MM804 Riemannian geometry and Einstein metrics, autumn 2011, 1st quarter.

Announcements

• The fourth lecture on Monday 19 September, 16-18, will be held in U49.
• The third lecture on Monday 12 September, 16-18, will be held in U49. 16-18, in U49.
• The second lecture on Monday 5 September, 16-18, will be held in U49.
• The first lecture will be on Monday 29 August, 16-18, in room U49B.

Lecturer

My name is Martin Svensson, and you can find my webpage including contact details here. You are always welcome to come and see me in my office. Should you wish to make sure that I am available, you may want to give me a call or send me an email before you come.

Literature

I will follow my lecture notes, that you can download here. These notes follow closely (parts of) the book by Kuhnel: Differential Geometry, Curves-Surfaces-Manifolds (AMS), which is available on the reference shelf in the maths library, where more books on Riemannian geometry can be found for those who are interested. I also recommend the excellent lecture notes by Sigmundur Gudmundsson, which can be downloaded here.

Course overview

The official course description can be found here, and the syllabus here. I expect us to cover the following topics:
• manifolds and submanifolds
• tangent spaces and tangent bundles
• Riemannian metrics
• vector fields
• covariant derivatives
• curvature
• tensors
• Einstein metrics
• Jacobi fields
• complete manifolds and the theorem of Bonnet-Meyers

Schedule

This course will run as a reading course. We will meet for the first lecture and decide on a suitable arrangement for the course.

Lectures

Below is a preliminary plan for the lectures. The last column in the table refers to the relevant sction of the lecture notes.

Week	Topic	Section
35	Differentiable manifolds, submanifolds of the Euclidean space, tangent space.	Chapter 1, 2.1-2.2
36	The first fundamental form, vector fields, Lie bracket, second fundamental form, covariant derivative, geodesics, parallel transport, the exponential map, tensors.	2.3-2.10
37	The curvature tensor, curvature of hypersurfaces, Gaussian curvature, sectional, Ricci and scalar curvature.	2.11-2.13
38	Abstract manifolds, tangent space, tangent bundle, vector fields, the Levi-Civita connection.	3.1-3.2
39	Einstein manifolds, first and second variation of arch length, Gauss lemma, Jacobi fields.	3.3-3.4
40	Jacobi field on manifolds with constant sectional curvature, completeness, the theorem of Bonnet-Meyers.	3.5, 4.1-4.2

Exercise sheets

• Exercise Sheet 1 (week 35, including set 1 of mandatory exercises).
• Exercise Sheet 2 (week 36).
• Exercise Sheet 3 (week 37, including set 2 of mandatory exercises).
• Exercise Sheet 4 (week 38).
• Exercise Sheet 5 (week 39).
• Exercise Sheet 6 (week 40).
• Exercise Sheet 7 (week 41, including exam questions).

Exam

Mandatory exercises to be handed out during the course. At the end of the course, there is an oral exam with grades according to the Danish 7 grade scale, external examiner. The mandatory exercises must be passed to take the oral exam.

Links

• The course in Blackboard.
• The web page for MM804 autumn 2010.
• The web page for MM804 autumn 2009.
• The web page for MM804 autumn 2008.
• Riemannian geometry on Wikipedia.

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Last modified: Wed Aug 17 20:24:16 CEST 2011

Martin Svensson (svensson@imada.sdu.dk)