Moving frames, Bäcklund’s theorem, and its affine extension

The sine-Gordon equation (SGE) is a partial differential equation which has received renewed interest in recent decades due to its connections to the physical phenomenon of solitons.Of particular interest to geometers is a well-established correspondence between solutions to the SGE and pseudospherical surfaces.Elaborating on the work of Chern and Terng [CT80], this thesis rederives an appropriate Bcklund transformation, allowing for the generation of new Euclidean pseudospherical surfaces from known ones.In doing so, we make use of the geometric structure of pseudospherical line congruences.Taking advantage of the aforementioned correspondence, we then demonstrate how known solutions to the SGE enable the discovery of novel ones.The thesis proceeds to consider an analogous procedure for surfaces in affine space, replacing the notion of a pseudospherical line congruence with the more abstract notion of a Weingarten congruence.This consideration then allows for the generalization of Bcklund's theorem and the Euclidean integrability theorem to affine space.i ContributionWhile the entirety of this thesis is my own writing, it leans heavily on several key sources.Foremost among these are Jeanne N. Clelland's textbook From Frenet to Cartan: The Method of Moving Frames [Cle17], and the paper An Analogue of Bcklund's Theorem in Affine Geometry by Shiing-Shen Chern and Chuu-Lian Terng [CT80].Clelland's influence is especially notable in the first three chapters of this thesis.Her impact can be not only be seen in much of the content within these portions, but within their structuring as a whole.Many of the theorems, definitions, and proofs laid out here are taken from her aforementioned text [Cle17], and I have attempted to be as transparent of their sourcing as possible without appearing overly repetitive.Unless otherwise stated, the content of Sections 1.3 to 1.7, 2.2 to 2.5 (prior to Lemma 2.13), and 3.1 to 3.3 should be assumed to source from [Cle17, ch.3,4,9,6].Similarly, the content of Chapter 4 is entirely based off of Chern and Terng's paper [CT80], along with my attempts to further elucidate their steps when possible.Several other sources have been relied upon as well, albeit to a lesser extent.I have indicated their presence as they appear throughout the thesis.

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