Abstract:
In general terms, this dissertation aims to study, from the geometric perspective of jet spaces
and exterior differential systems, the differential equations and the relationships that can be
established between their solutions. As we will see, from the process of reduction by symmetries
of exterior differential systems, we can systematically construct transformations that relate
solutions of equations obtained by a reduction process so that, in this way, we can establish
Bäcklund transformations between differential equations.
The work is subdivided into five chapters. Chapter 1 presents some prerequisites on smooth
manifolds, taking the opportunity to introduce the main notations and conventions used
throughout the text. Succesively, in Chapter 2, some basic results are shown in the context
of submersions, and, more specifically, some facts about fiber bundles are established that
will be useful for the development of the following chapters. Chapter 3, taking advantage
of the theoretical tools in previous chapters, introduces the jet spaces of sections of a fiber
bundle. Thus, the interpretation of differential equations as submanifolds of these geometric
environments is discussed, and based on this, some essential notions of symmetry are presented,
together with some basic results. Chapter 4 addresses the symmetry reduction of differential
equations form the point of view of the theory of exterior differential systems. Therefore, after
a short introduction to the formalism of exterior differential systems, their applications to the
study of differential equations are immediately discussed, aiming to elucidate above all the
aspects that play a central role in symmetry reduction. Finally, Chapter 5 is devoted to the
main topic of this work, which is the discussion of a useful method for obtaining Bäcklund
transformations through different reductions of the same exterior differential system.
