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Differential Geometry II Semester SoSe 2021
Lecturer Bernd Ammann
Type of course (Veranstaltungsart) Vorlesung
Contents In this lecture we will study semiRiemannian manifolds, mainly concentrating
on Lorentzian manifolds.
Lorentzian manifolds arise when one combines ndimensional space and time to
an (n+1)dimensional manifold. An understanding of Lorentzian manifold is the key ingredient to understand the theoretical aspects of general relativity.
A Lorentzian metric is a symmetric (0,2)tensor g on a manifold of
dimension n+1, such that in every p∈M there is a basis (e_{0},...,e_{n}) with g(e_{ij})=0 for i ≠ j, g(e_{0},e_{0})=1, and g(e_{i},e_{i})=1 for i>0. In other words, up to the minus sign, the definition coincides with the one of a Riemannian manifold.
Many aspects that you know from Riemannian geometry also hold for Lorentzian manifolds, we just have to add some signs at some places. These manifolds may be curved, and important notions of curvature are sectional curvature, Ricci curvature and scalar curvature. The famous Einstein equations are a statement about the Ricci curvature of the Lorentzian manifold describing our universe, e.g. vacuum spacetime is simply a Lorentzian manifold with vanishing Ricci curvature.
This allows to study important examples, as e.g. the Schwarzschild solution which is a (3+1)dimensional manifold with vanishing Riccicurvature, but nonzero sectional curvature. Another example are socalled RobertsonWalker spacetimes which are used to model the evolution of the universe.
Two major goals of the lecture will be the singularity theorems by Penrose and Hawking.
More details are available on the lecture's web page.
Literature
 C. Bär,
Vorlesungsskript "Lorentzgeometrie", SS 2004,
 Kriele, Marcus. Spacetime, Foundations of General Relativity
and Differential Geometry. Springer 1999
 B. O'Neill, SemiRiemannian geometry. With applications to relativity.
Pure and Applied Mathematics, 103. Academic Press
 Wald, Robert. General Relativity. University of Chicago Press
 Misner, C.W. and Thorne, K.S. and Wheeler, J.A.. Gravitation, Freeman
New York, 2003
 Hawking, S.W. and Ellis, G.F.R., The large scale structure of spacetime, Cambridge Monographs on Mathematical Physics, 1973
Recommended previous knowledge Linear algebra I and II, Analysis I to IV, Differential Geometry I
Time/Date Monday and Wednesday 810
Location via zoom
Course homepage http://www.mathematik.uniregensburg.de/ammann/lehre/2021s_diffgeo2/ (Disclaimer: Dieser Link wurde automatisch erzeugt und ist evtl. extern)
Registration Registration for course work/examination/ECTS: FlexNow
Additional comments There will be a weekly exercise class.
Modules BV, MV, MGAGeo
ECTS 9
