Infinitely many contact structures on S4m + 1
Infinitely many contact structures on S4m + 1
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S4m 1 上有无限多个接触结构
DOI：
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发表时间：
1999
期刊：
影响因子：
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通讯作者：
Ilya Ustilovsky
中科院分区：
文献类型：
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作者：
Ilya Ustilovsky
接触流形（M，M）是一个光滑的（2n−1）维流形M，具有完全不可积的超平面分布M.这意味着局部的α可以由满足α <$（dα）n−1 6= 0的1-形式α定义。这样的分布结构称为接触结构。当α是可余定向的（例如，当M是单连通的），α整体存在，称为切触形式。在这种情况下，TM的结构群以如下方式简化为U（n-1）× 1。首先，横截于θ的Reeb向量场R = Rα由以下方程确定：
A contact manifold (M,ξ) is a smooth (2n−1)-dimensional manifoldMwith a completely nonintegrable hyperplane distribution ξ ⊂ TM. This means that locally ξ is defined by a 1-form α satisfying α ∧ (dα)n−1 6= 0. Such a distribution ξ is called a contact structure. When ξ is coorientable (for instance,whenM is simply connected), α exists globally and is called a contact form. In this case, the structure group of TM reduces to U(n − 1) × 1 in the following way. First, the Reeb vector field R = Rα, which is transverse to ξ, is determined by the equations