Finite Weil Jets, Identity Transfer, and the Exponential Geometry of Proper Orthogonal Tensors

Abstract

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 $2k\pi$. 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.