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.

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.

Lower Bound of the First Eigenvalue for the Laplace Operator on Compact Riemannian Manifold

Let M be a compact m-dimensional Riemannian manifold, let d denote, its diameter, -R(R>O) the lower bound of the Ricci curvature, and λ_1 the first eigerivalue for the Laplacian on M. Then there exists a constant C_m=max{2^(1/m-1),2^(1/2)}, Such that λ_1≥π~2/d^2·1/(2-(11)/(2π~2))+11/2π~2e^cm、

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 .

混合曲率空间中的几何自适应元学习方法

元学习通过学习先验知识,能帮助模型快速适应新任务.在适应新任务的过程中,空间几何结构与数据几何结构的匹配程度对模型泛化起着重要作用.现实世界数据具有多样的非欧几何结构,例如自然语言具有非欧层级结构,人脸图像具有非欧环状结构等.已有研究表明,真实数据的非欧结构同黎曼流形的几何结构相匹配,从理论上提供了利用黎曼流形来建模数据的可行性.本文提出了混合曲率空间(mixed-curvature space)中的几何自适应元学习方法,利用多个混合曲率空间来表示数据,并生成与数据非欧结构相匹配的黎曼几何.本文构建了多混合曲率神经网络,将混合曲率空间的几何结构表示为曲率空间的曲率、数量和维度,由此通过梯度下降过程实现对数据非欧结构的几何自适应.本文进一步引入几何初始化生成策略和几何更新策略,通过少数几步迭代,空间几何结构即可快速匹配数据非欧结构,加速了梯度下降过程.本文在小样本分类和小样本回归等任务上进行了实验验证.与欧氏空间的元学习方法相比,本文方法在小样本分类任务上取得了约3%的准确率提升,在小样本回归任务上将均方误差减少了一半,验证了本文方法的有效性.

Eigenvalues for the Clamped Plate Problem of L_(ν)^(2) Operator on Complete Riemannian Manifolds

L_(ν) operator is an important extrinsic differential operator of divergence type and has profound geometric settings.In this paper,we consider the clamped plate problem of L_(ν)^(2)operator on a bounded domain of the complete Riemannian manifolds.A general formula of eigenvalues of L_(ν)^(2) operator is established.Applying this general formula,we obtain some estimates for the eigenvalues with higher order on the complete Riemannian manifolds.As several fascinating applications,we discuss this eigenvalue problem on the complete translating solitons,minimal submanifolds on the Euclidean space,submanifolds on the unit sphere and projective spaces.In particular,we get a universal inequality with respect to the L_(II) operator on the translating solitons.Usually,it is very difficult to get universal inequalities for weighted Laplacian and even Laplacian on the complete Riemannian manifolds.Therefore,this work can be viewed as a new contribution to universal estimate.

Cartan-Hadamard流形上关于Lorentz范数的Trudinger-Moser不等式

本文研究了Cartan-Hadamard流形上带Lorentz范数的Trudinger-Moser不等式.利用了相关格林函数的逐点估计以及O’Neil不等式,我们得到了该不等式的最佳常数,推广了相应欧氏空间上的结果.

拟常曲率Riemannian流行的一个Pinching定理

本文证明了积分不等式∫M∑i=1β≠n+1hi^2βj[3-1/p-1+n^1/2)S-na-1/2(n+1)(b-│b│）]*1≥0从而得到如下Pinching定理：若S≤[na+1/2(n+1)(b-│b│）]/（3-1/p-1+n^1/2)则M落在N的一个全测地子流行S^n+1中或S=[na+1/2(n+1)(b-│b│)]/(3-1/p-1+n^1/2)所得积分不等式优于白正国教授的结果而Pinching定理是丘成桐教授相应定理的推广。

An Interpolation of Hardy Inequality and Moser–Trudinger Inequality on Riemannian Manifolds with Negative Curvature

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and Moser-Trudinger inequality on M. Furthermore, the constant we obtain is sharp.

The extension for mean curvature flow with finite integral curvature in Riemannian manifolds

We investigate the integral conditions to extend the mean curvature flow in a Riemannian manifold. We prove that the mean curvature flow solution with finite total mean curvature at a finite time interval [0,T) can be extended over time T. Moreover,we show that the condition is optimal in some sense.

DARBOUX EQUATIONS AND ISOMETRIC EMBEDDING OF RIEMANNIAN MANIFOLDS WITH NONNEGATIVE CURVATURE IN R

The present paper is concerned with the existence of golbal smooth solutions for the homogeneous Dirichlet boundary value problem of the Darboux equation and the case degenerate onthe boundary is contained As some applications the smooth isometric embeddings of positivelyand nonnegatively curved disks into R^3 are constructed.

A Quantum Representation of the Homogeneous 5D Manifold and the Perelman Mappings of 5D onto Non-Homogeneous Lorentz 4D Manifolds

The expression of the Maxwell magnetic monopole was employed to correlate the space to space projection that gives rise to the Gell-Mann standard model, and space to time projection which gives the leptons;and how does it correlate to the Perelman mappings from the homogeneous 5D manifold to the Lorentz 4D manifold, together with correlating the physical consequences caused by the breaking of the Diagonal Long Range Order [DLRO] of the monopoles quantum states affected by the motion of massive particles in the Lorentz 4D boundary of the 5D manifold, which leads to gravitons and the gravity field via the General Relativity covariant Riemannian 4D curvatures metric equation.

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 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）

A Pinching Theorem for Riemannian Foliations with Parallel Mean Curvature in a Local-Symmetric Riemannian Manifold

We discuss the Riemannian foliations with parallel mean curvature in a local- symmetric Riemannian manifold,and obtain a pinching theorem about it.

The Extension of the H^k Mean Curvature Flow in Riemannian Manifolds

In this paper,the authors consider a family of smooth immersions Ft : Mn→Nn+1of closed hypersurfaces in Riemannian manifold Nn+1with bounded geometry,moving by the Hkmean curvature flow.The authors show that if the second fundamental form stays bounded from below,then the Hkmean curvature flow solution with finite total mean curvature on a finite time interval [0,Tmax)can be extended over Tmax.This result generalizes the extension theorems in the paper of Li(see "On an extension of the Hkmean curvature flow,Sci.China Math.,55,2012,99–118").

基于双曲率黎曼流体的网络攻击检测模型

基于深度学习的网络攻击检测方法以其强大的特征表示及提取能力得到了迅速发展。然而,传统欧氏空间中的深度学习网络攻击检测模型无法有效捕获具有复杂调用关系的网络拓扑结构。为高效建模网络攻击图中潜在的数据模式,提高网络攻击检测的准确性,提出一种基于双曲率黎曼流体的网络攻击检测模型。与现有方法不同的是,该模型将传统欧氏空间中的网络攻击检测模型迁移到异质化非欧式表示空间中,利用曲率黎曼几何空间所具备的大规模层次性和环形图结构模式建模能力,获取高质量的攻击图表示向量,进而提高网络攻击检测准确性。实验结果表明,基于双曲率黎曼流体的网络攻击检测模型Precision、Recall、F1-score分别为0.963、0.964、0.964,能够对网络攻击行为进行有效分析与检测。

RIGIDITY THEOREMS FOR HYPERSURFACES IN RIEMAN-NIAN MANIFOLD OF CONSTANT CURVATURE

In this letter we prove several global rigidity theorems for hypersurfaces in Riemannian manifold of constant curvature, which are generalizations of some wellknown theorems for convex hypersurfaces in En+1, Sn+1 and Hn+1. Our main results are as follows:

SUBMANIFOLDS WITH PARALLEL MEAN CURVATURE VECTOR

Let Sn+p be a unit （n+p） sphere and f: M→Sn+p an isometric immersion of an n-ditmensional Riemannian manifold M into Sn+p, If the length of the mean curvature vector ξ of f（M） is constant and the vector ξ/|ξ| is parallel im the normal bundle, then f（M） is called the submanifold

Manifolds of positive Ricci curvature,quadratically asymptotically nonnegative curvature,and infinite Betti numbers

In a previous paper(Jiang and Yang(2021)),we constructed complete manifolds of positive Ricci curvature with quadratically asymptotically nonnegative curvature and infinite topological type but dimensions greater than or equal to 6.The purpose of the present paper is to use a different technique to exhibit a family of complete I-dimensional(I≥5)Riemannian manifolds of positive Ricci curvature,quadratically asymptotically nonnegative sectional curvature,and certain infinite Betti numbers bj(2≤j≤I-2).