Differential Geometry - MAT00102H
Module summary
This module will develop the classical differential geometry of curves and surfaces.
Related modules
Pre-requisite 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.
Students on combined programmes at York who have not been able to take Vector Calculus MAT00047I can consider this module if they have a good mark (60% or more) in Multivariable Calculus and Matrices MAT00014C. In this case please consult the lecturer prior to choosing this module as some background reading may be required.
Elective Pre-Requisites
These pre-requisites only apply to students taking this module as an elective.
Semester 1
Prerequisites: A good knowledge of Vector Calculus,
such as that provided in second-year Maths or Physics modules on the subject.
Module will run
| Occurrence | Teaching period |
|---|---|
| A | Semester 1 2026-27 |
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 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 spaces; 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) | 100.0 |
Special assessment rules
None
Indicative reassessment
| Task | % of module mark |
|---|---|
| Closed/in-person Exam (Centrally scheduled) | 100.0 |
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