关于Einstein方程、Ramanujan方程和Chazy方程解之间关系的注记
A note on the relationship between solutions of Einstein, Ramanujan and Chazy equations
摘要：
弗里德曼-罗伯逊-沃克度规的爱因斯坦方程在宇宙学中起着基础性的作用。直接寻找爱因斯坦方程的精确解，即使在这种简单的度量情形下，有时也是一件困难的工作。因此，利用与爱因斯坦方程等价或相关的其它方程的已知解来构造爱因斯坦方程的解是很有用的。在这项工作中，我们建立了爱因斯坦方程与其他两个著名的方程，即拉马努金方程和Chazy方程的关系。这两个方程在数论中起着重要的作用。利用已知的解的Ramanujan和Chazy方程，我们找到相应的解决方案的爱因斯坦方程。建立了弗里德曼方程与三叶结、八字结方程之间的关系。
英文摘要：
The Einstein equation for the Friedmann-Robertson-Walker metric plays a fundamental role in cosmology. The direct search of the exact solutions of the Einstein equation even in this simple metric case is sometime a hard job. Therefore, it is useful to construct solutions of the Einstein equation using a known solutions of some other equations which are equivalent or related to the Einstein equation. In this work, we establish the relationship the Einstein equation with two other famous equations namely the Ramanujan equation and the Chazy equation. Both these two equations play an imporatant role in the number theory. Using the known solutions of the Ramanujan and Chazy equations, we find the corresponding solutions of the Einstein equation. The relationship between the Friedmann equation and the equations of the trefoil knot and figure-eight knot is established.
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