A Discussion on the Establishment That a Fibre Metric on the Positive Definite Real Inner-Product of a Properly Embedded Smooth Submanifold Be Always Extended to a Riemannian Metric on the Positive Definite Real Inner-Product

Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|s i.e. a positive definite real inner-product on TpM for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s can always be extended to a Riemannian metric on TM from a special perspective.

GEOMETRIC INEQUALITIES FOR CERTAIN SUBMANIFOLDS IN A PINCHED RIEMANNIAN MANIFOLD

This article gives some geometric inequalities for a submanifold with parallel second fundamental form in a pinched Riemannian manifold and the distribution for the square norm of its second fundamental form.

Embeddings of Almost Hermitian Manifold in Almost Hyper Hermitian Manifold and Complex (Hypercomplex) Numbers in Riemannian Geometry

Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold. Further, an almost hyper Hermitian structure has been constructed on the tangent bundle TM with help of the Riemannian connection of an almost Hermitian structure on a manifold M then, we consider an embedding of the almost Hermitian manifold M in the corresponding normal tubular neighborhood of the null section in the tangent bundle TM equipped with the deformed almost hyper Hermitian structure of the special form. As a result, we have obtained that any Riemannian manifold M of dimension n can be embedded as a totally geodesic submanifold in a Kaehlerian manifold of dimension 2n (Theorem 6) and in a hyper Kaehlerian manifold of dimension 4n (Theorem 7). Such embeddings are “good” from the point of view of Riemannian geometry. They allow solving problems of Riemannian geometry by methods of Kaehlerian geometry (see Section 5 as an example). We can find similar situation in mathematical analysis (real and complex).

Notes on the Heat Diffusion Semigroup in a Riemannian Manifold

In this paper, the properties of the heat diffusion semigroup {e^(t△)}_(t≥0) generated by the Hodge-deRham operator in a Riemannian manifold are discussed.

ON THE INVARIANT SUBMANIFOLDS OF RIEMANNIAN PRODUCT MANIFOLD

In this paper, the vertical and horizontal distributions of an invariant sub-manifold of a Riemannian product manifold are discussed. An invariant real space form in a Riemannian product manifold is researched. Finally, necessary and sufficient conditions are given on an invariant submanifold of a Riemannian product manifold to be a locally symmetric and real space form.

An Estimate on Riemannian Manifolds of Dimension 4

We give an estimate of type sup x inf on Riemannian manifold of dimension 4 for a Yamabe type equation.

The First Initial Boundary Value Problem for Parabolic Hessian Equations on Riemannian Manifolds

For a class of elliptic Hessian operators, one type of corresponding parabolic Hessian equations is studied on Riemannian manifolds. uniqueness of the admissible solution to the first initial boundary the equations are shown. The existence and value problem for

Complex Networks from Chaotic Time Series on Riemannian Manifold

Complex networks are important paradigms for analyzing the complex systems as they allow understanding the structural properties of systems composed of different interacting entities. In this work we propose a reliable method for constructing complex networks from chaotic time series. We first estimate the covariance matrices, then a geodesic-based distance between the covariance matrices is introduced. Consequently the network can be constructed on a Riemannian manifold where the nodes and edges correspond to the covariance matrix and geodesic-based distance, respectively. The proposed method provides us with an intrinsic geometry viewpoint to understand the time series.

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 .

A BRAY-BRENDLE-NEVES TYPE INEQUALITY FOR A RIEMANNIAN MANIFOLD

In this paper,for any local area-minimizing closed hypersurface∑with RcΣ=RΣ/ngΣ,immersed in a(n+1)-dimension Riemannian manifold M which has positive scalar curvature and nonnegative Ricci curvature,we obtain an upper bound for the area of∑.In particular,when∑saturates the corresponding upper bound,∑is isometric to S^(n)and M splits in a neighborhood of∑.At the end of the paper,we also give the global version of this result.

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 Uniqueness and Nonexistent Results for Some Nonlinear Partial Equations on Riemannian Manifolds

The paper studies a class of nonlinear elliptic partial differential equations on a compact Riemannian manifold （M,g） with some curvature restriction. The authors try to prove some uniqueness and nonexistent results for the positive solutions of the equations concerned.

Totally Umbilical Screen Transversal Lightlike Submanifolds of Semi-Riemannian Product Manifolds

We study totally umbilical screen transversal lightlike submanifolds immersed in a semi-Riemannian product manifold and obtain necessary and sufficient conditions for induced connection ▽ on a totally umbilical radical screen transversal lightlike submanifold to be metric connection. We prove a theorem which classifies totally umbilical ST-anti-invariant lightlike submanifold immersed in a semi-Riemannian product manifold.

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.

黎曼流形上的粒子群运动模型

自然界中生物群的协同运动模式多种多样,运动空间的复杂性对该多样性起着关键作用.为描述生物群的协调运动,人们基于欧氏空间提出了多种粒子群运动模型.然而,这些模型往往只能描述简单的运动模式.本文基于Morse势和黎曼流形提出了一个新模型,该模型可以描述任意运动空间中的群体运动,且易于工程实现.本文分别以球面、环面、Möbius带及类丘陵曲面为代表仿真分析了黎曼曲面的紧致性、亏格、可定向性及高斯曲率等性质对模型的影响.结果显示,紧致性导致粒子群聚合得更加紧密对称,非零亏格导致群体中粒子的运动速度趋于一致并阻止粒子群形成类涡流形态,不可定向性导致粒子群在任何条件下都趋于分散,而高斯曲率对粒子群运动行为的影响较小.此外,本文还研究了当曲面包含一个、两个及三个障碍物时模型的输出,结果显示,当势强度足够大时,粒子群能够趋向目标,也能包围目标或绕行障碍物.综上,相比已有模型,该模型能够描述更为丰富的运动模式.

基于黎曼流形的健身APP风险度量方法

随着智能设备的普及,其应用系统已成为恶意软件攻击的主要目标,存在巨大的网络安全隐患。健身App因其获取数据的隐私性和敏感性,面临的数据安全问题更加严峻,其安全度量模型成为解决这一挑战的关键点。目前的安全度量模型多数基于静态特征构建,未能全面考虑智能设备的动态网络行为。为了弥补这一不足,提出一种基于网络行为的健身App安全度量模型,运用协方差矩阵对网络空间进行转换,提高了对恶意软件攻击识别的准确率,根据健身App的动态网络行为特征,更全面地揭示了其安全状态,同时结合黎曼度量,有效描述了网络安全风险,并计算其值,从而构建出一个基于恶意软件攻击识别与黎曼流形的风险度量模型,以实现更安全的数据保护。

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.

特征提取中一类矩阵迹函数极值问题的黎曼优化算法

研究来源于特征提取中的一类鲁棒判别回归模型,该模型问题可以重构为Stiefel流形和线性流形组成的乘积流形约束下的一类矩阵迹函数极小化问题.整合紧流形和线性流形,结合乘积流形几何性质,本文设计适用于求解重构问题简化版本的一类基于Zhang-Hager技术拓展的乘积流形黎曼非线性共轭梯度法,并给出算法全局收敛性分析.数据实验表明所提算法对于问题求解是高效可行的,且与已有算法、其它黎曼梯度类算法及黎曼优化工具箱中已有的黎曼一阶和二阶算法相比在迭代解精度或迭代效率上有一定优势.

基于流形神经网络的协作频谱感知

针对传统频谱感知在复杂信号环境,如低信噪比和多路径衰落等情况下的性能局限,提出了一种基于黎曼流形神经网络的创新性协作频谱感知方案。该方法首先通过将多个协作用户的信号矩阵映射到黎曼流形上,生成具有几何特性的协方差矩阵。接着,利用黎曼流形神经网络进行高效的信号特征分类和频谱感知。黎曼流形神经网络不仅充分发挥了黎曼流形在非欧几里得数据结构上的优势,而且结合了神经网络强大的表达能力,从而在各种复杂环境下都展示出显著优越的频谱感知性能。通过一系列详细的仿真实验,证明了该方法在多样化环境下的优越性能,展示了其在实际无线通信系统中的潜在应用价值。

Well-Posedness of Stochastic Continuity Equations on Riemannian Manifolds

The authors analyze continuity equations with Stratonovich stochasticity,■ρ+divh[ρo(u(t,x)+∑_(i=1)^(N)a_(i)(x)w_(i)(t))]=0defined on a smooth closed Riemannian manifold M with metric h.The velocity field u is perturbed by Gaussian noise terms Wi(t),:WN(t)driven by smooth spatially dependent vector fields a1(x),...,aN(x)on M.The velocity u belongs to L_(t)^(1)W_(x)^(1,2)with divh u bounded in Lf,for p>d+2,where d is the dimension of M(they do not assume div_(h) u∈L_(t,x)^(∞)).For carefully chosen noise vector fields ai(and the number N of them),they show that the initial-value problem is well-posed in the class of weak L^(2) solutions,although the problem can be ill-posed in the deterministic case because of concentration effects.The proof of this“regularization by noise”result is based on a L^(2) estimate,which is obtained by a duality method,and a weak compactness argument.