有限Weil喷射、恒等转移与固有正交张量的指数几何

中文摘要 AI 辅助

设 $\mathbb{A}=\mathbb{R}1_{\mathbb{A}}\oplus\mathfrak{m}$ 为 Weil 代数，满足 $\mathfrak{m}^{N+1}=0$。我们通过有限 Weil 延拓研究固有正交群 $\mathrm{SO}(3,\mathbb{A})$ 的指数几何。对 $\mathfrak{so}(3,\mathbb{A})$ 的每个元素获得了精确的 Rodrigues 公式，并证明了指数映射 $\mathfrak{so}(3,\mathbb{A})\to\mathrm{SO}(3,\mathbb{A})$ 是满射。尽管每个固有正交 Weil 张量因此都是指数的且具有非归一化 Rodrigues 表示，但生成元分解为一个标量角度和一个单位轴可能在恒等元之上失败；该障碍对每个对数分支（包括标量角度 $2k\pi$）均被确立。解析机制是一个显式的有限正则性刚性定理：定义在 $U+\mathfrak{m}$ 上且具有标量值实迹的 $\mathbb{A}$-全纯映射唯一地由其有限 Weil 喷射决定。结合函子性，这实现了函数、矩阵、正交性和复合恒等式的精确转移。这些结果为理论及计算运动学中的对偶、多对偶、超对偶和超多对偶计算提供了严格基础。

英文摘要

Let $\mathbb{A}=\mathbb{R}1_{\mathbb{A}}\oplus\mathfrak{m}$ be a Weil algebra with $\mathfrak{m}^{N+1}=0$. We study the exponential geometry of the proper orthogonal group $\mathrm{SO}(3,\mathbb{A})$ through finite Weil prolongation. An exact Rodrigues formula is obtained for every element of $\mathfrak{so}(3,\mathbb{A})$, and the exponential map $\mathfrak{so}(3,\mathbb{A})\to\mathrm{SO}(3,\mathbb{A})$ is proved to be surjective. Although every proper orthogonal Weil tensor is therefore exponential and has an unnormalized Rodrigues representation, a factorization of a generator into one scalar angle and one unit axis may fail above the identity; the obstruction is established for every logarithmic branch, including scalar angles $2kπ$. The analytic mechanism is an explicit finite-regularity rigidity theorem: an $\mathbb{A}$-holomorphic map on $U+\mathfrak{m}$ with scalar-valued real trace is uniquely its finite Weil jet. Together with functoriality, this yields exact transfer of functional, matrix, orthogonality, and composition identities. The results provide a rigorous basis for dual, multidual, hyper-dual, and hyper-multidual calculations in theoretical and computational kinematics.

发表机构

• Gheorghe Asachi Technical University of Iaşi（雅西格奥尔基·阿萨奇工业大学）

机构由 AI 辅助整理，请以论文原文为准。

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14 pages