Unit information: Introduction to Geometry in 2016/17

Unit name	Introduction to Geometry
Unit code	MATH20004
Credit points	20
Level of study	I/5
Teaching block(s)	Teaching Block 2 (weeks 13 - 24)
Unit director	Dr. Mackay
Open unit status	Not open
Pre-requisites	Calculus 1, Analysis 1A & Linear Algebra & Geometry 1
Co-requisites	None
School/department	School of Mathematics
Faculty	Faculty of Science

Description including 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.

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.

Intended Learning Outcomes

By the end of the course the students should

• have developed an understanding of the basic notions and results in differential geometry, like local coordinates, tangents and normals, curvature and fundamental forms, Gauss-Bonnet.
• be able to apply the methods to the description and solution of problems from pure and applied mathematics.

Teaching Information

The course will be based on lectures, with weekly homework problems and a weekly problems class.

Assessment Information

100% Examination

Raw scores on the examinations will be determined according to the marking scheme written on the examination paper. The marking scheme, indicating the maximum score per question, is a guide to the relative weighting of the questions. Raw scores are moderated as described in the Undergraduate Handbook.

If you fail this unit and are required to resit, reassessment is by a written examination in the August/September Resit and Supplementary exam period.

Reading and References

Lecture notes will be provided and there is no required textbook.
The following books may be helpful.

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