实微分同胚群的全纯逼近
Holomorphic Approximation for Real Diffeomorphism Groups
AI总结：
该研究利用密度性质，证明n≥2时ℝⁿ的微分同胚可在Whitney Cᵏ拓扑中被ℂⁿ自同构逼近，多数线性代数群的分裂实形式满足相关充分条件，且其保体积微分同胚也可被全纯逼近。
AI中文摘要：
我们利用密度性质的概念证明，对于任意正整数k，n≥2时ℝⁿ的每个微分同胚都可在Whitney Cᵏ拓扑中被ℂⁿ的自同构逼近。更确切地说，我们找到了该微分同胚全纯逼近成立的充分条件，并证明多数线性代数群的分裂实形式满足这些条件。同样地，我们还证明了配备左不变体积形式的线性代数群的分裂实形式，其保体积微分同胚也可被全纯逼近。
英文摘要：
We show that every diffeomorphism of $\mathbb{R}^n$ for $n \ge 2$ can be approximated by an automorphism of $\mathbb{C}^n$ in the Whitney $C^k$-topology for any positive integer $k$ using the notion of density property. More precisely we find sufficient conditions for this holomorphic approximation of diffeomorphisms to hold and prove that the split real forms of most linear algebraic groups satisfy these conditions. In the same manner we also show holomorphic approximations of volume-preserving diffeomorphisms for the split real forms of linear algebraic groups equipped with the left-invariant volume form.
