AI 中文总结
研究$\mathbb{C}^2$中区域间光滑映射性质，重点关注$\mathbb{R}^4$的严格配对CR线性自同构，通过分析块矩阵秩等，将其集表征为$\mathrm{GL}(2,\mathbb{C})^2$的12维子流形，证明其几何和结构刚性。
AI 中文摘要
本文研究了$\mathbb{C}^2$中区域间光滑映射的代数和微分几何性质，根据其全纯和反全纯导数分量的常数秩进行分类。特别关注$\mathbb{R}^4$的严格配对CR（SPCR）线性自同构，其中两个块矩阵$A$和$B$的秩均为1。我们将SPCR线性自同构集表征为$\mathrm{GL}(2,\mathbb{C})^2$的12维子流形，并分析其潜在的CR结构，证明其几何和结构刚性。
英文摘要
This paper investigates the algebraic and differential geometric properties of smooth mappings between domains in $\mathbb{C}^2$, classifying them according to the constant ranks of their holomorphic and antiholomorphic derivative components. Particular emphasis is placed on strictly paired CR (SPCR) linear automorphisms of $\mathbb{R}^4$, where both block matrices $A$ and $B$ have rank 1. We characterise the set of SPCR linear automorphisms as a 12-dimensional submanifold of $\mathrm{GL}(2,\mathbb{C})^2$ and analyse its underlying CR structure, demonstrating its geometric and structural rigidity.
Comments11 pages
