Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles
Monotone Lagrange submanifolds of linear spaces and the Maslov class in cotangent bundles
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线性空间的单调拉格朗日子流形和余切丛中的 Maslov 类
DOI：
10.1007/bf02571385
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发表时间：
1991
影响因子：
0.8
通讯作者：
L. Polterovich
中科院分区：
文献类型：
--
作者：
L. Polterovich
1我们考虑下面两类辛流形：线性复空间·”和切长丛T* W。它们每个都被赋予标准的辛结构co。众所周知，co是正合的：~ o= dr。对于拉格朗日浸入f：W--* Vsetm（f）=[f*~]~ H1（W; R），令#（f）~ H1（W; Z）为Maslov类.用m（f）表示子群#（f）（H1（W; Z））~ Z的非负生成元。2本文研究了拉格朗日嵌入f：W--* V的不变量m（f），主要结果是：当W= S”~. T~，V= T* W和f是一个精确的拉格朗日嵌入（见Theor. 2和下文备注3和5）。对于W= T的情形，这个结果是由F.拉隆德和J. - C.”[10]而C。维泰博（IV的最终版本）。他们的方法不同，但都是基于C的以下结果。维泰博IV]：0< m（f）< n+ 1，对于每个拉格朗日嵌入f：T”--。C”。提到这个结果部分地证明了M. Audin [A2]指出，对于任何这样的嵌入，m（f）= 2。为了证明这个估计C.维泰博使用了一种基于对与I 12”中嵌入的拉格朗日环面相关联的某些特殊哈密顿系统的闭合轨道的研究的方法。我们的方法如下。首先我们证明了对于单调拉格朗日嵌入W~ 1~"O< m（f）< n+ 1（见定义Theor. 1和下面的注释2）。证明是基于Gromov的伪全纯曲线理论[-G]。然后使用F的论点。拉隆德和J. - C. Sikorav的结论，Maslov类是平凡的一些确切的拉格朗日子流形的连续丛。提到伪全纯曲线方法在lP]中用于研究拉格朗日曲面的Maslov类。本文的结果宣布在莫斯科研讨会的奇点理论的指导下，六阿诺德在1989年秋季。我非常感谢AB Givental和KM Khanin进行了有益的讨论，并感谢J. C.感谢Sikorav阅读手稿并提高理解力。一个拉格朗日浸入f：W~ 112”被称为单调（cf. IF 2]）如果
1 We consider below two kinds of symplectic manifolds, the linear complex space•" and the contangent bundle T* W. Each of them is endowed with the standard symplectic structure, co. It's well known that co is exact:~ o= d r. Denote one of these spaces by V. For a Lagrange immersion f: W--* Vsetm (f)=[f*~]~ H 1 (W; R) and let#(f)~ H 1 (W; Z) be the Maslov class. Denote by m (f) the non-negative generator of the subgroup#(f)(H1 (W; Z))~ Z. 2 In the present paper, we study the invariant m (f) for Lagrange embeddings f: W--* V. Our main result is that re (f)= 0 if W= S"~•...• T~, V= T* W and f is an exact Lagrange embedding (see Theor. 2 and Remarks 3 5 below). For the case W= T this result was obtained by F. Lalonde and J.-C. Sikorav [LS] and by C. Viterbo (the final version of IV]). Their methods are different, however, they are both based on the following result by C. Viterbo IV]: 0< m (f)< n+ 1 for every Lagrange embedding f: T"--. C". Mention that this result partially proves the hypothesis by M. Audin [A2] stating that m (f)= 2 for any such embedding. For the proof of this estimate C. Viterbo uses an approach based on the study of closed orbits of some special Hamiltonian system associated to an embedded Lagrange torus in I12". Our method is the following. First we prove that O< m (f)< n+ 1 for monotone Lagrange embeddings W~ 1~"(see the definition, Theor. 1 and Remark 2 below). The proof is based on Gromov's theory of pseudo-holomorphic curves [-G]. Afterwards using the arguments by F. Lalonde and J.-C. Sikorav we conclude that the Maslov class is trivial for some exact Lagrange submanifolds of the contangent bundles. Mention that the pseudo-holomorphic curves approach was used in lP] for study of the Maslov class of Lagrange surfaces. The results of this paper were announced at the Moscow Seminar of the Singularities Theory under the guidance of VI Arnold in autumn 1989. I am deeply grateful to AB Givental and KM Khanin for useful discussions, and to J.-C. Sikorav for reading the manuscript and improving remarks.Definition. A Lagrange immersion f: W~ 112" is called monotone (cf. IF2]) if