Geometry - Spring Semester 2005 (Matematik 5GE)

Information from SIS. The class.

Latest news

June 10th Assignment 4

May 10: Assignment 3

The schedule around Easter is this: Is there a lecture Tuesday March 22? NO, Thursday 24? NO, Tuesday 29? YES.

March 15: Smooth structures on S^7 (Anders Storm Hansen)

March 3: Assignment 2 (dvi) (pdf) is due April 5th.

Feb 24: Notes (dvi) (pdf)

Feb 15: Assignment 1 (dvi) (pdf) is due March 1st. (Updated Thu Feb 17 12:51:31 CET 2005)

Feb 10: Copies of the book can be found in the library, under the big table for new books. (Thanks to Tarje and Jes H).

Jan 25: Monday 7th Feb is the first day of the semester.

Jan 18: Lee's book is out of print! I think I'll still use the book, so we'll have to do with photocopies.

What is the course about?

This is supposed to be a course in Differential Geometry. I will assume that you know a little differential geometry such as what you learn in 3GE. I shall not assume that you have followed 4GE.

Riemannian geometry was born June 10th 1854 when Riemann presented his Habilitationsschrift Ueber die Hypothesen welche der Geometrie zu Grunde liegen in Goettingen.

If you have any suggestions for this course, let me know.

Textbooks

We will use

supplemented by

and

It could also be an idea to read

which is a little master piece.

Logbook

1. Topological manifolds, smooth manifolds, smooth maps
2. The tangent space, the differential of a smooth map.
3. Local coordinate expression for tangent vectors and differentials of smooth maps. Smooth vector bundles.
4. The tangent bundle, the cotangent bundle, tensor bundles. (Lee Chp 2)
5. Contraction of a tensor. Riemannian metrics (Lee Chp 3)
6. Connections

Research problems

(Ulrik) Which Riemannian n-manifolds are locally graphs of function R^n -> R? (Or graphs of functions R^n -> R^d?)

Suggestions from students

• Lee C. Loveridge: "Physical and Geometric Interpretations of the Riemann Tensor, Ricci Tensor, and Scalar Curvature"
• Robert M. Wald: "General Relativity". The University of Chicago Press, 1984
• Integration on manifolds and de Rham's theorem
• Morse theory
• Fibre bundles, classifying spaces, and characteristic classes.

Links

• The geometry Junkyard
• The mathematical Atlas 53: Differential geometry
• The mathematical paper og G.F.B. Riemann (1826-1866)
• Non-Euclidean geometry
• John Baez: General Relativity Tutorial
• VIDIGEO (Visual Interactive Differential Geometry)
• Sigmundur Gudmundsson: An introduction to Riemannian geometry
• Differential geometry notes
• Differential geometry notes
• John M. Lee
• Blackwell's Bookstore