Generalized DPW method and an application to isometric immersions of space forms
Generalized DPW method and an application to isometric immersions of space forms
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DOI：
10.1007/s00209-008-0367-9
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发表时间：
2006-04
影响因子：
0.8
通讯作者：
D. Brander;J. Dorfmeister
中科院分区：
文献类型：
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作者：
D. Brander;J. Dorfmeister
设G是复李群，Λ G表示从单位圆到G的映射群，属于适当类。一个从流形M到ΛG的可微映射F称为连通序的，如果F的Maurer-Cartan型族的圈参数λ中的Fourier展开式，即，具有以下形式。几何学中的大多数可积系统都与这样的映射有关。一般来说，DPW方法使用Birkhoff型分裂将调和映射约简到对称空间中，该对称空间可以由一定的阶映射表示，分别为一对较简单的阶和映射。相反，我们可以从任意一对和映射构造这样一个调和映射.这允许维尔斯特拉斯型描述调和映射到对称空间。我们扩展了这个方法，证明了对于一个大类的循环群，一个连接序映射，对于a < 0 <B，唯一地分裂成一对和映射。作为一个应用，我们证明了球面或双曲空间中具有平坦法丛的常非零截面曲率子流形分裂成平坦子流形对，从而将问题（至少局部地）简化为平坦的情况。为了充分扩展DPW方法来处理这个问题，需要一个更一般的岩泽型分裂的循环组，我们证明始终保持至少在本地。
LetGbe a complex Lie group andΛGdenote the group of maps from the unit circleintoG, of a suitable class. A differentiable mapFfrom a manifoldMintoΛG, is said to be ofconnection orderif the Fourier expansion in the loop parameter λ of the-family of Maurer-Cartan forms forF, namely, is of the form. Most integrable systems in geometry are associated to such a map. Roughly speaking, the DPW method used a Birkhoff type splitting to reduce a harmonic map into a symmetric space, which can be represented by a certain ordermap, into a pair of simpler maps of orderand, respectively. Conversely, one could construct such a harmonic map from any pair ofandmaps. This allowed a Weierstrass type description of harmonic maps into symmetric spaces. We extend this method to show that, for a large class of loop groups, a connection ordermap, fora< 0 <b, splits uniquely into a pair ofandmaps. As an application, we show that constant non-zero sectional curvature submanifolds with flat normal bundle of a sphere or hyperbolic space split into pairs of flat submanifolds, reducing the problem (at least locally) to the flat case. To extend the DPW method sufficiently to handle this problem requires a more general Iwasawa type splitting of the loop group, which we prove always holds at least locally.