Jet Bundles as Higher-Order Polarised $k$-Contact Manifolds

Abstract

We prove that the Cartan distribution $C_{\pi }^r$ on the jet bundle $J^r\pi$, for $\pi :E\to Q$, is an $N_{\pi }^r$-contact distribution, where $N_{\pi }^r$ is the corank of $C_{\pi }^r$. In other words, $C_{\pi }^r$ has non-degenerate distributional curvature, and admits, locally, a complementary distribution spanned by $N_{\pi }^r$ commuting infinitesimal symmetries. Consequently, there exists around every point of $J^r\pi$ an open $U\subset J^r\pi$ and an $R^{N_{\pi }^r}$-valued one-form $\eta \in {\Omega }^1(U,R^{N_{\pi }^r})$ such that $\ker \eta =C_{\pi }^r{|}_U$ and $\ker \eta \oplus \ker d\eta =TU$. This recovers the canonical structure of $J^r\pi$, 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 $N_{\pi }^r$-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.

66 pages. More detailed proofs and comments. Typos corrected. Reduction theory for classes of non-local symmetries added