THE LOGARITHMIC SOBOLEV INEQUALITY FOR A SUBMANIFOLD IN MANIFOLDS WITH ASYMPTOTICALLY NONNEGATIVE SECTIONAL CURVATURE

In this note,we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature.Like the Michale-Simon Sobolev inequality,this inequality contains a term involving the mean curvature.

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.

2-harmonic Submanifotds-in a Quasi Constant Holomorphic Sectional Curvature Space

In the present paper, the authors study totally real 2-harmonic submanifolds in a quasi constant holomorphic sectional curvature space and obtain a Simons＇ type inte- gral inequality of compact submanifoids as well as some pinching theorems on＇the second fundamental form.

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and codimension.Like the Michael-Simon Sobolev inequality,this inequality includes a term involving the mean curvature.This extends a recent result of Brendle with Euclidean setting.

Chern-Ricci curvatures, holomorphic sectional curvature and Hermitian metrics

We present some formulae related to the Chern-Ricci curvatures and scalar curvatures of special Hermitian metrics.We prove that a compact locally conformal Kähler manifold with the constant nonpositive holomorphic sectional curvature is K?hler.We also give examples of complete non-Kähler metrics with pointwise negative constant but not globally constant holomorphic sectional curvature,and complete non-Kähler metrics with zero holomorphic sectional curvature and nonvanishing curvature tensors.

Closed Geodesics and Volume Growth of Open Manifolds with Sectional Curvature Bounded from Below

In this paper, the relationship between the existence of closed geodesics and the volume growth of complete noncompact Riemannian manifolds is studied. First the authors prove a diffeomorphic result of such an n-m2nifold with nonnegative sectional curvature, which improves Marenich-Toponogov＇s theorem. As an application, a rigidity theorem is obtained for nonnegatively curved open manifold which contains a clesed geodesic. Next the authors prove a theorem about the nonexistence of closed geodesics for Riemannian manifolds with sectional curvature bounded from below by a negative constant.

Pluriclosed Manifolds with Constant Holomorphic Sectional Curvature

A long-standing conjecture in complex geometry says that a compact Hermitian manifold with constant holomorphic sectional curvature must be Kèahler when the constant is non-zero and must be Chern flat when the constant is zero.The conjecture is known in complex dimension 2 by the work of Balas-Gauduchon in 1985(when the constant is zero or negative)and by Apostolov±Davidov±Muskarov in 1996(when the constant is positive).For higher dimensions,the conjecture is still largely unknown.In this article,we restrict ourselves to pluriclosed manifolds,and confirm the conjecture for the special case of Strominger Kèahler-like manifolds,namely,for Hermitian manifolds whose Strominger connection(also known as Bismut connection)obeys all the Kaèhler symmetries.

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.

GEOMETRIC RIGIDITY THEOREM FOR SUBMANIFOLDS WITH POSITIVE CURVATURE

A geometric rigidity theorem for submanifolds with parallel mean curvature and positive curvature in a space form is proved. It is a generalization of the famous rigidity theorems due to S. T. Yau and others.

A Theorem on Infinitesimal I-isometry of Surfaces Immersed in a Space with Constant Curvature

In this paper we discuss the infinitesimal I-isometric de formations of surfaces immersed in a space with constant curvature. We obtain a sufficient condition for the de formation vector field to be zero vector field which is generalization of the results in [1] and [2].

Study on the methods of precision controlling for the manufacture of hull curvature sections

The precision controlling technology is a key step for the modern ship construction, with the precision controlling of the ship-hull curvature as one of bottlenecks in shipbuilding, where the initial is to present a compensation value for the ship-hull plate precisely. The compensation value of the curvature plate is composed of two parts: the construction compensation, which results in the process of heating construction of curvature plate, and the assembling compensation, which results in welding ribbed stiffeners onto the curvature plate. Based on the developed computation system for the local contraction value, this paper presents a method to establish the experimented samples for the assembling compensation by means of numerical experiments, and another method to establish the practical mathematical model for the construction compensation of curvature plate. Furthermore, it introduces the experimental measuring method for the assembling compensation of the curvature plate, based on which the related database system has been developed. Numerical examples are analyzed to demonstrate the process to establish mathematical model for the assembling compensation values.

ON A KHLER VERSION OF CHEEGER-GROMOLL-PERELMAN'S SOUL THEOREM

In this note, we will prove a Kahler version of Cheeger-Gromoll-Perelman＇s soul theorem, only assuming the sectional curvature is nonnegative and bisectional curvature is positive at one point.

RIGIDITY OF COMPACT MINIMAL SUBMANIFOLDS IN A LOCALLY SYMMETRIC AND CONFORMALLY FLAT RIEMANN MANIFOLD

The compact minimal submanifold in a locally symetric and conformally flat Riemann manifold is discussed in this paper.We get the Pinching constant for scalar curvature.The result of Li[2]is generallied,but the method is completely different. Meanwhile,we get better conclusion than that of [3].We also research the Pinching problem for sectional curvature on compact minimal submanifolds in a unit sphere, partially improving the results of S.T.Yan[4].

ISOTROPIC WEYL MANIFOLD WITH A SEMI-SYMMETRIC CONNECTION

In this work,it is proved that every isotropic Weyl manifold with a semi- symmetric connection is locally conformal to an Einstein manifold with a semi-symmetric connection.

ON THE INFINITESIMAL ISOMETRIC DEFORMATIONS OF SUBMANIFOLDS

In this paper, we consider the infinitesimal I- and Il-isometry deformations of submanifolds immersed in a space form N of constant curvature. We obtain some results which are new even in the case of N being the Euclidean space. At the same time, we generalize some classical results in E-3 Go the submanifolds immersed in a space form of constant curvature.

THE MAIN INVARIANTS OF A COMPLEX FINSLER SPACE

In this paper we extend the results obtained in [3], where are investigated the general settings of the two-dimensional complex Finsler manifolds, with respect to a local complex Berwahl frame. The geometry of such manifolds is controlled by three real invari- ants which live on T＇M： two horizontal curvature invariants K and W and one vertical curvature invariant I. By means of these invariants are defined both the horizontal and the vertical holomorphic sectional curvatures. The complex Landsberg and Berwald spaces are of particular into, rest. Complex Berwald spaces coincide with K/ihler spaces, in the two - dimensional case, We establish the necessary and sufficient condition under which K is a constant and we obtain a characterization for the Kghler purely Hermitian spaces by the fact K = W=constant and I = 0. For the class of complex Berwald spaces we have K =W = 0. Finally, a classitication of two-dimensional complex Finsler spaces for which the horizontal curvature satisfies a special property is obtained.

A VANISHING THEOREM ON L^2 HARMONIC FORMS

This paper is concerned with the L2 harmonic forms of a complete noncompact Riemannian manifold, i.e. If M has a pole Q, let 0 < p <p<n, and assume the radial section curvatures satisfy on M ?{Q}, where then Hp = {0}. If M has a soul, then similar result is obtained.

FUNDAMENTAL GROUPS OF CLOSED POSITIVELY CURVED MANIFOLDS WITH ALMOST DISCRETE ABELIAN GROUP ACTIONS

Let M be a closed n-manifold of positive sectional curvature. Assume that M admits an effective isometrical T1× Zpk-action with p prime. The main result of the article n＋1 for n 〉 5, then there exists a positive constant p（n）, is that ifk=lforn=3or k〉 n＋1/4 for n≥5,then there exists a positive constant p（n）,depending only on n, such that π1 （M） is cyclic if p ≥ p（n）.

Manifolds with Special Commuting Jacobi Operators

For a Riemannian manifold（Mn,g） with curvature tensor R,the Jacobi operator J（X） is give.In this paper,the flat Riemannian manifolds are characterized in terms of special commutation properties of their Jacobi operators.