PMAT 41353: Differential Geometry

Course Code    : PMAT 41353

Title                   : Differential Geometry

Pre-requisites   : PMAT 22293

Learning Outcomes:

At the end of this course the student should be able to

• demonstrate the fundamental knowledge of curves and surfaces in space

• identify the importance of the two factors curvature and torsion, and their intrinsic properties

• determine the local shape of the surface using first and second fundamental forms

• examine the theory of abstract manifolds

• use the theory, methods and techniques of the course to solve mathematical problems.

 Course Contents:

Theory of Curves: 

• Concept of a curve, Parametrized curves, Regular curves: Arc length, Tangent vectors, Normal and binormal vector, Curvature and torsion, Frenet-serret formulae (Frenet formulas), Frenet frame, Isoperimetric inequality for a plane curve, The four-vertex theorem, General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves, Canonical representation of a curve. Involutes and Evolutes, Bertrand curves, Theory of contact.

Theory of Surfaces: 

• Concept of a surface, Tangent Plane, Topological properties of a surface, Surface of revolution, Ruled surfaces, Length of arc on a surface, Vector element of an area, First and second fundamental forms, Curves on a surface, Direction coefficients, Direction ratios, Family of curves on a surface, Double family of curves. Umbilical point, Intrinsic properties of a surface, Geodesics. Principal curvatures and directions, Gaussian and Mean curvatures, Lines of curvature, Rodrigues formula,

Introduction to Riemannian geometry: 

• Riemannian Manifolds, Smooth Manifolds

 Method of Teaching and Learning : A combination of lectures, tutorials and presentations.

Assessment : Based on tutorials, tests, presentations and end of course examination.

 Recommended Reading:

1. Lipschutz, L. (1969). Differential Geometry, McGraw-Hill.

2. Willmore, T.J. (2013). An Introduction to Differential Geometry, Oxford University Press.

3. Do Carmo, M.P. (2016). Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey.