The University of Sydney - School of Mathematics and Statistics

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.