Differential Geometry (MSc) - MAT00068H
- Department: Mathematics
- Credit value: 10 credits
- Credit level: H
- Academic year of delivery: 2022-23
Module summary
This module is for postgraduate students only.
Related modules
Additional information
Pre-requisite modules: for Natural Sciences students must either have Vector Calculus and Linear Algebra, or Maths for Sciences III.
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Spring Term 2022-23 |
Module aims
The aim of the module is to describe how techniques from advanced calculus and linear algebra may be used to give meaning to the concept of "shape" for curves and surfaces in space.
Module learning outcomes
At the end of the module you should be able to:
- Understand the curvature and torsion of a space curve, how to compute them, and how they suffice to determine the shape of the curve.
- Understand the definition of a smooth surface, and the means by which many examples may be constructed.
- Understand the various different types of curvature associated to a surface, and how to compute them.
- Understand the first and second fundamental forms of a surface, how to compute them, and how they suffice to determine the local shape of the surface.
- Appreciate the distinction between intrinsic and extrinsic aspects of surface geometry.
Module content
Syllabus
The geometry of smooth curves. Curvature; torsion; the Frenet formulas; congruence, and the fundamental theorem of space curves.
Smooth surfaces. Charts and atlases; tangent planes; the inverse function theorem; the regular value theorem; smooth mappings and their differentials; diffeomorphisms and local diffeomorphisms.
The geometry of smooth surfaces. First fundamental form (Riemannian metric); shape operator; normal curvature and principal curvatures; Gauss and mean curvatures; second fundamental form; local isometries; the "Theorema Egregium" of Gauss.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
Pass/fail
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the undergraduate student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
Indicative reading
J McCleary, Geometry from a Differentiable Viewpoint, Cambridge University Press.
C Baer, Elementary Differential Geometry, Cambridge University Press
A N Pressley, Elementary Differential Geometry, available as both a book and an e-book