Quantitative symplectic geometry
Quantitative symplectic geometry
复制标题
DOI：
10.1017/cbo9780511755187.002
复制
发表时间：
2005-06
期刊：
影响因子：
--
通讯作者：
K. Cieliebak;H. Hofer;J. Latschev;F. Schlenk
中科院分区：
文献类型：
--
作者：
K. Cieliebak;H. Hofer;J. Latschev;F. Schlenk
辛流形(M，ω)是具有非退化闭2-形式ω的光滑流形M。根据达布定理，这样的流形局部地看起来像是具有标准辛形的ℝ2n≅ℂn中的开集，因此辛流形没有局部不变量。这与黎曼流形形成鲜明对比，对于黎曼流形，黎曼度量允许各种曲率不变量。然而，辛流形确实允许许多全局数值不变量，其中最突出的是所谓的辛容。辛性是由I.Ekeland和H.Hofer在1990年引入的(尽管第一个辛性实际上是由M.Gromov构造的)。从那时起，定义了许多新的能力，并在中对它们进行了进一步的研究。关于辛能力的调查是。不同的能力以不同的方式定义，因此能力之间的关系经常导致辛几何和哈密顿动力学的不同方面之间令人惊讶的关系。这一点在第二节中得到了说明，在那里我们讨论了一些辛容量的例子，并描述了它们存在的一些结果。在第三节中，我们试图更好地理解所有辛容的空间，并进一步讨论辛容的一些一般性质。
A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux’s Theorem such a manifold looks locally like an open set in some ℝ2n≅ ℂnwith the standard symplectic form and so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities. Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer (although the first capacity was in fact constructed by M. Gromov). Since then, lots of new capacities have been defined and they were further studied in. Surveys on symplectic capacities are. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in Section 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In Section 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities.