Tight contact structures via dynamics
Tight contact structures via dynamics
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通过动力学实现紧密接触结构
DOI：
10.1090/s0002-9939-99-05377-0
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发表时间：
1999
期刊：
影响因子：
--
通讯作者：
John B. Etnyre
中科院分区：
文献类型：
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作者：
R. Ghrist;John B. Etnyre
研究了在封闭可定向三流形上实现紧密接触结构的问题。通过应用Hofer等人的定理，人们可以从（Reeb）流横向到接触结构的动力学特性中推断出紧密性。我们详细介绍了两种经典结构，Dehn手术和分支覆盖，如何在动力约束的链接上以这种方式进行，以保持横向紧密接触结构。1. 接触几何和动力学有关接触结构几何和动力学的基本定义和定理的更彻底的处理见，例如，b[1]。3流形M上的接触结构5是TM中完全不可积的2平面场。更具体地说，在每个点p E M处，我们有一个2-平面~p C TpM，它平滑地随p变化，具有5在Frobenius意义上无处可积的性质，即存在（局部）一个定义的1-形式ac（其核为J），使得a a doa为70。如果a是全局定义的，则~称为可定向的，而a为~的contact 1-form。我们采用对可定向接触结构的一般限制。接触几何中有趣的（和困难的）问题都是全局性质的：达布定理（参见，例如，[23,1]）意味着所有的接触结构都是局部接触同构的，或者是保持平面场的微分同构。对于接触流形(M)中的曲面E也有类似的结果，如下所示。一般来说，TpE n $p将是TpE中的一行。这个线场积分成一个奇异叶理SE，称为s的特征叶理。我们可以证明，就像达布定理的单点情况一样，>E决定了。最近在三维接触几何中出现了一种基本的二分法。如果M中存在一个嵌入盘D，其特征叶理D1包含一个极限环，则接触结构S是超扭的。如果~没有过度扭曲，则称为紧。Eliashberg[6]对闭合3流形上的超扭接触结构进行了完全分类，将超扭接触结构的几何化约为平面场的同伦类代数。这种对紧密接触结构的洞察来得很慢。构建紧密结构的唯一通用方法是Stein填充（见[14,7]），其独特性由编辑于1998年1月28日收到。1991数学学科分类。初级53C15, 57M12；二次58 f05。
We consider the problem of realizing tight contact structures on closed orientable three-manifolds. By applying the theorems of Hofer et al., one may deduce tightness from dynamical properties of (Reeb) flows transverse to the contact structure. We detail how two classical constructions, Dehn surgery and branched covering, may be performed on dynamically-constrained links in such a way as to preserve a transverse tight contact structure. 1. CONTACT GEOMETRY AND DYNAMICS For a more thorough treatment of the basic definitions and theorems related to the geometry and dynamics of contact structures see, e.g., [1]. A contact structure 5 on a 3-manifold M is a totally nonintegrable 2-plane field in TM. More specifically, at each point p E M we have a 2-plane ~p C TpM that varies smoothly with p, with the property that 5 is nowhere integrable in the sense of Frobenius, i.e., there exists (locally) a defining 1-form ac (whose kernel is J) such that a A doa 7 0. If a is globally defined, ~ is called orientable and a a contact 1-form for ~. We adopt the common restriction to orientable contact structures. The interesting (and difficult) problems in contact geometry are all of a global nature: Darboux's Theorem (see, e.g., [23, 1]) implies that all contact structures are locally contactomorphic, or diffeomorphic preserving the plane fields. A similar result holds for a surface E in a contact manifold (M, ) as follows. Generically, TpE n $p will be a line in TpE. This line field integrates to a singular foliation SE called the characteristic foliation of S. One can show, as in the single-point case of Darboux's Theorem, that >E determines the germ of . along S. There has recently emerged a fundamental dichotomy in three dimensional contact geometry. A contact structure S is overtwisted if there exists an embedded disk D in M whose characteristic foliation D1 contains a limit cycle. If ~ is not overtwisted, then it is called tight. Eliashberg [6] has completely classified overtwisted contact structures on closed 3-manifolds the geometry of overtwisted contact structures reduces to the algebra of homotopy classes of plane fields. Such insight into tight contact structures is slow in coming. The only general method for constructing tight structures is by Stein fillings (see [14, 7]), and the uniqueness Received by the editors January 28, 1998. 1991 Mathematics Subject Classification. Primary 53C15, 57M12; Secondary 58F05.