Introduction to Geometry

Unit aims

The aim of this unit is to introduce fundamental concepts in geometry in a hands-on and rigorous way, focused on curves and surfaces, and to lay the foundations for more advanced courses in later years.

Unit description

Geometry is central to mathematics, both as a subject in its own right and as an essential viewpoint on nearly every area of pure and applied mathematics.  This course focuses on the geometry of curves and surfaces in R2 and R3, continuing on from calculus.  

A key concept is the curvature of a surface, which describes locally how it bends, whether it is flat like Euclidean space, positively curved like a sphere, or, less-familiarly, negatively curved like a saddle.  A major goal is to prove the Gauss-Bonnet theorem, which links the curvature of a surface to its overall shape.

Relation to other units

The unit complements MATH20901 Multivariable Calculus and MATH20006 Metric Spaces, and leads into later units in geometry such as MATH32900 Differentiable Manifolds (from 18/19 Fields, Forms & Flows) and MATH30001 Topics in Modern Geometry.

Learning objectives

By the end of the course, students should

  • have developed an understanding of basic notions and results in differential geometry, like tangents, normals, curvature and Gauss-Bonnet
  • know and be able to use basic properties of Euclidean, spherical and hyperbolic geometries
  • understand, be able to prove and apply the methods in the course to the description and solution of problems from pure and applied mathematics.

Transferable skills

Clear logical reasoning, ability to write coherent and correct arguments, assimilation and use of new ideas, problem solving.

Syllabus

Topics covered will include:

  • Curves in two and three dimensional space, curvature and Frenet frames
  • Euclidean, Spherical and hyperbolic geometry
  • Regular surfaces in space, parametrisations, tangent planes, first fundamental form
  • Gauss map, 2nd fundamental form, Gaussian curvature
  • Local isometries and Gauss' Theorem Egregium
  • Local and global Gauss-Bonnet theorem
  • Optional topics: Minimal surfaces and geodesics

Reading and References

Lecture notes will be provided and there is no required textbook.

The following books may be helpful:

  • Elementary Differential Geometry, A. Pressley, Springer-Verlag, 2010.
  • Curved Spaces, P. M. H. Wilson, CUP, 2008.
  • Differential Geometry of Curves and Surfaces, M. P. Do Carmo, Prentice-Hall, 1976.
  • Elementary Differential Geometry, B. O'Neill, Academic Press, 2006.
  • Lectures on Classical Differential Geometry, D. J. Struik, Addison-Wesley, 1961

Unit code: MATH20004
Level of study: I/5
Credit points: 20
Teaching block (weeks): 2 (13-24)
Lecturer: Dr John Mackay

Pre-requisites

MATH11005 Linear Algebra and Geometry, MATH11007 Calculus 1, MATH10003 Analysis 1A

Co-requisites

None

Methods of teaching

Lectures, problem classes, homework problems, quizzes and solutions.  Formative assessment will be provided by problem sheets and quizzes that will be set and marked throughout the course.

Assessment methods

The pass mark for the unit is 40.

The final assessment mark will be 100% based on a 2 hour 30 minute examination in May/June. 

Note: Calculators are NOT allowed in the examination.

For information resit arrangements, please see the re-sit page on the intranet.

Please use these links for further information on relative weighting and marking criteria.

Further exam information can be found on the Maths Intranet.

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