The Infimum of the Energy of Unit Vector Fields on Odd-Dimensional Spheres

We construct a one-parameter family of unit smooth vector fieldsglobally defined on the sphere ^2k+1 for k 2, with energyconverging to the energy of the unit radial vector field, which isdefined on the complementary of two antipodal points. So we prove thatthe infimum of the energy of globally defined unit smooth vector fieldsis

A Girsanov Type Theorem on the Path Space Over a Compact Riemannian Manifold

Abstract

This article establishes a Girsanov type theorem on path spaces over compact Riemannian manifolds, generalizing the classical Girsanov theorem for Euclidean spaces.     

Scalar Curvature, Killing Vector Fields and Harmonic One-Forms on Compact Riemannian Manifolds

It is well known that no non-trivial Killing vector field existson a compact Riemannian manifold of negative Ricci curvature;analogously, no non-trivial harmonic one-form exists on a compactmanifold of positive Ricci curvature. One can consider the following,more general, problem. By reducing the assumption on the Riccicurvature to one on the scalar curvature, such vanishing theoremscannot hold in general. This raises the question: "What informationcan we obtain from the existence of non-trivial Killing vectorfields (or, respectively, harmonic one-forms)?" This paper givesanswers to this problem; the results obtained are optimal. 2000Mathematics Subject Classification 53C20 (primary), 53C24 (secondary).     

紧致黎曼流形上一类非线性热方程的梯度估计

In this paper,we study gradient estimates for the nonlinear heat equation ut-△u =au log u,on compact Riemannian manifold with or without boundary.We get a Hamilton type gradient estimate for the positi...     

Energy of Radial Vector Fields on Compact Rank One Symmetric Spaces

We consider the energy (or the total bending) of unit vector fields oncompact Riemannian manifolds for which the set of its singularitiesconsists of a finite number of isolated points and a finite number ofpairwise disjoint closed submanifolds. We determine lower bounds for theenergy of such vector fields on general compact Riemannian manifolds andin particular on compact rank one symmetric spaces. For this last classof spaces, we compute explicit expressions for the total bending whenthe unit vector field is the gradient field of the distance function toa point or to special totally geodesic submanifolds (i.e., for radialunit vector fields around this point or these submanifolds).     

Constructing a Brownian Sheet with Values in a Compact Riemannian Manifold

Mathematical Notes -     

The Vector Space of Conformal Killing Forms on a Riemannian Manifold

T. Kashiwada and S. Tachibana defined conformal Killing p-forms on a Riemannian manifold of dimension p \geqslant 1$$ " align="middle" border="0"> and generalized some results on conformal Killing vector fields to the case of such forms. In this paper, conformal Killing p-forms are defined with the help of natural differental operators on Riemannian manifolds and representations of orthogonal groups. The geometry of the vector space of conformal Killing p-forms and of its two subspaces of coclosed conformal Killing p-forms and of closed conformal Killing p-forms are considered. Some local and global results due to Tachibana and Kashiwada about conformal Killing and Killing p-forms are generalized. An application to Hermitian geometry is given. Bibliography: 30 titles.     

关于局部紧域的加法特征的一点注记

In this note, we give an elementary and constructive proof for that the additive character group of a locally compact field is isomorphic to itself as an additive topological group.     

局部对称共形平坦黎曼流形中紧致子流形的一个刚性定理

研究局部对称共形平坦黎曼流形N^n＋p（p≥2）中具有平等平均曲率向量的紧致子流形M^n的余维可约性问题，在n≥8的条件下得到了量佳拼挤常数．     

HOLOMORPHIC VECTOR FIELDS AND CHARACTERISTIC FORMS ON A HERMITIAN MANIFOLD

Let M be a compact Hermtian manifold, dim_cM=m, Ω be the curvature form of the Hermitian connection. F is a U(m)-invariant polynomial of degree k相似文献   

Monotone and Accretive Vector Fields on Riemannian Manifolds

The relationship between monotonicity and accretivity on Riemannian manifolds is studied in this paper and both concepts are proved to be equivalent in Hadamard manifolds. As a consequence an iterative method is obtained for approximating singularities of Lipschitz continuous, strongly monotone mappings. We also establish the equivalence between the strong convexity of functions and the strong monotonicity of its subdifferentials on Riemannian manifolds. These results are then applied to solve the minimization of convex functions on Riemannian manifolds.     

A Global Newton Method for the Nonsmooth Vector Fields on Riemannian Manifolds

Journal of Optimization Theory and Applications - This paper proposes and analyzes a globalized version of the Newton method for finding a singularity of the nonsmooth vector fields. Basically, the...     

局部对称共形平坦黎曼流形中的紧致子流形

本文讨论局部对称共形平坦黎曼流形中紧子流形问题．改进了［１］的结果并将［２］中关于球面子流形的一个结果推广到局部对称共形平坦黎曼流形子流形．     

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 thatλ_1≥π~2/d~2·1/(2-(11)/(2π~2))+11/2π~2e~cm、(?)     

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Let p_M(t,x,y) be the minimal heat kernel of a d-dimenional compact Riemannian manifold M for any time t\in(0,1] and x,y\in M. Using the horizontal Brown bridge on M, we prove that, for any nonnegative integers n and m, there is a constant C depending on n,m and the manifold M, such that |\nabla^n_x\nabla^m_y\ln p_M(t,x,y)|\leq C[d(x,y)/t+1/\sqrt{t}\,]^{n+m}$, which generalizes the conclusion of the higher derivatives of the logarithmic heat kernel \ln p_M(t,x,y) about single variable in \ncite{1}.     

Bernstein-type Theorems in a Riemannian Manifold with an Irrotational Killing Vector Field

Mediterranean Journal of Mathematics - The minimal hypersurface equation for a graph in a Riemannian manifold which admits a nowhere zero Killing vector field, whose orthogonal distribution is...     

局部对称黎曼流形中具有平行平均曲率向量的子流形

设Nn+p是截面曲率KN满足的n+p维局部对称完备黎曼流形,p≥2.M是Nn+p的具有平行平均曲率向量的n维紧致子流形.本文讨论了这类子流形关于第二基本形式模长平方的积分不等式及其Pinching问题.