Flat Tori of Finite Type in \(S^3\)
Alan McCarthy (UNSW)
Abstract
A torus in the three sphere (\(S^3\)) is said to be flat if it's Gaussian curvature is identically zero. Flat surfaces in \(S^3\) are of particular interest as they are the only complete surfaces in \(S^3\) with constant curvature that are not spheres. In this talk I will explain in more detail what I mean by 'flat', why the Gaussian curvature of a surface in \(S^3\) is not exactly the same as the Guassian curvature of a surface in \(R^3\). A summary will be given of the classification of flat tori in \(S^3\) in terms of their asymptotic curves due to Kitagawa, Bianchi and Spivak. I will also give a brief overview of my research into finite type flat tori and will explain why these objects are of interest.