE BUNDLES(RELATED TO A COHOMOGENEITY ONE POINT OF VIEW ON A YAU CONJECTURE)

In this article,we study Kahler metrics on a certain line bundle over some compact Kahler manifolds to find complete Kahler metrics with positive holomorphic sectional(or bisectional)curvatures.Thus,we apply a strategy to a famous Yau conjecture with a co-homogeneity one geometry.

RANDERS SPACES WITH SCALAR FLAG CURVATURE

Let(M, F) be an n-dimensional Randers space with scalar flag curvature. In this paper, we will introduce the definition of a weak Einstein manifold. We can prove that if(M, F) is a weak Einstein manifold, then the flag curvature is constant.

ON HOLOMORPHIC CURVES OF CONSTANT CURVATURE IN THE COMPLEX GRASSMANN MANIFOLD G(2,5)

In this article, it is proved that there doesn’t exist any nonsingular holomorphic sphere in complex Grassmann manifold G（2, 5） with constant curvature k = 4/7, 1/2, 4/9. Thus, from [7] it follows that if φ ： S2 → G（2, 5） is a nonsingular holomorphic curve with constant curvature K, then, K = 4, 2, 4/3, 1 or 4/5.

ON COMPLETE SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE IN NEGATIVE PINCHED MANIFOLDS

A rigidity theorem for oriented complete submanifolds with parallel mean curvature in a complete and simply connected Riemannian （n ＋ p）-dimensional manifold N^n＋p with negative sectional curvature is proved. For given positive integers n（≥ 2）, p and for a constant H satisfying H 〉 1 there exists a negative number τ（n,p, H） ∈ （-1, 0） with the property that if the sectional curvature of N is pinched in [-1, τ-（n,p, H）], and if the squared length of the second fundamental form is in a certain interval, then N^n＋p is isometric to the hyperbolic space H^n＋P（-1）. As a consequence, this submanifold M is congruent to S^n（1√H^2 - 1） or the Veronese surface in S^4（1/√H^2-1）.

Open Manifolds with Nonnegative Ricci Curvature and Large Volume Growth

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

A STABILITY RESULT FOR TRANSLATINGSPACELIKE GRAPHS IN LORENTZ MANIFOLDS

In this paper,we investigate spacelike graphs defined over a domain Ω⊂M^(n) in the Lorentz manifold M^(n)×ℝ with the metric−ds^(2)+σ,where M^(n) is a complete Riemannian n-manifold with the metricσ,Ωhas piecewise smooth boundary,and ℝ denotes the Euclidean 1-space.We prove an interesting stability result for translating spacelike graphs in M^(n)×ℝ under a conformal transformation.

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.

Characterizations of Null Holomorphic Sectional Curvature of GCR-Lightlike Submanifolds of Indefinite Nearly Khler Manifolds

We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite nearly Kahler manifold and obtain characterization theorems for holo- morphic sectional and holomorphic bisectional curvature. We also establish a condi- tion for a GCR-lightlike submanifold of an indefinite complex space form to be a null holomorphically fiat.

Adaptive Neighboring Selection Algorithm Based on Curvature Prediction in Manifold Learning

Recently manifold learning algorithm for dimensionality reduction attracts more and more interests, and various linear and nonlinear,global and local algorithms are proposed. The key step of manifold learning algorithm is the neighboring region selection. However,so far for the references we know,few of which propose a generally accepted algorithm to well select the neighboring region. So in this paper,we propose an adaptive neighboring selection algorithm,which successfully applies the LLE and ISOMAP algorithms in the test. It is an algorithm that can find the optimal K nearest neighbors of the data points on the manifold. And the theoretical basis of the algorithm is the approximated curvature of the data point on the manifold. Based on Riemann Geometry,Jacob matrix is a proper mathematical concept to predict the approximated curvature. By verifying the proposed algorithm on embedding Swiss roll from R3 to R2 based on LLE and ISOMAP algorithm,the simulation results show that the proposed adaptive neighboring selection algorithm is feasible and able to find the optimal value of K,making the residual variance relatively small and better visualization of the results. By quantitative analysis,the embedding quality measured by residual variance is increased 45. 45% after using the proposed algorithm in LLE.

A Note on Gap Phenomena of K＆#168;ahler Manifolds with Nonnegative Curvature

In this paper, we study the complex structure and curvature decay of Khler manifolds with nonnegative curvature. Using a recent result obtained by Ni-Shi-Tam, we get a gap theorem of Ricci curvature on Khler manifold.

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.

The Manifolds with Ricci Curvature Decay to Zero

The paper quotes the concept of Ricci curvature decay to zero. Base on this new concept, by modifying the proof of the canonical Cheeger-Gromoll Splitting Theorem, the paper proves that for a complete non-compact Riemannian manifold M with Ricci curvature decay to zero, if there is a line in M, then the isometrically splitting M = R × N is true.

Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.

Ricci Curvature of Certain Submanifolds in Kenmotsu Space Forms

In this paper, we obtain some sharp inequalities between the Ricci cur- vature and the squared mean curvature for bi-slant and semi-slant submanifolds in Kenmotsu space forms. Estimates of the scalar curvature and the k-Ricci curvature, in terms of the squared mean curvature, are also proved respectively.

RIGIDITY OF COMPACT SURFACES IN HOMOGENEOUS 3-MANIFOLDS WITH CONSTANT MEAN CURVATURE

In this paper, we establish a rigidity theorem for compact constant mean curva- ture surfaces of the Berger sphere in terms of the surfaces＇ geometric invariants. This extends the previous similar result on compact minimal surfaces of the Berger sphere.

Small Excess and the Topology of Open Manifolds with Ricci Curvature Negatively Lower Bounded

在这份报纸，我们由使用比较几何学的方法学习在开的 manifolds 和他们的拓扑学的过量之间的关系。我们证明弯曲否定地降低的与 Ricci 歧管的完全的开的 Riemmannian 跳了如果 conjugate 半径被围住从，具有有限拓扑的类型被它的 conjugate 半径的某功能被一个积极常数和它的过量下面围住，它改进一些结果在[4 ] 。

WEYL CURVATURE OF A FINSLER SPACE

The Weyl curvature of a Finsler metric is investigated.This curvature constructed from Riemannain curvature.It is an important projective invariant of Finsler metrics.The author gives the necessary conditions on Weyl curvature for a Finsler metric to be Randers metric and presents examples of Randers metrics with non-scalar curvature.A global rigidity theorem for compact Finsler manifolds satisfying such conditions is proved.It is showed that for such a Finsler manifold,if Ricci scalar is negative,then Finsler metric is of Randers type.

正曲率齐性Finsler空间的分类:偶数维情形下的一种新方法(英文)

本文介绍了正曲率齐性Finsler流形的分类。在偶数维的情形下,给出了一种新方法,证明了偶数维光滑陪集空间上有正曲率齐性Finsler度量,当且仅当其上面有正曲率齐性黎曼度量。

The holomorphic d-scalar curvature on almost Hermitian manifolds

In this paper,we study the existence of conformal metrics with the constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed,connected almost Hermitian manifolds of dimension n>6.In addition,we obtain an application and a variational formula for the associated conformal invariant.

SOME RESULTS ON PRODUCT COMPLEX FINSLER MANIFOLDS

Let （M, F ） be the product complex Finsler manifold of two strongly pseudoconvex complex Finsler manifolds （M 1 , F 1 ） and （M 2 , F 2 ）. In this paper, we obtain the relationship between the Chern Finsler connection coefficients Γ i ; k associated to F and the Chern Finsler connection coefficients Γ a ; c , Γα ; γ associated to F 1 , F 2 , respectively. As applications we prove that, if both （M 1 , F 1 ） and （M 2 , F 2 ） are strongly Ka¨hler Finsler （complex Berwald, or locally complex Minkowski, respectively） manifolds, so does （M, F ）. Furthermore, we prove that the holomorphic curvature K F = 0 if and only if K F1 = 0 and K F2 = 0.