Finite Weil Jets, Identity Transfer, and the Exponential Geometry of Proper Orthogonal Tensors
Abstract
Let 𝔸=ℝ1𝔸⊕m be a Weil algebra with mN+1=0. We study the exponential geometry of the proper orthogonal group SO(3,𝔸) through finite Weil prolongation. An exact Rodrigues formula is obtained for every element of so(3,𝔸), and the exponential map so(3,𝔸)→SO(3,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 𝔸-holomorphic map on U+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.
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BibTeX
@misc{condurache2026finite,
title = {{Finite Weil Jets, Identity Transfer, and the Exponential Geometry of Proper Orthogonal Tensors}},
author = {Daniel Condurache},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.28505},
eprint = {2609.28505},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.28505},
}
RIS
TY - UNPB
TI - Finite Weil Jets, Identity Transfer, and the Exponential Geometry of Proper Orthogonal Tensors
AU - Daniel Condurache
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.28505
UR - https://arxiv.org/abs/2609.28505
AB - Let 𝔸=ℝ1_𝔸⊕m be a Weil algebra with m^N+1=0. We study the exponential geometry of the proper orthogonal group SO(3,𝔸) through finite Weil prolongation. An exact Rodrigues formula is obtained for every element of so(3,𝔸), and the exponential map so(3,𝔸)→SO(3,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 𝔸-holomorphic map on U+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.
ER -