自对偶爱因斯坦空间和一般天体方程。作为坐标的本征函数
Self-dual Einstein spaces and the general heavenly equation. Eigenfunctions as coordinates
摘要：
本征函数被证明构成特权坐标的自对偶爱因斯坦空间与基本的控制方程被揭示为一般的天体方程。这里发展的形式主义可以用来连接各种已知的天体方程的算法。特别是，Plebanski的第一和第二天方程之间的经典连接检索和解释的特征函数。此外，最近推出的TED方程，构成了一个4+4-维可积的一般天球方程的行波约化的连接被发现。这些都是通过（部分）勒让德变换。作为一个特殊的应用，我们证明了一大类的自对偶爱因斯坦空间所管辖的兼容系统的dispersionless广田方程是真正的四维的（通用）度量不承认任何（适当的或非适当的）共形Killing向量。这推广了已知的联系之间的一个特定类别的自对偶爱因斯坦空间和无色散广田方程编码三维爱因斯坦-外尔几何。
英文摘要：
Eigenfunctions are shown to constitute privileged coordinates of self-dual Einstein spaces with the underlying governing equation being revealed as the general heavenly equation. The formalism developed here may be used to link algorithmically a variety of known heavenly equations. In particular, the classical connection between Plebanski's first and second heavenly equations is retrieved and interpreted in terms of eigenfunctions. In addition, connections with travelling wave reductions of the recently introduced TED equation which constitutes a 4+4-dimensional integrable generalisation of the general heavenly equation are found. These are obtained by means of (partial) Legendre transformations. As a particular application, we prove that a large class of self-dual Einstein spaces governed by a compatible system of dispersionless Hirota equations is genuinely four-dimensional in that the (generic) metrics do not admit any (proper or non-proper) conformal Killing vectors. This generalises the known link between a particular class of self-dual Einstein spaces and the dispersionless Hirota equation encoding three-dimensional Einstein-Weyl geometries.
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