Jet Bundles as Higher-Order Polarised $k$-Contact Manifolds
Let $\pi:E\to Q$ be a fibred manifold, with $\dim Q=n$ and rank $m$. We prove that the Cartan distribution $C^r_\pi$ on $J^r\pi$ is an $N^r_\pi$-contact distribution, where $N^r_\pi=m\binom{n+r-1}{r-1}$, by giving a natural local construction of an $N^r_\pi$-contact form. This recovers the canonical structure of $J^r\pi$ and the Spencer contractions, among other structures. It also yields a natural local Hamiltonian structure on $J^r\pi$, recovering and extending the standard theory of characteristics to general Lie symmetries of the Cartan distribution. We introduce new classes of polarisations for $k$-contact distributions. This leads to our main recognition theorem, which shows that a polarised $k$-contact manifold is locally equivalent to a finite-order jet bundle with its 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, the symbol spaces, the vertical and horizontal differentials, holonomic submanifolds, and initial conditions for differential equations are reconstructed via $k$-contact geometry. For instance, adapted coordinates become $k$-contact Darboux coordinates, solutions of PDEs are treated as polarised Legendrian submanifolds, jet prolongations are recovered as polarised Legendrian prolongations, and so on. The resulting formalism gives a deeper geometric understanding of several parts of jet geometry, provides a uniform intrinsic language for constructions that are awkward in a single fixed jet presentation, such as B\"acklund transformations, and allows jet theory to be extended to new problems. In particular, our techniques provide very general reduction methods for PDEs. The results are applied to PDEs with mathematical and physical relevance.