MA3D9 Geometry of Curves and Surfaces
Lecturer: Felix Schulze
Term(s): Term 2
Status for Mathematics students: List A
Commitment: 30 lectures
Assessment: 100% 3-hour examination
Formal registration prerequisites: None
Useful background: Some familiarity with MA260 Norms, Metrics and Topologies or MA222 Metric Spaces and MA254 Theory of ODEs may be useful.
Synergies: This module goes well with the following modules:
- MA3H5 Manifolds
- MA3F1 Introduction to Topology
- MA3B8 Complex Analysis
- MA3H6 Algebraic Topology
- MA3G1 Theory of Partial Differential Equations
- PX436 General Relativity
Leads to: The following modules have this module listed as assumed knowledge or useful background:
Content: This will be an introduction to some of the ``classical'' theory of differential geometry, as illustrated by the geometry of curves and surfaces lying (mostly) in 3-dimensional space. The manner in which a curve can twist in 3-space is measured by two quantities: its curvature and torsion. The case a surface is rather more subtle. For example, we have two notions of curvature: the gaussian curvature and the mean curvature. The former describes the intrinsic geometry of the surface, whereas the latter describes how it bends in space. The gaussian curvature of a cone is zero, which is why we can make a cone out of a flat piece of paper. The gaussian curvature of a sphere is strictly positive, which is why planar maps of the earth's surface invariably distort distances. One can relate these geometric notions to topology, for example, via the so-called Gauss-Bonnet formula. This is mostly mathematics from the first half of the nineteenth century, seen from a more modern perspective. It eventually leads on to the very general theory of manifolds.
Aims: To gain an understanding of Frenet formulae for curves, the first and second fundamental forms of surfaces in 3-space, parallel transport of vectors and gaussian curvature. To apply this understanding in specific examples.
Books: John McCleary, Geometry from a differential viewpoint Cambridge University Press 1994.
Dirk J. Struik, Lectures on classical differential geometry Addison-Wesley 1950.
M Do Carmo, Differential geometry of curves and surfaces, Prentice Hall.