Darboux Intergrability Of Wave Maps Into 2-Dimensional Riemannian Manifolds
Abstract (Summary)The harmonic map equations can be represented geometrically as an exterior differential system (EDS), E. Using this representation we study the harmonic maps from 2D Minkowski space into 2D Riemannian manifolds. These are also known as wave maps. In this case, E is invariant under conformal transformations of Minkowski space. The quotient of E by these conformal transformations, E/G, is an s=0 hyperbolic system. The main result of our study is that the prolonged EDS, E(k), is Darboux integrable if and only if the prolonged quotient EDS, E/G(k+1), is Darboux integrable. We also find invariants determining the Darboux integrability of both systems. Analyzing these invariants leads to three additional results. First, Darboux integrability of E, without prolongation, requires that the range manifold have zero scalar curvature. Second, after one prolongation there are two inequivalent metrics for which E(1) /super> is Darboux integrable. Third, prolonging to E(2) does not provide any further metrics with Darboux integrable wave maps.
School:Utah State University
School Location:USA - Utah
Source Type:Master's Thesis
Date of Publication:12/01/2008