A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds
A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds
复制标题
DOI：
10.3934/dcds.2008.20.801
复制
发表时间：
2006-09
影响因子：
1.1
通讯作者：
A. Agrachev;U. Boscain;M. Sigalotti
中科院分区：
文献类型：
--
作者：
A. Agrachev;U. Boscain;M. Sigalotti
我们考虑一个推广的黎曼几何，自然出现在控制理论的框架。设X和Y是二维流形M上的两个光滑向量场。如果$X$和$Y$处处线性无关，则它们在$M$上定义了一个经典的黎曼度量（它们是标准正交的度量），并且它们给了$M$度量空间的结构。如果$X$和$Y$在$M$的某个地方线性依赖，那么相应的黎曼度量有奇点，但是在一般的条件下度量结构仍然是很好定义的。能够以这种方式局部定义的度量结构称为准黎曼结构。它们是\ar s的特殊情况，它们自然地定义在$M$上的光滑向量场空间的子模中。几乎黎曼结构表现出有趣的现象，特别是那些涉及曲率，存在的共轭点，和拓扑的流形之间的关系。本文的主要结果是Gauss-Bonnet公式推广到几乎黎曼结构。
We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent, then they define a classical Riemannian metric on $M$ (the metric for which they are orthonormal) and they give to $M$ the structure of metric space. If $X$ and $Y$ become linearly dependent somewhere on $M$, then the corresponding Riemannian metric has singularities, but under generic conditions the metric structure is still well defined. Metric structures that can be defined locally in this way are called almost-Riemannian structures. They are special cases of \ar s, which are naturally defined in terms of submodules of the space of smooth vector fields on $M$. Almost-Riemannian structures show interesting phenomena, in particular those which concern the relation between curvature, presence of conjugate points, and topology of the manifold. The main result of the paper is a generalization to almost-Riemannian structures of the Gauss-Bonnet formula.