MM516, MM816 Geometry of Surfaces, Spring 2011

Messages

• The first lecture will be on Tuesday 1 February, 10-12, in U27A.

Course description

The official course description can be found here (for MM516).

Schedule

The factulty schedule for the course.

Literature

Course book: Andrew Pressley, Elementary differential geometry, Second Edition, Springer Undergraduate Mathematics Series (SUMS), Springer-Verlag London Ltd., London, 2010. In addition, we will use some notes, which will be handed out during the lectures.

As extra reading, I recommend the following:

Sigmundur Gudmundsson, An introduction to Gaussian Geometry. Lecture notes, Lund University.

William S. Massey, A Basic Course in Algebraic Topology, Graduate Texts in Mathematics 127, Springer-Verlag New York Inc., 1997.

Manfredo P. do Carmo, Differential Geometry of Curves and Surfaces, Prentice-Hall, 1976.

John Oprea, Differential Geometry and its Applications, Prentice Hall, 1997.

Alfred Gray, Modern Differential Geometry of Curves and Surfaces with Mathematica, Second Edition, CRC Press, New York, 1998.

Wolfgang Kühnel, Differential Geometry: Curves - Surfaces - Manifolds, American Mathematical Society, Student Mathematical Library, volume 16, 2002.

These books can be found on the reference shelf in the mathematical library.

Lectures

Here is a temporary list of subjects covered during the lectures:

Week	Section	Subjects
5	Notes, 4.1, 4.2, 4.3, 5.6	The Inverse Function Theorem, smooth surfaces, change of coordinates, smooth maps between surfaces.
6	4.3-4.5, 6.1-6.2, 7.1-7.3	Tangents and derivatives, orientability, Gauss map, first fundamental form, isometries, second fundamental form, Weingarten map, normal and geodesic curvature.
7	8.1-8.2, 8.6	Principal curvatures, Gaussian and mean curvature, principal directions, Gaussian curvature on compact surfaces.
8	10.1-10.2, 13.1-13.2, 13.4	The equations of Gauss and Codazzi-Mainardi, Teorema Egregium, The theorem of Gauss-Bonnet.
9	Massey (chapter 1)	Topological surfaces, surfaces from polygons, quotient topology, connected sum of surfaces.
10	Massey (chapter 1)	Classification of compact topological surfaces, Euler characteristic, orientability.
11	9.5, 10.3 (page 257-258)	Geodesic coordinates, surfaces of constant curvature.

Weekly sheets

• Weekly sheet 1 (week 5)
• Weekly Sheet 2 (week 6), including mandatory exercises.
• Weekly Sheet 3 (week 7), including mandatory exercises.
• Weekly Sheet 4 (week 8), including mandatory exercises.
• Weekly Sheet 5 (week 9).
• Weekly Sheet 6 (week 10).
• Weekly Sheet 7 (week 11), including exam questions.

Notes

• The Inverse Function Theorem.
• Hopf's Umlaufsatz.
• Surfaces with all points umbilic.

Exam

Oral exam, marks according to the Danish 7 mark scale. External evaluation. Mandatory assignments, which must be passed in order to attend the oral exam. Pass/Fail, internal evaluation by teacher.

Links

• The course in Blackboard.
• Andrew Pressley's home page at King's College, London.
• History of Mathematics Archive at St. Andrew's.
• Many interesting surfaces in 3D-XplorMath.
• David Eppsteins Geometry Junkyard.
• Mathworld has several articles about surfaces.
• Mathematical Curves and Surfaces by Roger Assouly.
• Beautiful pictures of surfaces by John M. Sullivan. See also Ken Brakke's home page.
• The theme of Mathematics Awareness Month 2005 is Mathematics and the Cosmos with lots of geometry.

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Last modified: Sun Mar 13 14:55:03 CET 2011

Martin Svensson (svensson@imada.sdu.dk)