Differential Geometry II — Lorentzian Geometry

This lecture will be held in the summer term 2021. It builds upon the lecture Differential Geometry I by Clara Löh held during the winter term 2020/21.
Information about the content of this lectures will be published in January or February 2021.

Content In this lecture we will study semi-Riemannian manifolds, mainly concentrating on Lorentzian manifolds. More details will follow soon.

The image was obtained from the web page linked here Picture created by Alain Riazuelo, IAP/UPMC/CNRS under the license CC-BY-SA 3.0.

Permission by Gourgoulhon, 2013

Then we...

Time and Location

Exercise Sheets

• Exercise Sheet No. 1,
• Exercise Sheet No. 2,
• Exercise Sheet No. 3,
• Exercise Sheet No. 4,
• Exercise Sheet No. 5,
• Exercise Sheet No. 6,
• Exercise Sheet No. 7,
• Exercise Sheet No. 8,
• Exercise Sheet No. 9,
• Exercise Sheet No. 10,
• Exercise Sheet No. 11,
• Exercise Sheet No. 12,

Related Websites

• Link to the lecture's G.R.I.P.S. page / Link zur G.R.I.P.S.-Seite
• Page in the "KVV" / Seite im kommentierten Vorlesungsverzeichnis

Literature

Literature directly associated to the lecture

• C. Bär, Vorlesungsskript "Lorentzgeometrie", SS 2004,
• Kriele, Marcus. Spacetime, Foundations of General Relativity and Differential Geometry. Springer 1999
• B. O'Neill, Semi-Riemannian 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 space-time, Cambridge Monographs on Mathematical Physics, 1973

Literature about mathematical foundations

Scripts whose knowledge is required for the lecture

• B. Ammann, Lineare Algebra I, WS 2007/08
• B. Ammann, Analysis I+II, 2018/19
• B. Ammann, Analysis III, WS 2019/20
• B. Ammann, Analysis IV, SS 2020
• C. Löh, Differential Geometry I, Lecture Notes, Regensburg Winter term 2020/21

Alternative scripts

• C. Bär, Differential Geometry (Unpublished Lecture Notes), Summer Term 2013.
  
• C. Bär, Differentialgeometrie (Vorlesungsskript in deutsch), Summer Term 2006.
  
• Boothby, W. M., Introduction to differential manifolds and Riemannian geometry, Academic Press 1986

Literature about the physics associated to the subject

• C. Bär, Script to the lecture 'Relativity Theory', Summer Term 2013
• Fischer, Helmut und Kaul, Helmut. Mathematik für Physiker, Band 3. Teubner, 2003.
• d'Inverno, Ray. Einführung in die Relativitätstheorie, deutsche Ausgabe, (Ed. G. Schäfer, übers. O. Richter), VCH Weinheim, 1995
• Chrusciel, Piotr. Lectures on Mathematical Relativity
• Gourgoulhon, Jamarillo, New theoretical approaches to black holes
• Gourgoulhon, Talk about "Black holes: from event horisons to trapping horizons"

Books

• M. do Carmo, Riemannian Geometry, Birkhäuser
• Cheeger, Ebin, Comparison theorems in Riemannian Geometry
• F. Warner, Foundations of differentiable manifolds and Lie groups, Springer
• T. Sakai, Riemannian Geometry, Transl. Math. Monogr., AMS
• W. Kühnel, Differentialgeometrie, Vieweg
• J. Lee, Introduction to topological manifolds, Springer
• J. Lee, Introduction to smooth manifolds, Springer
• J. Lee, Riemannian manifolds, Springer

Other links

• An animation of a black hole hier. More on the background of this animation can be found here.
• web site of Eric Gourgoulhon