MOBIUS STRUCTURES AND TWO DIMENSIONAL EINSTEIN-WEYL GEOMETRY
MOBIUS STRUCTURES AND TWO DIMENSIONAL EINSTEIN-WEYL GEOMETRY
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莫比乌斯结构和二维爱因斯坦-韦尔几何
DOI：
10.1515/crll.1998.111
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
D. Calderbank
中科院分区：
文献类型：
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作者：
D. Calderbank
共形2-流形拥有一个迷人的丰富和优雅的理论，可以从许多方面来看：它是复分析中的黎曼曲面理论，或者代数几何中的复曲线理论。本文从纯微分几何的观点出发，介绍了共形2-流形可能具有的两种几何结构，并研究了它们之间的关系。这些结构与黎曼曲面理论的射影结构和空结构密切相关。第一个结构可以被看作是一个不可积或非全纯版本的复射影结构，并将被称为MM奥比乌斯结构。共形2-流形上的一个可积或在MM上的奥比乌斯结构诱导出一个复射影结构：该流形具有一个其过渡函数是复MM奥比乌斯变换的图集。然而，与常见用法9]相反，本文讨论的MM奥比斯结构不一定是可积的：它们具有曲率，类似于共形3-流形的Cotton-York张量，其消失等价于可积性。MM obius结构也不同于真实的投射结构，在很大程度上与在更高维度中的共形和真实的投射结构不同。(In一维，MM obius和真实的射影结构一致，并且总是可积的。）这里感兴趣的另一个主题是Einstein-Weyl几何3，8，14]。这是一个共形流形的几何，它配备了一个相容的（或共形的）无挠联络，使得这个联络的里奇张量的对称无迹部分为零。这些流形以一种自然的方式概括了爱因斯坦流形，最近已经进行了一些详细的研究（见2，6，8]和其中的参考文献）。在[11]中，Ped-ersen和Tod提出了紧二维Einstein-Weyl流形的分类问题|Tod [13]对紧致三维Einstein-Weyl流形的可能几何进行了（局部）分类。然而，在二维情况下，给出的Einstein-Weyl流形的定义是空洞的，Pedersen和Tod没有给出另一种定义。本文的主要目的之一就是明确地给出这样一个定义，并给出紧可定向例子的分类。
Conformal 2-manifolds possess a fascinatingly rich and elegant theory which can be viewed in many ways: it is the theory of Riemann surfaces in complex analysis, or of complex curves in algebraic geometry. In this paper, a purely diierential geometric point of view will be taken, the aim being to introduce two geometric structures that a conformal 2-manifold might be equipped with, and to study the relationship between them. These structures are closely related to the projective and aane structures of Riemann surface theory. The rst structure can be viewed as a nonintegrable or nonholomorphic version of a complex projective structure, and will be called a MM obius structure. An integrable or at MM obius structure on a conformal 2-manifold induces a complex projective structure: the manifold possesses an atlas whose transition functions are complex MM obius transformations. However, contrary to common usage 9], the MM obius structures discussed herein are not necessarily integrable: they possess a curvature, analogous to the Cotton-York tensor of a conformal 3-manifold, whose vanishing is equivalent to integrability. MM obius structures are also diierent from real projective structures, in much the same way as conformal and real projective structures diier in higher dimensions. (In one dimension, MM obius and real projective structures do coincide and are always integrable.) The other topic of interest here is Einstein-Weyl geometry 3, 8, 14]. This is the geometry of a conformal manifold equipped with a compatible (or conformal) torsion free connection, such that the symmetric tracefree part of the Ricci tensor of this connection vanishes. These manifolds generalise Einstein manifolds in a natural way, and have been investigated in some detail recently (see 2, 6, 8] and references therein). In 11], Ped-ersen and Tod posed the problem of classifying compact two dimensional Einstein-Weyl manifolds|the possible geometries of compact three dimensional Einstein-Weyl mani-folds have been classiied (locally) by Tod 13]. However, the deenition just given of an Einstein-Weyl manifold is vacuous in the two dimensional case and Pedersen and Tod did not ooer an alternative deenition. One of the main goals of this paper is to give explicitly such a deenition and present a classiication of the compact orientable examples.