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Riemannian Geometry 2020/21
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Riemannian Geometry
Roland Steinbauer, Fall term 2020/21
Hours/ECTS credits: 2/3
Time and Place: Tue. 13:15-14:45 online.
Start: 6.10.2020
Covid-Info: Although this course was prioritized by the Faculty as a class croom lecture it was necessary to go virtual following regulations of the University. It is exclusively accessible via Moodle.
General: Riemannian geometry is the study of smooth manifolds which carry a Riemannian metric, i.e. a scalar product on each tangent space. This allows to define local notions of angle, length, volume and curvature and hence to transfer the bulk of the classical elementary differential geometry of surfaces into the setting of abstract manifolds. In particular, global properties of the manifold can be studied by integrating the local contributions.
Riemannian geometry has its birthplace in Bernhard Riemann's habilitation lecture Ueber die Hypothesen, welche der Geometrie zu Grunde liegen (On the Hypotheses on which Geometry is based) of 1854. Especially since the second half of the 20th century it developped into one major branch of differential geometry with strong ties to group and representation theory as well as analysis and algebraic and differential topology. Finally it also lies at the mathematical foundations of Albert Einstein's General Relativity.
Aim and Contents: This is a first course on Riemannian geometry and provides a general introduction into the field. The natural major topics are
- (Semi-)Riemannian metrics and manifolds
- The Levi-civita connection
- Geodescis, the exponential map and convexity
- Arclength and Riemannian distance
- The Hopf Rinow theorem
- Curvature
- The Einstein equations
Traget audience: This lecture course primarily addresses master students in the geometry and topology branch of the curriculum but is especially open to students of theoretical physics with a twist towards general relativity.
Position in the curriculum: The course is part of the core module (alternative Pflichtmodulgruppe) "Differential geometry" (code: MGED) in the area of specialisation "Geometry and Topology”.
Literature, course material: This course is strongly based on the first three chapters of the standard text book "Semi-Riemannnian Geometry (With Applications to Relativity)" by Barrett O'Neill (Volume 103 of Pure and Applied Mathematics, Academic Press, San Diego, 1983).
I will also provide an updated version of the script "Riemannian Geometry" by Michael Kunzinger and myself in due time. Meanwhile you can look at the draft version acompanying my 2016 lecture course.
The final version of the lecture notes can be downloaded here.
Further literature which I find particularly useful includes
- M. do Carmo, Riemannian Geometry (Birkhäuser, Basel, 1992)
- S. Gallot, D. Hulin, J. Lafontaine, Riemannian Geometry (3rd edition, Springer, Berlin, 2004)
- the books of Brikel Clarke, Boothby, and Kühnel as well as the lecture notes of Andreas Kriegl.
Exams will exclusively be oral and by personal appointment starting from end Jannuary. Please drop me an email approxiamtely 2 weeks prior to the desired date. The duartion of the exam is approximately 45 minutes.
Continuation: In springterm I have scheduled a 4-hours corse on Lorentzian Geometry, which is a direct continuation of this course and leads up to a proof of the celebrated singularity theorems of Penrose and Hawking.
Lorentzian Geometry
Roland Steinbauer, Summer term 2021
Hours/ECTS credits: 4/6
Time and Place: Tue. 9:45-11:15, Thu. 9:00-10:30 online.
Start: 2.3.2021
General: Lorentzian geometry is the differential geometry of Lorentzian manifolds, that is semi-Riemannian manifolds with an indefinite metric of index 1. Its particular importance comes from the fact that Lorentzian manifolds, also called spacetimes, act as the stage for General Relativity (GR), Albert Einstein's eminent theory of space, time and gravity. In fact GR's fundamental statment is contained in its field equations, also called Einstein equations, which state that the gravitational field is a property of spacetime and that its energy matter content is proportional to its curvature. In this sense GR is but the study of 4-dimensional Lorentzian manifolds which satisfy the Einstein equations.
The decisive difference between Riemannian and Lorentzian manifolds is that a Riemannian metric encodes the topological structure of the manifolds (as eg. seen from the Hopf-Rinow theorem) while a Lorentzian metric does not even induce a metric in the toplogical sense via its length functional. Instead it gives rise to the causal structure: the vectors in each tangent space fall into one of the distinct classes of timelike, null and spacelike vectors according to the sign of their norm.
Aim and Contents: We study Lorentzian manifolds, and, in particular, their local and global causal structure with the goal to cover the famous singularity theorems of Penrose and Hawking. These are truly milestones in the development of GR, in particular the 1965-paper by Roger Penrose which won him the 2020-Nobel Price in physics. (You can find the paper here, and here you can find an appraisal written on the accassion of the centennial of GR in 2015). The theorems assert that under physically realistic conditions spacetimes generically become singular in the sense that they contain an incomplete causal geodesic. Moreover, they do not make any use of the field equations but only suppose a condition on the Ricci curvature along with some causality conditions, and so they are actually pure geometric results. If time permits we will cover as a further highlight a rather recent result by Bernal and Sanchez on the structure of globally hyperbolic spacetimes.
More specifically the topics of the course will be
- Prerequisites: sectional curvature, semi-Riemannian submanifolds
- Basic examples of spacetimes (Minkowski, (anti-)de Sitter, and Robertson-Walker spaces, Schwarzschild half-plane)
- Basic causality theory (local causality, causality conditions)
- Calculus of variations (Jacobi fields, focal and conjugate points)
- Global hyperbolicity (Cauchy hypersurfaces, developments, and horizons)
- The singularity theorms of Penrose and Hawking
- The stucture of globally hyperbolic spacetimes
Prerequisite for this course is a solid konwledge of manifolds and basic semi-Riemannian geometry as is available e.g. form the course in fall term.
Traget audience: This lecture course primarily addresses master students in the geometry and topology branch of the mathematics curriculum and the students of the Vienna Master Class Mathematical Physics.
Position in the curriculum: The course is part of the compulsary module "Electives in Geometry and Toplogy" (code: MGEV) in the area of specialisation "Geometry and Topology” in the master curriculum mathematics.
Mode: Due to Covid-regulations of the University this course will be fully online. You can access it exclusively via univie's moodel platform under the link https://moodle.univie.ac.at/course/view.php?id=217695. The individual sessions will be held as video conferences using the software collaborate and will be recorded.
In case you are interested in the course but not eligible to moodle, please drop me an email.
Literature, course material: My main sources will be
- Barrett O'Neill, Semi-Riemannnian Geometry (With Applications to Relativity) (Volume 103 of Pure and Applied Mathematics, Academic Press, San Diego, 1983), chapters 10 and 14.
- Christian Bär, Lorentzgeometrie, Vorlesungsskriptum, Universität Potsdam, 2006.
Exams will exclusively be oral and by personal appointment starting at the end of the semester. Please drop me an email approxiamtely 2 weeks prior to your desired date. In view of current university wide regulations exams will probably have to be online but changes might be possible. The duration of the exam is 45-60 minutes.