具有Morse奇点的光滑叶理
On smooth foliations with Morse singularities
摘要：
设$M$是光滑流形，$\F$是余维流形，$C^\infty$是$M$上的叶理，具有Morse型的孤立奇点.对$(M,\F)$的研究和分类是一个具有挑战性（和困难）的问题。在这种情况下，由于Reeb \cite{Reeb}的一个经典结果指出，一个流形承认一个叶理正好有两个中心型奇点是一个球。特别是，如果叶理由函数给出，则这是真的。沿着这些路线的结果由于Eells和柯伊伯\cite{Ku-Ee}分类流形具有实值函数承认正好三个非退化奇点。在本文中，我们证明了上述结果的一个推广。为了做到这一点，我们首先描述了可能的安排对奇点和相应的余维不变集，然后我们给出了一个消除过程，适当的中心鞍和一些鞍鞍配置（连续指数）。在第二部分中，我们调查，如果其他经典的结果，如Haefliger和Novikov（紧叶）定理，证明了正规叶理，仍然成立的存在奇点。在这个目的，在奇异集，$Sing(\F)$的叶状$\F$，我们认为{ \em{weakly stable} }组件，我们定义为那些组件承认一个邻居，所有的叶子是紧凑的。如果$Sing(\F)$只允许弱稳定分支，由光滑嵌入曲线与$S^1$同构，我们能够推广Haefliger定理。最后，存在一个封闭的曲线，横向叶理，使我们国家诺维科夫型的结果。
英文摘要：
Let $M$ be a smooth manifold and let $\F$ be a codimension one, $C^\infty$ foliation on $M$, with isolated singularities of Morse type. The study and classification of pairs $(M,\F)$ is a challenging (and difficult) problem. In this setting, a classical result due to Reeb \cite{Reeb} states that a manifold admitting a foliation with exactly two center-type singularities is a sphere. In particular this is true if the foliation is given by a function. Along these lines a result due to Eells and Kuiper \cite{Ku-Ee} classify manifolds having a real-valued function admitting exactly three non-degenerate singular points. In the present paper, we prove a generalization of the above mentioned results. To do this, we first describe the possible arrangements of pairs of singularities and the corresponding codimension one invariant sets, and then we give an elimination procedure for suitable center-saddle and some saddle-saddle configurations (of consecutive indices). In the second part, we investigate if other classical results, such as Haefliger and Novikov (Compact Leaf) theorems, proved for regular foliations, still hold true in presence of singularities. At this purpose, in the singular set, $Sing(\F)$ of the foliation $\F$, we consider {\em{weakly stable}} components, that we define as those components admitting a neighborhood where all leaves are compact. If $Sing(\F)$ admits only weakly stable components, given by smoothly embedded curves diffeomorphic to $S^1$, we are able to extend Haefliger's theorem. Finally, the existence of a closed curve, transverse to the foliation, leads us to state a Novikov-type result.
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