Date of Award
Doctor of Philosophy (PhD)
Franz Pedit, Chair
Rob Kusner, Member
Peter Norman, Member
discrete differential geometry, integrable, klein, lie, quaternion, touch
The Grassmannian space GC(2, 4) embedded in CP5 as the Klein quadric of twistor theory has a natural interpretation in terms of the geometry of “round” 2-spheres in S4. The incidence of two lines in CP3 corresponds to the contact properties of two 2- spheres, where contact is generalized from tangency to include “half-tangency:” 2-spheres may be in contact at two isolated points. There is a connection between the contact properties of 2-spheres and soliton geometry through the classical Ribaucour and Darboux transformations. The transformation theory of surfaces in S4 is investigated using the recently developed theory of “Discrete Differential Geometry” with results leading to the conclusion that the discrete conformal maps into C of Hertrich-Jeromin, McIntosh, Norman and Pedit may be defined in terms a discrete integrable system employing halftangency in S4.
Shapiro, George, "On the Discrete Differential Geometry of Surfaces in S4" (2009). Open Access Dissertations. Paper 135.