论文标题
Morse功能和接触凸表面
Morse functions and contact convex surfaces
论文作者
论文摘要
让$ f $是封闭的表面$σ$上的摩尔斯功能，这样零是常规的值，因此$ f $既不承认正式minima也不承认负面最大值。在此说明中，我们表明$σ\ times \ mathbb {r} $接纳了$ \ mathbb {r} $ - 不变触点表格$α= fdt+β$，其特征沿零部分的特征性叶片为（负）较弱的梯度，类似于$ f $。该证明是独立的，并给出了任何$ \ mathbb {r} $的明确构造 - $σ\ times \ times \ mathbb {r} $中的不变触点结构，直至isotophy。作为一个应用程序，我们给出了$ \ mathbb {r} $的同型分类的替代几何证明 - 就其分配集而言，不变的联系结构。
Let $f$ be a Morse function on a closed surface $Σ$ such that zero is a regular value and such that $f$ admits neither positive minima nor negative maxima. In this expository note, we show that $Σ\times \mathbb{R}$ admits an $\mathbb{R}$-invariant contact form $α=fdt+β$ whose characteristic foliation along the zero section is (negative) weakly gradient-like with respect to $f$. The proof is self-contained and gives explicit constructions of any $\mathbb{R}$-invariant contact structure in $Σ\times \mathbb{R}$, up to isotopy. As an application, we give an alternative geometric proof of the homotopy classification of $\mathbb{R}$-invariant contact structures in terms of their dividing set.
