具有负数量曲率的紧致黎曼流形的Killing向量场（英文）

本文研究了具有负数量曲率的紧致黎曼流形上的Killing向量场.利用Bochner方法,得到在此类流形上非平凡的Killing向量场的存在的必要条件.这个结果拓广了文献[6]中的定理1.     

Riemann流形上Killing向量场的零点

命M是偶维定向紧致Riemann流形。M上Killing向量场的零点首先由S.Kobayashi所研究[1],[2].P.F.Baum和J.Cheeger用R.Bott[4],[5]的方法研究了Killing向量场的零点的状况与Riemann流形Pontrjagin数之间的关系。他们工作的关键部分是作出相应的Pontrjagin形式的超渡式。但是他们是凑出了这样一个超渡式,作法上不很自然。本文中我们将沿用传统的陈-Weil的办法自然地得出这个超渡式。     

近切触流形的φ~*-解析向量场（英文）

本文引入了近切触流形(M,φ,ξ,η,g)中φ~*-解析向量场的概念,并研究了其性质.利用近切触流形的性质,证明了切触度量流形中的φ~*-解析向量场v是Killing向量场且φv不是φ*-解析的.特别地,如果近切触流形M是正规的,得到v与ξ平行且模长为常数.另外,证明了3维的切触度量流形不存在非零的φ~*-解析向量场.     

拟常曲率流形中保持曲率的无穷小变换

1 如所知,在一个 Riemann 流形中,若由′σ~α=σ~α+v~α(σ)dt (1)确定的无穷小变换满足(?)(v)a_(λμ)=2(?)a_(λμ) (2)式中 a_λ是度量张量,(?)是某纯量函数,(?)(v)是关于无穷小变换 v 的李导数,则(1)称为无穷小共形变换,而向量场 v 称为共形 Killing 向量场。如果(?)=const,则称 v 为无穷小位似变换.特别,当(?)=0时,(1)成为无穷小等距变换.在这个情形下,(2)化为 Kil-     

微分几何中的BOCHNER技巧(上)

这篇报告的标题涉及微分几何中一个一般的方法,它是由S.Bochner首创的([B1],[B2])。三十多年前,Bochner用这一技巧证明:Riemann流形上某些几何上有兴趣的对象(例如Killing向量场、调和形式、旋量场)必定平行或者为零。今天,它已成为几何学者们的基本术语之一。尽管这一技巧表现的形式简单,但很难把它讲清楚。较好的办法或许是给出运用它的一个典型例子。让我们实际考虑这方面Bochner的第一批定理之一:具负Ricci曲率的紧Riemann流形M,不存在非零的Killing向量场。定理证明略     

三维Minkowski空间中基于Killing向量场的磁力曲线

本文在三维Minkowski空间中,研究了基于Killing向量场的磁力曲线.首先给出了三维Minkowski空间中对应于平移和旋转的所有Killing向量场,并对不同的Killing向量所对应的磁力曲线进行了分类,最后给出了一些磁力曲线的图像.     

具有平行平均曲率向量场的子流形

本文讨论了具有平行平均曲率向量场的某一类子流形，得到了这类子流形成为全脐点子流形及其余维数减小的充分条件。     

近切触流形的φ*-解析向量场

本文引入了近切触流形（M,ø,ξ,η,g）中φ^*-解析向量场的概念,并研究了其性质.利用近切触流形的性质,证明了切触度量流形中的φ^*-解析向量场v是Killing向量场且φv不是φ^*-解析的.特别地,如果近切触流形M是正规的,得到v与ξ平行且模长为常数.另外,证明了3维的切触度量流形不存在非零的φ^*-解析向量场.     

(α, β)- 空间中的Killing 向量场

本文证明了非Riemannian (α, β)- 空间中的Killing 向量场最大维数是n(n - 1)/2 + 1. 并且给出了具有最大维数Killing 向量场的非Riemannian (α, β)- 空间的度量形式. 最后, 若进一步假定α 是一个齐性Riemannian 度量, 则可确定(α, β)- 空间的第二空隙. 最后给出几个低维流形上Killing 场空间维数的例子, 这表明在(α, β) 情形下Killing 场空间维数的空隙被压缩.     

具有虚Killing旋子的Lorentz Spin流形的一些新结果

利用Lorentz Spin流形上虚Killing旋子的Dirac流V_的性质,通过对其不同情况得讨论,得到了具有虚Killing旋子的Lorentz Spin流形的一个分类定理.此外证明了2-形式dV_~b是共形killing 2-形式.     

Strongly and weakly affine vector fields on Finsler manifolds

In this paper, we investigate the affine vector fields on both compact and forward complete Finsler manifolds. We first give definitions of the affine transformation and the affine vector field. Unexpectedly, we find two kinds of affine fields, which are named as the strongly and weakly affine vector fields. Based on these definitions, we prove some rigidity theorems of affine fields on compact and forward complete Finsler manifolds.     

Affine Connection Complexes

Affine connections of 1-forms, rather than vector fields, induce complexes that project to the de Rham complexes of the underlying manifolds. This observation provides short direct proofs of Bianchi identities, existence and uniqueness of Levi-Civita connections, and symmetries of the Riemann curvature tensor.     

The geometry of closed conformal vector fields on Riemannian spaces

In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T.K. Pan. Then we explain why it is so difficult to find examples, other than trivial ones, of spaces having at least two closed, conformal and homothetic vector fields. We then focus on isometric immersions, firstly generalizing a theorem of J. Simons on cones with parallel mean curvature to spaces furnished with a closed, Ricci null conformal vector field; then we prove general Bernstein-type theorems for certain complete, not necessarily cmc, hypersurfaces of Riemannian manifolds furnished with closed conformal vector fields. In particular, we obtain a generalization of theorems J. Jellett and A. Barros and P. Sousa for complete cmc radial graphs over finitely punctured geodesic spheres of Riemannian space forms.     

Harmonic maps between compact Hermitian manifolds

In this paper, we generalize the Bochner-Kodaira formulas to the case of Hermitian com- plex (possibly non-holomorphic) vector bundles over compact Hermitian (possibly non-K¨ahler) mani- folds. As applications, we get the complex analyticity of harmonic maps between compact Hermitian manifolds.     

Invariant monotone vector fields on Riemannian manifolds

Various concepts of invariant monotone vector fields on Riemannian manifolds are introduced. Some examples of invariant monotone vector fields are given. Several notions of invexities for functions on Riemannian manifolds are defined and their relations with invariant monotone vector fields are studied.     

丛上的相交数与Euler数

本文主要研究了带边流形上丛截面的相交数．应用作倍流形上的向量场的延拓，我们得到了带边流形上的Euler数用相交数的表示．     

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).     

On a differential equation characterizing Euclidean spheres

A characterization of Euclidean spheres out of complete Riemannian manifolds is made by certain vector fields on complete Riemannian manifolds satisfying a partial differential equation on vector fields.     

Killing vector fields of constant length on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on complete smooth Riemannian manifolds. Various examples are constructed of the Killing vector fields of constant length generated by the isometric effective almost free but not free actions of S ¹ on the Riemannian manifolds close in some sense to symmetric spaces. The latter manifolds include “almost round” odd-dimensional spheres and unit vector bundles over Riemannian manifolds. We obtain some curvature constraints on the Riemannian manifolds admitting nontrivial Killing fields of constant length.     

Topological Equivalence of Morse–Smale Vector Fields with beh2 on Three-Dimensional Manifolds

For the Morse–Smale vector fields with beh2 on three-dimensional manifolds, we construct complete topological invariants: diagram, minimal diagram, and recognizing graph. We prove a criterion for the topological equivalence of these vector fields.