Bäoklund haps and related connections

Abstract

The aim of this thesis is to show a link between solutions of Differential equations, and the integral submanifolds of sets of forms defined on jet bundles. The original idea behind Baoklund maps was discovered by A, V. Baoklund around 1875 during research into pseudospherlcal surfaces i.e. surfaces of constant negative curvature. The central idea of this thesis is the Backlund map, which is a smooth map of jet bundles parameterised by the target manifold of its co-domain. The original system of differential equations appears as a system of integrability conditions for the Baoklund map. The map induces an horizontal distribution on its domain from the natural contact structure of its co-domain, which makes possible a geometrical description in tenns of a connection, called here the BBcklund connection the system of integrability conditions reappears as the vanishing of the curvature of this oonnection. The paper by Backlund was very obscurely written and perhaps for that reason was ignored for nearly a hundred years. Development of applied mathematics, hydrodynamios, mechanlos and fluid mechanics published work raise the Interest of Baoklund maps and related topics. Chapter I gives a brief account of Jet-Bundles (Pirani) and contact module on Jet-Bundles, Chapter II summarises different ways of describing integrability conditions associated with Baoklund maps. It also explains hash-operator, use of contact module and some examples. Chapter III gives the idea of prolongation and explains with some examples. Chapter IV discusses the Idea of connections associated with Backlund maps, given by pull back of contact module of forms on J(^1)(M,N(_2)) determine a cart mi connection. Then shows that the solution of differential equation corresponds of curvature tensor of their connection. I conclude the introduction with a summary of my notation and conventions. All objects and maps are assumed to be in C; in the application they are generally real-analytic. If f:M→N is a map, then the domain of f is an open set in M, not necessarily the whole of M. If M is any set then Δ(_M) denotes the original map M→M x M by m → m (m,m) for every mEM. If o is a map of manifold then ϕ* is the induced map of forms and functions. If 0 is any collection of exterior forms the ϕ *0 means , d0 means I(0) means the ldeal generated by 0, where denotes ∆ the exterior product and xj0 means where X is any vector field and ┘ denotes the interior product of a vector field and a form. Projection of a cartesian product on the i-th factor is nenoted by piri. The end of an example is denoted by □.