Differentialgeometrie (Bachelor)

Veranstaltungstitel:	Differentialgeometrie (65-055)
Veranstalter:	David Lindemann, email: david.lindemann at uni-hamburg.de

Inhalt:	Einführung in die (pseudo)-Riemannsche Geometrie

Ort und Zeit:	Vorlesung: ~~Mittwochs 16:15-17:45 Uhr (H2) und Donnerstags 10:15-11:45 Uhr (H4) (Geomatikum)~~ Übung: ~~Donnerstags 12:15-13:45 Uhr (Raum 142, Geomatikum)~~ Exercise Classes: Mondays 15:00 starting on 17.5., online via bbb, link: https://bbb1.physnet.uni-hamburg.de/b/dan-zkt-f94 Until further notice: online, see below!
Important info:	Due to the current situation the course will be, until the lockdown is lifted, held online. This will (most likely) work as follows. I will, starting on Monday (20. April), upload videos of the lectures. These are basically beamer presentations + sketches on a piece of paper filmed with a document camera. The video files will be available for direct download on this website and (if it works as expected) on lecture2go, I will embed links. Should all servers fail horribly I will torrent the lectures so that they will at any time be available. The course will be held in English. This is contrary to what I was planning to do (ask the audience what language they prefer) but I think this is in the current situation the best way to proceed. The main reason is that, at least in the past, many participants in the DG beginners course are students of the master program "Mathematical Physics" who are more often than not not fluent in German. Since the lectures are on video you should, even if your English is not perfect, have no problem following the course. In case you should still have problems understanding my (definitely far from perfect) English, do not hesitate to contact me so that I can take the time to explain to you the stuff you did not get. ANY kind of feedback regarding the lecture videos is greatly appreciated. If you prefer giving feedback anonymously simply make a throwaway e-mail account (although I am not sure which of these providers go directly into the spam folder...). Handing in the exercises for this course is NOT mandatory to get the permission for the oral exam. However, the exercises are considered part of this course and I expect you to try solving them. To make this 100% clear: I might ask you some question in the exam that was dealt with in the exercises, e.g. "Is object A a smooth manifold?". There will be one exercise sheet per week (roughly on Fridays) and you can hand them in (in English or German) via e-mail until the deadline if you want them to be graded. The grading/checking is done by me and by Danu Thung who was supposed to do the exercise classes. Most importantly, in particular because gauging how well students of the course understand what I am trying to teach will be tricky, to say the least: Do not hesitate to ask ANY kind of questions! There are no stupid questions in my opinion, and asking questions will not influence your grade in the exam. You can always send me e-mails with questions (even though exercise sheets past their deadline will not be graded, you can of course still ask questions regarding these exercises), and once I tested the video conferencing systems (jitsi or bbb) this will also be a possibility. I do not use skype/zoom or any other kind of spyware, sorry... I will post updates on this site regarding individual online meetings as soon as possible. The first version of the lecture notes will go online with the first video. The lecture notes are WIP, as of writing this the first (comparatively big) chapter is finished. It will be regularly updated. Update 23.04.2020: The first video should be available on lecture2go now. Apparently, this involves re-encoding so the quality of the videos will suffer. I will, of course, post the upcoming lecture videos here as well and I advise you to use the direct download links and simply download the lectures. Update 24.04.2020: I wanted to fix the slightly overmodulated audio for the second lecture and messed up an other mic setting in the process. So you might need to turn up the volume a bit. Once I figure out ffmpeg a bit better I'll normalize the audio on the second lecture. Fixed. Update 27.04.2020: lecture2go seems to be refusing to work for me, either won't save metadata or complain about missing metadata. As soon as my tracker req gets through and the issue is fixed, I'll start uploading the new videos there as well. Fixed. Update 04.05.2020: The next lecture (5, Smooth submanifolds) will be uploaded a day or so later than planned (this night) because I was sick over the weekend, sorry. If you want to prepare for it in advance a little bit, recall the constant rank theorem from real analysis. "Fixed" Update 07.05.2020: 1) A request from Danu Thung: Try to hand in your solutions in pdf-Format. Using LaTex is encouraged! 2) I have been informed of the problem that it is difficult to find groups for studying or preparing the solutions for the exercise sheets together. I am not allowed to simply put the list of students in this course online, so I would like to suggest the following: Let me know if you want me to publish your e-mail address so that other students can contact you. I will upload a list of these e-mail addresses in stine. If anyone has a better idea please let me know. Update 11.05.2020: I uploaded (in stine) a list of e-mails from students who would like to cooperate with others on solving the exercises. If you have already send me an e-mail asking to be added to that list and I somehow missed it simply write me an other e-mail. If you also want your e-mail address to be part of the list (or deleted from it) let me know. Update 15.05.2020: Starting next monday there will be online exercise classes by Danu Thung on our departments BigBlueButton-Server. Timeslot: Mondays 15:00, link: https://bbb1.physnet.uni-hamburg.de/b/dan-zkt-f94 Update 25.05.2020: The evaluation of the lecture courses has started. I kindly ask you to fill out the form at https://evasys-online.uni-hamburg.de/evasys/online.php?pswd=Q7A2A until 07.06.20, that is the 7th of June, so that I can get some additional feedback. Thanks in advance! Update 26.05.2020: There will be a short break of lecture videos, roughly 1 week, since I need to work on the lecture notes and preparing the videos takes longer than anticipated. Don't worry though, this does not mean that I will skip anything. Some future videos might be a bit longer... Update 29.05.2020: No exercise sheet today, enjoy your free week! Instead, you might want to go over lectures 1-11 again and, if you find any typos in the slides or lecture note, let me know. I guarantee that you will remember finding an error, so error hunting is definitely an effective way to study. Update 10.06.2020: The evaluation has been extended, PLEASE! fill out the form if you haven't done so already: https://evasys-online.uni-hamburg.de/evasys/online.php?pswd=Q7A2A until 19.06.20, that is the 19th of June. Update 23.06.2020: I just realized that you need the Levi-Civita connection for exercise sheet 8, and I didn't introduce it yet. Since the new lecure (nr. 16) is not out yet, I have uploaded the new WIP17 lecture notes. The missing proofs will be added most likely tomorrow. You can either do exercise 4 by simply using Prop. 2.92 of WIP17, or prepare it next week if you want it to be corrected. Note that this exercise is not difficult and I will most likely tell you what to do in the next lecture anyway, it's more a test to check that you can calculate with Christoffel symbols and indices. Update 24.06.2020: Fixed an error in slides of lecture 15 in the Christoffel symbols in polar coordinates example. Update 26.06.2020: In case you are wondering where the next lecture is: It took a little longer than expected to prepare and I didn't want to split it in 2 lectures since the topics nicely fit together and, if torn apart in this format (i.e. videos, meaning that I cannot easily say some informal stuff at the end of each lecture like "yeah this looks like it doesn't make any sense but I promise it will hah!" and at the same time gauge your reaction) the first lecture would simply look like playing around. So the next lecture might be a bit longer, buckle up... (you can already take a look at the slides) Update 01.07.2020: Updated exercise sheet 9, now the third exercise is actually a new one... Update 04.07.2020: I was too slow! Lecture 17 tomorrow (a.k.a. today), you can already take a peak at the new slides. My plan as of now is to upload two more lectures (18 and 19) next week and the week after, first one about curvature, second one about curvature on submanifolds. Afterwards, there will be sort of "bonus" lectures, i.e. we (as in everyone teaching a lecture course) were allowed to upload during the first two weeks of the summer break. As this is likely to be met with, lets say, varying amount of enthusiasm, these lectures will not be part of the exam. This also allows me to be a bit less strict with how I present the material, so look forward to "cool" stuff. Update 12.07.2020: Lecture 18 will be up ~~this night~~ nope, tomorrow, sorry (slides are up though). Lecture 19 will be up hopefully next Friday and as already announced be the last one to part of the exam. I will upload the 11th exercise sheet with Lecture 19, containing problems for both Lectures 18 and 19. Since on next Monday (13.7.) we have the last exercise class, the last exercise sheet is primarily meant for you to prepare for the exam. If you want me to grade it, send it to me until 14th of August. Update 13.07.2020: Todays exercise class will be supervised by myself, DIFFERENT LINK: https://mathbbb.physnet.uni-hamburg.de/b/dav-iwe-its Update 25.07.2020: The last lecture took (a.k.a. takes) a little longer to prepare than anticipated (again, sorry). I will hopefully be able to upload the video and a bunch of exercises tomorrow (yes, I know it's summer """""break""""" already but at least I'm still in the added buffer period of two weeks...). You can already take a look at the updated lecture notes. Also note that I've asked the Studienbuero Mathematik about the possibility of a "live" tutorial, but as of now these are discouraged. I will keep you up to date on this issue as I would prefer to do at least one of these before the exam. In case you missed it before, you have always the possibility to contact me via e-mail for a bigbluebutton meeting! Do not hesitate to ask questions, in particular when you are preparing for the exam. Update 29.07.2020: I checked the overall time of the lectures and got roughly 40h 25min, so about 27 normal irl lectures. This is the exact amount that we would have had if not for the special situation this semester. Also, this was 100% not planned in any way whatsoever. Update 01.08.2020: The last exercise sheet 11 is now online. If you want me to have a look at your solutions, please send them to me (not Danu) by the 17th of August. Also make sure to try solving the exercises in the lecture notes!


Lecture notes and slides:	lecture notes WIP30: dg_lindemann_ss2020_WIP30.pdf [WIP3: fixed condition when a tangent vector is tangential to a submfd. of \(\mathbb{R}^n\), fixed target space of \(F\) in Exercise 1.40, WIP4: in equation (1.9) it now correctly states \(\mathrm{supp}(b)\subset U\) instead of (the not sufficient, e.g. for trivial extensions) condition \(b\|_{M\setminus U}\equiv 0\), also fixed subsequently, WIP5: fixed some typos, WIP6: completed proof of Prop. 1.62, WIP7: some typos, WIP8: typos, grammar, WIP9: added some definition and lemma, typos, WIP10: typos, additional remarks, \(\phi\)-related stuff, WIP13: started new section, WIP21: added section on geodesics, WIP23: added section on curvature, WIP26: added section on ps.-R. submanifold geometry, WIP28: first proofreading up to page 45, fixed a lot of typos, WIP29: first proofreading finished! some comments still remain. no real error found so far, WIP30: more typos fixed] beamer slides of lecture 1: DG2020L1_slides.pdf [changelog: the \(\varphi_i\) on p. 5 are homeomorphisms, not just any maps] beamer slides of lecture 2: DG2020L2_slides.pdf beamer slides of lecture 3: DG2020L3_slides.pdf beamer slides of lecture 4: DG2020L4_slides.pdf beamer slides of lecture 5: DG2020L5_slides.pdf beamer slides of lecture 6: DG2020L6_slides.pdf beamer slides of lecture 7: DG2020L7_slides.pdf beamer slides of lecture 8: DG2020L8_slides.pdf beamer slides of lecture 9: DG2020L9_slides.pdf beamer slides of lecture 10: DG2020L10_slides.pdf beamer slides of lecture 11: DG2020L11_slides.pdf beamer slides of lecture 12: DG2020L12_slides.pdf beamer slides of lecture 13: DG2020L13_slides.pdf beamer slides of lecture 14: DG2020L14_slides.pdf beamer slides of lecture 15: DG2020L15_slides.pdf beamer slides of lecture 16: DG2020L16_slides.pdf beamer slides of lecture 17: DG2020L17_slides.pdf beamer slides of lecture 18: DG2020L18_slides.pdf beamer slides of lecture 19: DG2020L19_slides.pdf


Lecture notes and videos:	all lecture videos can be watched on lecture2go: https://lecture2go.uni-hamburg.de/l2go/-/get/l/5510 direct download links below lecture 1 (video, ~500MB): DG_lindemann_L1_h264.mkv alternativ encode of lecture 1 using HEVC (smaller filesize, ~270MB): DG_lindemann_L1_h265.mkv lecture 2 (video, ~550MB): DG_lindemann_L2_h264_v2.mkv alternativ encode of lecture 2 using HEVC (smaller filesize, ~300MB): DG_lindemann_L2_h265.mkv lecture 3 (video, ~600MB): DG_lindemann_L3_h264.mkv alternativ encode of lecture 3 using HEVC (smaller filesize, ~250MB): DG_lindemann_L3_h265.mkv lecture 4 (video, ~700MB): DG_lindemann_L4_h264.mkv alternativ encode of lecture 4 using HEVC (smaller filesize, ~250MB): DG_lindemann_L4_h265.mkv lecture 5 (video, ~640MB): DG_lindemann_L5_h264.mkv alternativ encode of lecture 5 using HEVC (smaller filesize, ~340MB): DG_lindemann_L5_h265.mkv lecture 6 (video, ~550MB): DG_lindemann_L6_h264.mkv alternativ encode of lecture 6 using HEVC (smaller filesize, ~400MB): DG_lindemann_L6_h265.mkv lecture 7 (video, ~600MB): DG_lindemann_L7_h264.mkv alternativ encode of lecture 7 using HEVC (smaller filesize, ~420MB): DG_lindemann_L7_h265.mkv lecture 8 (video, ~650MB): DG_lindemann_L8_h264.mkv alternativ encode of lecture 8 using HEVC (smaller filesize, ~450MB): DG_lindemann_L8_h265.mkv lecture 9 (video, ~540MB): DG_lindemann_L9_h264.mkv alternativ encode of lecture 9 using HEVC (smaller filesize, ~310MB): DG_lindemann_L9_h265.mkv lecture 10 (video, ~620MB): DG_lindemann_L10_h264.mkv alternativ encode of lecture 10 using HEVC (smaller filesize, ~330MB): DG_lindemann_L10_h265.mkv lecture 11 (video, ~740MB): DG_lindemann_L11_h264.mkv alternativ encode of lecture 11 using HEVC (smaller filesize, ~440MB): DG_lindemann_L11_h265.mkv lecture 12 (video, ~615MB): DG_lindemann_L12_h264.mkv lecture 13 (video, ~760MB): DG_lindemann_L13_h264.mkv lecture 14 (video, ~540MB): DG_lindemann_L14_h264.mkv lecture 15 (video, ~750MB): DG_lindemann_L15_h264.mkv lecture 16 (video, ~760MB): DG_lindemann_L16_h264.mkv lecture 17 (video, ~1.1GB): DG_lindemann_L17_h264.mkv lecture 18 (video, ~800MB): DG_lindemann_L18_h264.mkv lecture 19 (video, ~800MB): DG_lindemann_L19_h264.mkv


Exercise sheets:	DGU1SS2020.pdf DGU2SS2020.pdf DGU3SS2020.pdf DGU4SS2020.pdf DGU5SS2020.pdf DGU6SS2020.pdf DGU7SS2020.pdf DGU8SS2020.pdf DGU9SS2020_v2.pdf DGU10SS2020.pdf DGU11SS2020.pdf


Arbeitsaufwand:	9 Leistungspunkte (4+2 SWS)

Art der Prüfung:	mündliche Prüfung / oral exam Please contact me via e-mail so we can fix the date and time. In general you are free to choose when (and of course if) you want to take the exam, just let me know at least 2 weeks in advance. As least as far as I am concerned, there is no deadline for when you take the exam, so as far as I am concerned you can take the exam this semester, next semester, whatever semester you want. You might want to check with your Studienbüro if there is some sort of deadline by which you must have completed all your exams, as I have no idea whether this is the case or not (read: don't blame me if you miss a deadline that I don't have any control over). The earliest possible date is 01.08.2020 (1st of August).

Verwendbarkeit des Moduls:	Wahlpflichtmodul in den Bachelor-Studiengängen Mathematik, Technomathematik und Wirtschaftsmathematik, sowie den Bachelor-Lehramtsstudiengängen der Mathematik. Dieses Modul kann außerdem in die Master-Studiengänge der Mathematik und der Mathematischen Physik, und in Physikstudiengängen eingebracht werden. Bitte wenden Sie sich für Details an Ihr zuständiges Studienbüro.
Modulbeschreibung:	In der Vorlesung werden wir die grundlegenden Begriffe der modernen Differentialgeometrie behandeln. In der Differentialgeometrie studiert man glatte Mannigfaltigkeiten, das heißt geometrische Objekte, die grob gesagt lokal aussehen wie \(\mathbb{R}^n\) (und deren globale toopologische Eigenschaften nicht zu schräg sind, siehe z. B. die sogenannte ''long line''). Beispiele wie die \(n\)-Sphäre \(S^n\) oder glatte Flächen im \(\mathbb{R}^3\) kennen Sie schon aus der Analysis (oder den Vorlesungen Mathe für Physiker 1-3). Auf glatten Mannigfaltigkeiten werden wir eine Reihe an Konstruktionen und Strukturen studieren, z.B. Vektorfelder, Metriken und verschiedene Krümmungsbegriffe. Außerdem eignen sich glatte Mannigfaltigkeiten als Raum für gewöhnliche und partielle Differentialgleichungen, die im Vergleich mit Gebieten im \(\mathbb{R}^n\) andere globale topologische Eigenschaften zulassen (z.B. PDEs auf der Kleinschen Flasche oder auf den reell-projektiven Räumen \(\mathbb{R} P^n\)). Ein Fokus dieser Vorlesung werden Untermannigfaltigkeiten und induzierte geometrische Strukturen sein. Außerdem werden wir einige Themen aus dem Blickwinkel der Variationsrechnung betrachten, z.B. Geodäten as kritische Punkte des Energiefunktionals. Diese Vorlesung eignet sich ausdrücklich auch für Studentinnen und Studenten der Physikstudiengänge (und des Master-Studienganges ''Mathematische Physik''), da die Differentialgeometrie eine fundamentale theoretische Grundlage für viele moderne Theorien in der Physik darstellt (insbesondere ART, Eichtheorien wie Yang-Mills-Theorie, SuSy, SuGra,...).
Vorwissen/Voraussetzungen:	Lineare Algebra 1-2 und Analysis 1-3, alternativ Mathe für Physiker 1-3 Sie sollten in der Lage sein, mit partiellen Ableitungen umzugehen und sich an den Satz von der impliziten Funktion erinnern. Linear algebra, calculus, and some real analysis You should be able to deal with partial derivatives and remember the implicit function theorem.

Literatur:	B. O'Neill, Semi-Riemannian Geometry With Applications to Relativity (1983), Pure and Applied Mathematics, Vol. 103, Academic Press, NY. O. Goertsches, Differentialgeometrie, Vorlesungsskript (SS 2014) link V. Cortés, Differentialgeometrie, Vorlesungsskript (SS 2019) (uploaded in Stine) J.M. Lee, Riemannian manifolds - An Introduction to Curvature (1997), Springer Graduate Texts in Mathematics, Vol. 176 J.M. Lee, Introduction to Smooth Manifolds (2003), Springer Graduate Texts in Mathematics, Vol. 218 R.W. Sharpe, Differential Geometry - Cartan's Generalization of Klein's Erlangen Program (1997), Springer Graduate Texts in Mathematics, Vol. 166 G.E. Bredon, Topology and Geometry (1993), Springer Graduate Texts in Mathematics, Vol. 139 V.I. Arnold, Ordinary Differential Equations, third edition (1992), Springer Universitext C. Bär, Differential Geometry, lecture notes (2013)
*Weitere Information / Additional information:*	Falls Sie an einem der Angegebenen Termine für Vorlesung oder Übung eine Überschneidung mit einer anderen Veranstaltung haben und an beiden teilnehmen möchten, schreiben Sie mir bitte möglichst vor Beginn des Semesters eine e-mail. If you want to attend this course but its timeslot conflicts with some other course you also want to attend, please tell me in advance if possible. Diese Vorlesung wird entweder auf Deutsch oder Englisch gehalten. Dies wird in der ersten Vorlesung (entgültig!) festgelegt. Da diese Vorlesung eine Bachelor-Vorlesung ist, würde eine Stimme gegen Englisch ausreichen, damit die Vorlesung auf Deutsch gehalten wird. This lecture course will be taught in either german or english language. This will be determined in the first session (no takebacks afterwards). Be aware that since this is a Bachelor-course, the default language is german and one person disagreeing with english would be enough for the course to be taught in german.