Curvature and circles: A complex and real approach

Abstract

The first part of this thesis discusses complex curvature as presented by Needham and comparisons are made of this curvature to that of real curvature. Using the fact that complex numbers can be represented in the form of a vector, the use of the real curvature formula is easily used to help find similarities and differences with Needham's formula. In the second part, there is a discussion of independent subsets of the interval (0,1) and Cartesian products of the unit circle in the complex plane. Using Curtis' discussion of Monogenic Groups, elementary proofs are presented for a topological corollary of Kronecker's Approximation Theorem in two dimensions.