Differential Geometry from Differential Equations
Differential Geometry from Differential Equations
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微分方程的微分几何
DOI：
10.1007/s002200100548
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发表时间：
2000
期刊：
影响因子：
--
通讯作者：
E. Newman
中科院分区：
文献类型：
--
作者：
S. Frittelli;C. Kozameh;E. Newman
我们首先展示了如何从形式的一般三阶颂歌出发，构造四维空间上的自然洛伦兹共形度量。当函数满足特殊的微分条件时，共形度量具有共形Killing场，从而允许共形度量映射到空间上的三维洛伦兹度量，或者等价地映射到原始微分方程解的空间上。然后将这种构造推广到一对微分方程组zss=S(z，zs，zt，zst，S，t)和ztt=T(z，zs，zt，zst，S，t)，其中zss关于andt的导数为zandzt。在这种情况下，从SandT可以自然地再次构造六维空间(z，zs，zt，zst，S，t)上的洛伦兹共形度量。当SandT满足类似于三阶常微分方程的微分条件时，6-空间具有一对共形Killing场，它允许通过到(z，zs，zt，zst)的四空间的映射和选择共形因子来构造四维洛伦兹度规。事实上，所有的四维洛伦兹度量都可以用这种方式来构造。这种构造，加上关于SandT的进一步条件，因此包括了爱因斯坦方程的所有(局部)解。
We first show how, from the general 3rd order ODE of the form , one can construct a natural Lorentzian conformal metric on the four-dimensional space . When the function satisfies a special differential condition the conformal metric possesses a conformal Killing field, , which in turn, allows the conformal metric to be mapped into a three dimensional Lorentzian metric on the space ) or equivalently, on the space of solutions of the original differential equation. This construction is then generalized to the pair of differential equations,zss=S(z,zs,zt,zst,s,t) andztt=T(z,zs,zt,zst,s,t), withzsandztthe derivatives ofzwith respect tosandt. In this case, fromSandT, one can again, in a natural manner, construct a Lorentzian conformal metric on the six dimensional space (z,zs,zt,zst,s,t). When theSandTsatisfy differential conditions analogous to those of the 3rdorder ode, the 6-space then possesses a pair of conformal Killing fields, and which allows, via the mapping to the four-space of (z,zs,zt,zst) and a choice of conformal factor, the construction of a four-dimensional Lorentzian metric. In fact all four-dimensional Lorentzian metrics can be constructed in this manner. This construction, with further conditions onSandT, thus includes all (local) solutions of the Einstein equations.