Aims & Learning Objectives:
Aims: This will be a self-contained course which uses little more than elementary vector calculus to develop the local differential geometry of curves and surfaces in 33. In this way, an accessible introduction is given to an area of mathematics which has been the subject of active research for over 200 years.
At the end of the course, the students will be able to apply the methods of calculus with confidence to geometrical problems. They will be able to compute the curvatures of curves and surfaces and understand the geometric significance of these quantities.
Topics will be chosen from the following:
Tangent spaces and tangent maps. Curvature and torsion of curves: Frenet-Serret formulae. The Euclidean group and congruences. Curvature and torsion determine a curve up to congruence.
Global geometry of curves: isoperimetric inequality; four-vertex theorem.
Local geometry of surfaces: parametrisations of surfaces; normals, shape operator, mean and Gauss curvature. Geodesics, integration and the local Gauss-Bonnet theorem.