MTH3110 - Differential geometry - 2018

This unit will explore the metric structure of curves and surfaces, primarily in 3-dimensional Euclidean space. The major focus is on the various concepts of curvature and related notions, and the relationships between them. Curvature and torsion of a curve. First and second fundamental forms of a surface. Geodesic and normal curvatures of a curve on a surface. Gaussian, mean and principal curvatures of a surface. Important theorems relating these concepts. Links will be drawn with many other areas of mathematics, including real and complex analysis, linear algebra, differential equations, and general relativity.

On completion of this unit students will be able to:

1. Explain the significance of intrinsic measures of curvature, for curves and surfaces in 3-dimensional space.
2. Perform calculations of curvature and related quantities for curves and surfaces in 3-dimensional spaces.
3. Explain and apply important concepts and theorems about the geometry of curves and surfaces in 3-dimensional space.
4. Apply results about differential geometry to write proofs and solve problems about curves and surfaces in 3-dimensional space.
5. Recognise many of the links between differential geometry and other areas of mathematics and physics, such as real and complex analysis, linear algebra, differential equations, and general relativity.
6. Communicate mathematical ideas relating to differential geometry in a clear, precise and rigorous manner.

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.

Three hours of lectures and one 2-hour support class per week

See also Unit timetable information

Applied mathematics

Mathematical statistics

Mathematics

Pure mathematics