- Department: Mathematics
- Credit value: 20 credits
- Credit level: H
- Academic year of delivery: 2023-24
- See module specification for other years: 2024-25
This module will develop the classical differential geometry of curves and surfaces.
Pre-requisite modules
Co-requisite modules
- None
Prohibited combinations
- None
This is a first course in the differential geometry of curves and surfaces which requires only a background in vector calculus and some abstract linear algebra.
Occurrence | Teaching period |
---|---|
A | Semester 2 2023-24 |
This module will develop the classical differential geometry of curves and surfaces.
By the end of the module, students will be able to:
Compute the curvature and torsion of a space curve, and use them to classify curves geometrically.
Formulate the definition of a smooth surface, and differentiate smooth functions defined on them.
Apply the Regular Value Theorem to construct a wide variety of examples of smooth surfaces and identify their tangent spaces.
Formulate the first and second fundamental forms of a surface, and compute them using local coordinates.
Formulate the different notions of curvature of a surface, and use them to classify the points of a surface according to their geometric type.
Calculate the geodesic curvature of a curve on a surface, the total curvature of a polygon on a surface, and relate the total curvature of a closed surface to its topology.
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; the Gauss-Bonnet theorem for compact surfaces.
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
None
Task | % of module mark |
---|---|
Closed/in-person Exam (Centrally scheduled) | 100 |
Current Department policy on feedback is available in the student handbook. Coursework and examinations will be marked and returned in accordance with this policy.
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