Resumo:
This work aims to study the Darboux integrability of some Euler-Lagrange exterior differential systems that geometrically describe harmonic maps. In particular, we consider harmonic maps from the 2-dimensional Minkowski space to 2-dimensional Riemannian manifolds and prove that locally and up to isometries, there are only four such systems that are Darboux integrable up to 1-prolongation. Therefore, we classify, up to point transformations, the corresponding hyperbolic systems of Euler-Lagrange partial differential equations.
The work is subdivided into five chapters. Chapter 1 reviews the main preliminaries, including the theory of bundles, jet spaces, and exterior differential systems. In particular, we present the notion of Darboux integrability of an exterior differential system. Chapter 2 presents the Poincaré-Cartan variational formalism, introducing fundamental concepts for this study, including multi-contact geometry and the Poincaré-Cartan form. In Chapter 3, we calculated the Euler-Lagrange exterior differential system for harmonic maps between pseudo-Riemannian manifolds. In Chapter 4, we restrict and adapt the calculations from Chapter 3 to harmonic maps from 2-dimensional Minkowski space to 2-dimensional Riemannian manifolds. We then study the Darboux integrability of Euler-Lagrange exterior differential systems describing such maps and find conditions for this kind of integrability in terms of the Gauss curvature of the codomain manifolds. Finally, in Chapter 5, elucidating the classification results obtained by R.Ream, J.N. Clelland and P.J. Vassiliou, we classify the metrics and systems of differential equations of the corresponding harmonic applications, whose Euler-Lagrange exterior differential systems are Darboux integrable up to 1-prolongation.
