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.

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.

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^＋.

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].

The Riemannian Structure of the Three-Parameter Gamma Distribution

In this paper, we will utilize the results already known in differential geometry and provide an intuitive understanding of the Gamma Distribution. This approach leads to the definition of new concepts to provide new results of statistical importance. These new results could explain Chen [1-3] experienced difficulty when he attempts to simulate the sampling distribution and power function of Cox’s [4,5] test statistics of separate families of hypotheses. It may also help simplify and clarify some known statistical proofs or results. These results may be of particular interest to mathematical physicists. In general, it has been shown that the parameter space is not of constant curvature. In addition, we calculated some invariant quantities, such as Sectional curvature, Ricci curvature, mean curvature and scalar curvature.

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.

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.

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.

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.

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.

Some Remarks on the Existence of the Cusp Metric on R^3

In this paper, we show that there exists no complete rotationally symmetric cusp metric on R^3. Also we consider the existence of the complete rotationally symmetric cusp metric with other situations.

ON THE NON-EXISTENCE OF L^2-EIGENVALUES OF MANIFOLDS OF COLUM TYPE

In[4],Li proved that Yau’s conjecture“For non-compact connected completeRiemannian manifold M.M has no L;-eigenvalues if its sectional curvature K;≥0”holds if M can be represented as a Riemannian product M=R;×N.Acturally,heproved（without the restriction K;≥0）

If the Redshift Depends on the Pressure then the Acceleration of the Universe Can Be Explained

We propose a model where the Hubble's law is slightly changed. We propose new interpretation of the covariant divergence of the energy-impulse vector and this produce a new correction to redshift. Acceleration of the expansion of the Universe appeared as a pure observational effect. High values of the mass density are consistent with the experimental data on Supernova Ia within this FRW model without the cosmological constant (Λ=0).

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.

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 Harmonic Functions on a Complete Asymptotic Flat Riemannian Manifold

Let be a simply connected complete Riemannian manifold with dimension n≥3 . Suppose that the sectional curvature satisfies , where p is distance function from a base point of M, a, b are constants and . Then there exist harmonic functions on M .

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.

ON THE SECTIONAL CURVATURE OF A RIEMANNIAN MANIFOLD

In this paper the author establishes the following1.If M^n(n≥3)is a connected Riemannian manifold,then the sectional curvatureK(p),where p is any plane in T^x(M),is a function of at most n(n-1)/2 variables.Moreprecisely,K(p)depends on at most n(n-1)/2 parameters of group SO(n).2.Lot M^n(n≥3)be a connected Riemannian manifold.If there exists a point x ∈ Msuch that the sectional curvature K(p)is independent of the plane p∈T_x(M),then M is aspace of constant curvature.This latter improves a well-known theorem of F.Schur.

On the Convergence Rate of Bergman Metrics

We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler n-manifolds(M,L,g,x)with Vol(B1(x))>v and|sec|≤K on M.Relying on Tian’s peak section method(Tian in J Differ Geom 32(1):99-130,1990),we show that the C^(1,α)convergence of Bergman metrics is uniform.In the end,we discuss the sharpness of our estimates.

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].