Jet Bundles as Higher-Order Polarised k-Contact Manifolds
Abstract
We prove that the Cartan distribution Crπ on the jet bundle Jrπ, for π:E→ Q, is an Nrπ-contact distribution, where Nrπ is the corank of Crπ. In other words, Crπ has non-degenerate distributional curvature, and admits, locally, a complementary distribution spanned by Nrπ commuting infinitesimal symmetries. Consequently, there exists around every point of Jrπ an open U⊂ Jrπ and an RNrπ-valued one-form η∈Ω1(U,RNrπ) such that kerη= Crπ|U and kerη⊕ker dη=TU. This recovers the canonical structure of Jrπ, algebraic Spencer contractions, and other mathematical entities in a simple unifying manner while providing a local Hamiltonian structure and other advantages. We will call a manifold endowed with a k-contact distribution a k-contact manifold. New types of polarisations for k-contact distributions lead to our main recognition theorem, which shows that a polarised k-contact distribution is locally equivalent to a Cartan distribution precisely when its polarisation is of jet type. This characterises finite-order jet geometry as polarised Nπr-contact geometry of jet type. Moreover, the highest-order vertical polarisation, holonomic submanifolds, characteristics for Lie symmetries and their brackets, and other structures are reconstructed and generalised via k-contact geometry. Solutions of PDEs become polarised Legendrian submanifolds, jet prolongations are polarised k-contact Legendrian prolongations, etc. Our k-contact formalism gives a common language for constructions that are awkward in a single jet presentation, and extends jet geometry to new problems. Our techniques provide new general reduction methods for PDEs retrieving standard and non-local reductions as particular cases, and are applied to PDEs.
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BibTeX
@misc{lucas2026bundles,
title = {{Jet Bundles as Higher-Order Polarised $k$-Contact Manifolds}},
author = {Javier de Lucas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2606.09263},
eprint = {2606.09263},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.09263},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Jet Bundles as Higher-Order Polarised k-Contact Manifolds
AU - Javier de Lucas
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2606.09263
UR - https://arxiv.org/abs/2606.09263
AB - We prove that the Cartan distribution C^r_π on the jet bundle J^rπ, for π:E→ Q, is an N^r_π-contact distribution, where N^r_π is the corank of C^r_π. In other words, C^r_π has non-degenerate distributional curvature, and admits, locally, a complementary distribution spanned by N^r_π commuting infinitesimal symmetries. Consequently, there exists around every point of J^rπ an open U⊂ J^rπ and an R^N^r_π-valued one-form η∈Ω^1(U,R^N^r_π) such that kerη= C^r_π|_U and kerη⊕ker dη=TU. This recovers the canonical structure of J^rπ, algebraic Spencer contractions, and other mathematical entities in a simple unifying manner while providing a local Hamiltonian structure and other advantages. We will call a manifold endowed with a k-contact distribution a k-contact manifold. New types of polarisations for k-contact distributions lead to our main recognition theorem, which shows that a polarised k-contact distribution is locally equivalent to a Cartan distribution precisely when its polarisation is of jet type. This characterises finite-order jet geometry as polarised N_π^r-contact geometry of jet type. Moreover, the highest-order vertical polarisation, holonomic submanifolds, characteristics for Lie symmetries and their brackets, and other structures are reconstructed and generalised via k-contact geometry. Solutions of PDEs become polarised Legendrian submanifolds, jet prolongations are polarised k-contact Legendrian prolongations, etc. Our k-contact formalism gives a common language for constructions that are awkward in a single jet presentation, and extends jet geometry to new problems. Our techniques provide new general reduction methods for PDEs retrieving standard and non-local reductions as particular cases, and are applied to PDEs.
ER -