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Differential Geometry - MAT00102H

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  • Department: Mathematics
  • Module co-ordinator: Dr. Chris Wood
  • Credit value: 20 credits
  • Credit level: H
  • Academic year of delivery: 2024-25
    • See module specification for other years: 2023-24

Module summary

This module will develop the classical differential geometry of curves and surfaces.

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations

  • None

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:

  1. Compute the curvature and torsion of a space curve, and use them to classify curves geometrically.

  2. Formulate the definition of a smooth surface, and differentiate smooth functions defined on them.

  3. Apply the Regular Value Theorem to construct a wide variety of examples of smooth surfaces and identify their tangent spaces.

  4. Formulate the first and second fundamental forms of a surface, and compute them using local coordinates.

  5. Formulate the different notions of curvature of a surface, and use them to classify the points of a surface according to their geometric type.

  6. 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.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Differential Geometry		 3 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Closed exam : Differential Geometry		 3 hours 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


The information on this page is indicative of the module that is currently on offer. The University is constantly exploring ways to enhance and improve its degree programmes and therefore reserves the right to make variations to the content and method of delivery of modules, and to discontinue modules, if such action is reasonably considered to be necessary by the University. Where appropriate, the University will notify and consult with affected students in advance about any changes that are required in line with the University's policy on the Approval of Modifications to Existing Taught Programmes of Study.