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Differential geometry

Lecture cycle 2016/2017

Roland Steinbauer

News [latest on top]
2017-03-06: New version of the script on Riemannian geometry online.
2017-02-05: Details of the spring term course on Lorentzian geometry online
Sept. 2016: Details of the fall term course on Riemannian geometry online
June 2016: Material for the last week of the lecture course online. Details on the exam online.

English: This page contains all information on the three-part course on differential geometry by Roland Steinbauer (to be) taught from summer term 2016 until summer term 2017. It covers the entire module "Differential Geometry (MGED)" of the master curriculum in mathematics: the lecture course plus the accompanying problem session "Analysis on Manifolds" in the summer term 2016 and the lecture course "Riemannian Geometry" in the fall term 2016/17. In addition it contains the specialised course "Lorentzian Geometry" (code MGEV) in summerterm 2017.

Deutsch: Auf dieser Seite finden Sie alle Informationen zum 3-teiligen "Differentialgeometrie Zyklus" von Roland Steinbauer (Sommersemester 2016 bis Sommersemester 2017). Dieser deckt zunächst das gesamte Modul "Differentialgeometrie (MGED)" des Masterstudiums Mathematik ab: Vorlesung und begleitendes Prosemiar "Analysis auf Mannigfaltigkeiten" im Sommersemester 2016, Vorlesung "Riemann'sche Geometrie" im Wintersemester 2016/17. Darüber hinaus findet im Sommersemester 2017 die Vertiefungsvorlesung (Prüfungspass-Code MGEV) "Lorentzgeometrie" statt.

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Lorentzian Geometry

Spring term 2017

Roland Steinbauer

Lecture course: Lorentzian Geometry

Course number: 250099
Hours/ECTS credits: 4/6
Time and Place: Mo. 12.15-13.45, Do. 11.30-13.00 SR 11 (OMP 1)
Start: 2.3.2017

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 and say that under physically quite reasonable conditions spacetimes generically become singular in the sense that they contain an incomplete causal geodesic. Since they, however, do not make any use of the field equations but only suppose a condition on the Ricci curvature along with some causality conditions they are actually pure geometric results. A further highlight will be a rather recent result by Bernal and Sanchez on the structure of globally hyperbolic spacetimes.

More specifically the topics of the course will be

• Basic examples of spactimes (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 first two courses of this lecture-cycle.

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 compulsary module "Electives in Geometry and Toplogy" (code: MGEV) in the area of specialisation "Geometry and Topology” in the master curriculum mathematics.

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 Postdam, 2006.

Further hints on literature will be given during the course.

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 the desired date. The duartion of the exam is approximately 60 minutes.

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Riemannian Geometry

Fall term 2016/17

Roland Steinbauer

Lecture course: Riemannian Geometry

Course number:250070
Hours/ECTS credits: 2/3
Time and Place Tue. 10.00-11.30 SR 11, OMP 1
Start: 4.10.2016

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 an introduction into the field based on the lecture course Analysis on manifolds from the spring therm. 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
• Semi-Riemannian submanifolds

Prerequisite for this course is a solid konwledge of manifolds and tensors as is available e.g. form the course Analysis on manifolds.

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 typed notes for the course mainly based on handwritten notes of Michael Kunzinger acompanying his 2008 fall semesetr course Differentialgeometrie 2. These notes will be made available here prior to the respective lectures at least in a draft version.

• Version 0.98 (2017-03-06): pre-final; replacing all prior versions

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 in the below list from the spring term.

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.

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Analysis auf Mannigfaltigkeiten

Sommersemester 2016

Roland Steinbauer

Vorlesung: Analysis auf Mannigfaltigkeiten

Lehrveranstaltungsnummer:250079
Lehrveranstaltungstyp: VO
Semesterwochenstunden/ECTS: 4/6
Zeit und Ort: Di. 11:30-13:00, Mi. 9:45-11:15, Hs. 2, OMP1
Beginn: 1.3.2016

Allgemeines: Die Differentialgeometrie benutzt Techniken der Differentialrechnung zum Studium geometrischer Probleme, ist also Synthese aus Analysis und Geometrie. An ihrem Ursprung (18. und 19. Jahrhundert) steht die Theorie der ebenen und der Raumkurven, sowie der zweidimensionalen Flächen im Raum. Ihr zentraler Begriff ist der der differenzierbaren Mannigfaltigkeit, eine Verallgemeinerung des Flächenbegriffs und die "moderne" Differentialgeometrie ist ganz allgemein das Studium geometrischer Strukturen auf differenzierbaren Mannigfaltigkeiten. Seit Ende des 19. Jahrhunderts hat sich die Differentialgeometrie zu einem zentralen Gebiet der Mathematik entwickelt, das viele Querbezüge zu anderen mathematschen Gebieten sowie zur (theoretischen) Physik aufweist.

Inhalt: Diese Vorlesung bietet eine Einführung sowohl in die klassische Differentialgeometrie der Kurven und (Hyper)flächen als auch (schwerpunktmäßig) in die Analysis auf Mannigfaltigkeiten. Genauer werden folgende Themen behandelt:

• Kurven
• Differenzierbare Mannigfaltigkeiten
  • Teilmannigfaltigkeiten des R^n
  • Abstrakte Mannigfaltigkeiten
  • Topologische Eigenschaften von Mannigfaltigkeiten
  • Differentiation, Tangentialraum
  • Vektorbündel, Tangentialbündel, Vektorfelder
  • Tensoren
  • Differentialformen
  • Integration, Satz von Stokes
• Hyperflächen

Voraussetzung zum erfolgreichen Besuch der Lehrveranstaltung sind vor allem fundierte Kenntnisse der mehrdimensionalen Analysis, sowie Kenntnisse der Toplogie, der linearen Algebra und aus den gewöhnlichen Differentialgleichungen.

Zielpublikum: Prinzipiell alle, die am Thema Interesse und Freude haben, insbesondere Masterstudierende der Mathematik aber etwa auch Studierende der Physik.

Positionierung im Curriculum: Masterstudium Mathematik: Alternative Pflichtmodulgruppe „Standardausbildung Geometrie” im Studienschwerpunkt „Geometrie und Topologie”.

Literatur/Skriptum: Die Vorlesung wird sich stark am Skriptum Differentialgeometrie von Michael Kunzinger orientieren. Am Ende des Semesters ist eine korrigierte Neuauflage des Skriptums geplant.
Darüber hinaus gibt es natürlich eine schwer zu überblickende Fülle ausgezeichneter einführender Lehrbücher zum Thema. Eine unvollständige Liste von Texten, die meine Vorbereitung beeinflussen ist:

• R. Abraham, J.E. Marsden, Foundations of Mechanics.
• R.L. Bishop, S.I. Goldberg, Tensor Analysis on Manifolds.
• F. Brickel, R.S. Clark, Differentiable Manifolds. An Introduction.
• W. M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry.
• A. Cap, Differentialgeometrie 1 (Skriptum).
• M. do Carmo, Differential forms and applications.
• A. Kriegl, Differentialgeometrie (Skriptum).
• W. Kühnel, Differentialgeometrie. Kurven - Flächen - Mannigfaltigkeiten.
• P.W. Michor, Topics in Differential Geometry (Skriptum,).

Die Prüfung ist ausschließlich mündlich bei freier Terminvereinbarung. Bitte melde dich/melden Sie sich ca. 2 Wochen vor dem Wunschtermin per E-mail. Im Juli sind leider keine Prüfungen möglich, Termine für Anfang-Mitte Ausgust müssen noch im Juni fixiert werden. Die Prüfung dauert ca. 45-60 Minuten.

Proseminar zu "Analysis auf Mannigfaltigkeiten"

Lehrveranstaltungsnummer:250081
Lehrveranstaltungstyp: PS
Semesterwochenstunden/ECTS: 1/2
Zeit und Ort: Mi 12-12:45, SR 09, OMP 1
Beginn: 9.3.2016

Inhalt: Das Proseminar bildet mit der dazugehörigen Vorlesung eine untrennbare Einheit: Der behandelte Stoff ist identisch, es werden bloss die beiden jeweils passenden Teile des Lernprozesses in der Vorlesung bzw. im Proseminar ablaufen. Ein Verständnis der einschlägigen Begriffe kann daher nur auf Basis beider Veranstaltungen entstehen.

Aufgabensammlung:

• Blatt 1
• Blatt 1-2
• Blatt 1-3
• Blatt 1-4
• Blatt 1-5
• Blatt 1-7
• Blatt 1-10
• Blatt 1-11 (Blatt 11 = Aufgaben für den 29.6.)

Die Anmeldung erfolgt in der ersten Vorlesungseinheit am 1.3.2016.

Beurteilung: Das Proseminar ist prüfungsimmanente Lehrveranstaltung im Sinne des Curriculums. Die Leistung wird anhand von Tafelpräsentationen und der Mitarbeit bei Aufgabenlösungen sowie dem Ankreuzen der vorbereiteten Lösungen zu wöchentlichen Aufgaben beurteilt.