论文标题
关于退化的para-cr结构：尚丹还原和均匀模型
On degenerate para-CR structures: Cartan reduction and homogeneous models
论文作者
论文摘要
本文努力研究了Levi Denerate Cr几何形状的最新作品，努力研究更广泛，更灵活的Para-Cr结构，在复杂的共轭下，不这样的不知道的限制放宽了。我们考虑$ 5 $维度的Para-Cr结构，其Levi形式的等级为$ 1 $，并且在参数和变量方面都是$ 2 $ nondementer的。消除参数，这些结构可以用PDE $ z_y = f（x，y，y，z，z_x）$ $ $ \，\，\＆\，\，$ z__ {xxx} = $ f_ {z_xz_x} \ neq 0 $，是完全可集成的$ d_x^3 f =δ_yh $， 在高级cartan的等价方法中，我们确定了所有相关的均匀模型以及它们的对称性： （i）$ z_y = \ tfrac14（z_x）^2 \ quad \＆\ quad z__ {xxx} = 0 $; （ii）$ z_y = \ tfrac14（z_x）^2 \ quad \＆\ quad z__ {xxx} =（z__ {xx}）^3 $; （iiia）$ z_y = \ tfrac14（z_x）^b \，\，\，\＆\，\，\，z_ {xxx} =（2-b）\ frac {（z__ {xx}）^2}^2} {z_x} {z_x} {z_x} $ for $ z_x> 0 $ for $ z_x> 0 $ for $ z_x> for $ z_x> for $ b \ b \ b \ b \ b \ b \ b \ b \ b \ in [1,2）$; （iiib）$ z_y = f（z_x）\ quad \＆\ quad z_ {xxx} = h（z_x）\ big（z__ {xx} \ big）^2 $，其中函数$ f $由隐式公式确定：\ [（z_x^2+f（z_____x） 2b \，\ Mathrm {arctan} \ tfrac {bz_x-f（z_x）}} {z_x+bf（z_x）} \ right）= 1+b^2 \]和wery：\ [h（z_x）：= = = = = = = = = = \ frac {（b^2-3）z_x-4bf（z_x）} {（f（z_x）-bz_x）^2}，\]，对于任何真正的$ b> 0 $。
Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider $5$-dimensional para-CR structures whose Levi forms are of constant rank $1$ and that are $2$-nondegenerate both with respect to parameters and to variables. Eliminating parameters, such structures may be represented modulo point transformations by pairs of PDEs $z_y=F(x, y, z, z_x)$ $\,\,\&\,\,$ $z_{xxx}=H(x,y,z,z_x,z_{xx})$, with $F$ independent of $z_{xx}$ and $F_{z_xz_x} \neq 0$, that are completely integrable $D_x^3 F=Δ_y H$, Performing at an advanced level Cartan's method of equivalence, we determine all concerned homogeneous models, together with their symmetries: (i) $z_y=\tfrac14 (z_x)^2\quad \&\quad z_{xxx}=0$; (ii) $z_y=\tfrac14 (z_x)^2\quad \& \quad z_{xxx}=(z_{xx})^3$; (iiia) $z_y=\tfrac14 (z_x)^b\,\, \& \,\,z_{xxx} = (2-b)\frac{(z_{xx})^2}{z_x}$ with $z_x>0$ for any real $b\in[1,2)$; (iiib) $z_y = f(z_x)\quad \& \quad z_{xxx}=h(z_x)\big(z_{xx}\big)^2$, where the function $f$ is determined by the implicit equation: \[ (z_x^2+f(z_x)^2)\, \mathrm{exp} \left( 2b\,\mathrm{arctan}\tfrac{bz_x-f(z_x)}{z_x+bf(z_x)} \right) = 1+b^2 \] and where: \[ h(z_x) := \frac{(b^2-3)z_x-4bf(z_x)}{(f(z_x)-bz_x)^2}, \] for any real $b>0$.
