Date of Award:
12-2008
Document Type:
Thesis
Degree Name:
Master of Science (MS)
Department:
Mathematics and Statistics
Committee Chair(s)
Mark Fels
Committee
Mark Fels
Committee
Ian Anderson
Committee
Charles Torre
Abstract
The harmonic map equations can be represented geometrically as an exterior differential system (EDS), ε. 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 ε by these conformal transformations, E/G, is an s=0 hyperbolic system.
The main result of our study is that the prolonged EDS, ε(k), is Darboux integrable if and only if the prolonged quotient EDS, ε(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 ε, without prolongation, requires that the range manifold have zero scalar curvature. Second, after one prolongation there are two inequivalent metrics for which ε(1) is Darboux integrable. Third, prolonging to ε(2) does not provide any further metrics with Darboux integrable wave maps.
Checksum
da5a5341f75c5e13891fae246b77ba00
Recommended Citation
Ream, Robert, "Darboux Integrability of Wave Maps into 2-Dimensional Riemannian Manifolds" (2008). All Graduate Theses and Dissertations, Spring 1920 to Summer 2023. 203.
https://digitalcommons.usu.edu/etd/203
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