PMAT 41783: Differential Geometry

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Course Code    : PMAT 41783

Title                   : Differential Geometry

Pre-requisites   : PMAT 22583

 

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.

 Course Contents:

Theory of Curves: Concept of a curve, Arc length, Curvature and torsion, Frenet-serret formulae, General helix, intrinsic equations, Fundamental existence and uniqueness theorems for space curves, Canonical representation of a curve. Involutes and Evolutes, Theory of contact.

 Theory of Surfaces: Concept of a surface, 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, Geodesies. 

Curvature: Principle curvature and directions, Gaussian and mean curvatures, Lines of curvature Rodrigues formula.

Introduction to Riemannian geometry

 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-
  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.
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