Differential Geometry - MAT00102H
Module summary
This module will develop the classical differential geometry of curves and surfaces.
Related modules
Additional information
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.
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 2 2024-25 |
Module aims
This module will develop the classical differential geometry of curves and surfaces.
Module learning outcomes
By the end of the module, students will be able to:
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Compute the curvature and torsion of a space curve, and use them to classify curves geometrically.
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Formulate the definition of a smooth surface, and differentiate smooth functions defined on them.
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Apply the Regular Value Theorem to construct a wide variety of examples of smooth surfaces and identify their tangent spaces.
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Formulate the first and second fundamental forms of a surface, and compute them using local coordinates.
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Formulate the different notions of curvature of a surface, and use them to classify the points of a surface according to their geometric type.
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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.
Module content
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.
Indicative assessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100 |
Module feedback
Current Department policy on feedback is available in the 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