Foundations on k-contact geometry

k-Contact geometry is a generalisation of contact geometry to analyse field theories. We develop an approach to k-contact geometry based on distributions that are distributionally maximally non-integrable and admit, locally, k commuting supplementary Lie symmetries: the k-contact distributions. We related k-contact distributions with Engel, Goursat and other distributions, which have mathematical and physical interest. We give necessary topological conditions for the existence of globally defined Lie symmetries, k-contact Lie groups are defined and studied, and we study and propose a k-contact Weinstein conjecture for co-oriented k-contact manifolds. Polarisations for k-contact distributions are introduced and it is shown that a polarised k-contact distribution is locally diffeomorphic to the Cartan distribution of the first-order jet bundle over a fibre bundle of order k. We relate k-contact manifolds to presymplectic and k-symplectic manifolds on fibre bundles of larger dimension and define types of submanifolds in k-contact geometry. We study Hamilton-De Donder-Weyl equations in Lie groups for the first time. A theory of k-contact Hamiltonian vector fields is developed, and we describe characteristics of Lie symmetries for first-order partial differential equations in a k-contact Hamiltonian manner. We use our techniques to analyse Hamilton-Jacobi and Dirac equations. Other potential applications of k-contact distributions to non-holonomic and control systems are briefly described.