Geometry in dimension 4
Student No.:50
Time:Mon./Thu. 13:00-14:50
Instructor:Lars Andersson  [Albert Einstein Institute]
Place:Conference Room 3, Floor 2, Jin Chun Yuan West Building (近春园西楼2层第三会议室)
Starting Date:2012-4-16
Ending Date:2012-5-10

Course description:

Riemannian and Lorentzian geometry in dimension 4 exhibits many interesting special features and coincidences which do not generalize directly to higher dimensions. These include the relation between tensors and 2-spinors which provides a powerful calculational tool. In this course I will introduce the 2-spinor and related formalisms in both the Lorentzian and the Riemannian cases, and discuss some important examples of special 4-geometries, focussing on Ricci flat spaces. The classes of spaces which will be discussed include Petrov type D spaces (both Riemannian and Lorentzian) and self-dual spaces in the Riemannian case. The Kerr family of black hole spacetimes is perhaps the most important example of a Petrov type D space and an understanding of the characterizations of Kerr as well as its hidden symmetries can be expected to be of importance for the black hole stability problem. On the Riemannian side, the Euclidean signature counterpart to the Kerr space and the gravitational instantons provide a rich zoo of examples.

Topics include hidden symmetries, conserved quantities and characterizations.



Differential geometry


Reference for the course:

R. Penrose, W. Rindler: Spinors and spacetime, vol I, II, CUP 1986
Lawson, H. B., Michelsohn, M.-L., Spin Geometry, PUP 1990