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Differential Geometry - MAT00006H

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  • Department: Mathematics
  • Module co-ordinator: Dr. Graeme Wilkin
  • Credit value: 10 credits
  • Credit level: H
  • Academic year of delivery: 2022-23
    • See module specification for other years: 2021-22

Related modules

Pre-requisite modules

Co-requisite modules

  • None

Prohibited combinations


Additional information

Prerequisite modules: students must have taken courses in Vector Calculus and Linear Algebra

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.

Assessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Differential Geometry		 2 hours 100

Special assessment rules

None

Reassessment

Task Length % of module mark
Closed/in-person Exam (Centrally scheduled)
Differential Geometry		 2 hours 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


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.