洛伦兹CR结构
Lorentzian CR structures
摘要：
综述了四维可重正化的一般协变拉格朗日模型（含一阶导数）的数学。洛伦兹CR流形是由四个特殊条件确定的四维复流形的全实子流形。定义的四元组允许定义一类洛仑兹度量，它允许两个测地和剪切自由同余。这些度量允许使用Weyl张量和Flaherty伪复杂结构对结构进行分类。嘉当程序允许定义三个相对不变量。洛仑兹CR结构可以看作是一对两个超曲面型的三维CR结构，它们可以在SU（1，2）群的基础上密切联系。在SU（2，2）群基础上的密切关系揭示了庞加莱群，它可以与自然界中的观察群等同.静态轴对称洛仑兹CR结构的例子进行了计算。对于每一个Lorentzian CR流形，找到了其周围复流形的一类Kaehler度量，从而导出了子流形上的相容Lorentzian度量类.然后洛伦兹CR流形成为相应的环境（Kaehler）辛流形中的拉格朗日子流形。洛仑兹CR流形可以被认为是运动方程的Einstein-Infeld-Hofman推导的背景下的动力学过程。
英文摘要：
The mathematics of a 4-dimensional renormalizable generally covariant lagrangian model (with first order derivatives) is reviewed. The lorentzian CR manifolds are totally real submanifolds of 4(complex)-dimensional complex manifolds determined by four special conditions. The defining tetrad permits the definition of a class of lorentzian metrics which admit two geodetic and shear free congruences. These metrics permit the classification of the structures using the Weyl tensor and the Flaherty pseudo-complex structure. The Cartan procedure permits the definition of three relative invariants. Viewed as a pair of two hypersurface-type 3-dimensional CR structures, the lorentzian CR structures may be osculated on the basis of SU(1,2) group. An osculation on the basis of SU(2,2) group reveals the Poincare group which may be identified with the observed group in nature. Examples of static axially symmetric lorentzian CR structures are computed. For every lorentzian CR manifold, a class of Kaehler metrics of the ambient complex manifold is found, which induce the class of compatible lorentzian metrics on the submanifold. Then the lorentzian CR manifold becomes a lagrangian submanifold in the corresponding ambient (Kaehler) symplectic manifold. The lorentzian CR manifolds may be considered as dynamical processes in the context of the Einstein-Infeld-Hofman derivation of the equations of motion.
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