Differential Geometry (M3.4)

LECTURER : CC Remsing

Prerequisites

Linear algebra, Advanced calculus. Some exposure to geometry would be advantageous.

Description

Roughly speaking, differential geometry is concerned with understanding shapes and their properties in terms of (differential and integral) calculus. Differential geometry is a subject with a long and wonderful history, but also a subject which has found new relevance in many areas ranging from machinery design to gravitation and cosmology to the study of DNA.

An elementary course on differential geometry provides a perfect transition to higher mathematics and its applications. It is a subject which allows students to see mathematics for what it is - a unified whole mixing together geometry, calculus, linear algebra, differential equations, complex variables, calculus of variations and topology.

Topics

Curves (in the plane and in the space),  curvature, surfaces, the first fundamental form, isometries, the second fundamental form, the Gauss and Weingarten maps, the normal and geodesic curvatures,  the Gaussian and mean curvatures, principal curvatures, flat surfaces, surfaces of constant mean curvature,  geodesics, the Gauss and Codazzi-Mainardi equations, Gauss' remarkable theorem, surfaces of constant Gaussian curvature.

Textbook

"Elementary Differential Geometry" by Pressley (Springer, 2010).

Tutorials

ONE tutorial per week.

Tests/Exams

TWO class tests and ONE final examination.