# TMA4190 Manifolds spring 2016

## Last news

15 Juli | MERK: Kont. eksamen, som er muntleg, blir arrangert i slutten av august. Dei som planlegg å ta eksamen, ta kontakt med forelesar. Om du er oppmeldt, men trekker deg seinare, gi også beskjed til forelesar så fort som mulig. |
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10 June | Here is a copy of the exam yesterday, together with a discussion of (possible) solutions of the problems : pdf Have a nice summer all of you! |

31 May | For practice, try some of the exercises also used for this purpose in May 2015, see under"Øvingar/Exercises". |

11 May | I propose to have the Spørretime (Question time) on Monday 6 June, 3pm-4pm (or longer) in 13th floor(coffee room). But this plan may have to be changed, so please watch up in case there is a new message. |

11 May | How are you doing? I hope your exams and preparations are going well. Under "Gamle eksamensoppgaver" you will also find the exam given in 2015 and the solution proposal of the lecturer. |

24 April | This is the last week with lectures (Monday and Tuesday). The hour we have on Thursday (practice session) is cancelled. However, we plan to have an extra meeting - spørretime (questions, problem solving..) at the beginning of June, say. Watch up for later announcement on this page. Recall that the exam date is Tuesday 9 June. |

17 April | We continue with Chap.10 ( vector bundles) and Chap.11 ( cotangent bundle) and Chap.13 (Riemannian metric). This is the next last week, so we shall continue with these topics also in the last week |
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10 April | In this coming week we shall go through the two first sections of Chap. 9 , called "integral curves" and "flows", namely the pages 205 -217. (The remainder of the chapter is not part of our curriculum). |

3 April | This week we focus mostly on Chap.8: Vector fields, and a little from Chap.9: integral curves. But we also look briefly at Whitney's theorems in Chap.6. |
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14 March | There will be no practice session on Thursday 17 March. But Exercise 8 (week 11) is assigned to this week. So make sure you work on these exercises, as usual. The first meeting after Easter is at the practice session on Thursday 31 March, to which Exercise 9 (week 13) is assigned. It is expected that you also find the opportunity to look at these exercises prior to 31 March. During April there will be 8 lectures (week 14-17), where we aim at covering the following topics: |
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From Chap. 6 we shall only briefly discuss Sard's theorem and Whitney's embedding theorem, without proofs. Then we shall concentrate on vector fields, their integral curves and flows (see chap. 8 and 9), next discuss vector bundles in more generality , including the cotangent bundle of a manifold (Chap.10 and 11), and finally, hopefully we have some time for Riemannian metrics (chap.13).|

13 March | The topic for this week will Chapter 5 on submanifolds, but we shall not pay much attention to manifolds with boundary.Next, we shall have a brief look at Sard's theorem, from the first 3 sections of Chap.6. Then we start on Chap.8 and 9; wait and see. |
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6 March | This week we shall work hard on the main issues of chapter 4,where the focus is on the differential dF of F at the various points,first of all what we can say about the behavior of F on an open subset where the rank happens to be constant. Chapter 5 is also closely related to this, so we shall also touch upon the concept of a submanifold. |
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29 February | We shall finish Chapter 3 and continue with chapter 4 (where the section "Smooth covering maps" is left out). Here the main topics are smooth maps of constant rank, with immersions, embeddings, and submersions as the important special cases. |
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22 February | We are now working with Chapter 3, about tangent vectors, tangent spaces, the differential (derivative) of a smooth function, and the tangent bundle of a smooth manifold. After this week we shall be finished with most of Chap.3, and then we turn to Chapter 4. (The subsection called "Smooth Covering Maps" will be dropped). |
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8 February | This week we shall concentrate on the concept of tangent vector, tangent space, and finally tangent bundle. A smooth map between manifolds induces a linear map (its derivative) between the corresponding tangent spaces and a smooth map between their tangent bundles. All this is explained in Chap.3, showing several equivalent but different ways of defining the above notions. Also, dont ignore the new exercises for the Thursday session 9.15-10. They are chosen from Chap. 2. |

26 January | The practice session hour will be on Thursdays 9.15-10, room R20, to begin with at least. Unfortunately there is some overlapping with the courses MA4150 (algebra) and TMA4165 (diff.likn. dyn. system), and some practice sessions in other courses. I also propose an office hour on Thursdays 10.15 -11 at my office in 12th floor, which may help those who dont attend the practice hour 9.15-10 but still want some advice. |

24 January | During the following two weeks the lectures are concerned with the notion of smooth manifolds and smooth maps between them. It is important to investigate various concrete examples, and you should study carefully the text in Chapter 1 and 2 of Lee's book. |

22 January | Exercise set 1 is available. Test yourself as much as you can by trying to solve/understand the exercises. |

We have an 1 hour practice session, so we must decide upon time and meet next week. We should decide during the lecture on Monday. How about Wednesday afternoon? (At 1,2,3,4,5 pm ?). Or Thursday before noon? |

^ ** 18 January ** | The lecture on Tuesdays will be in room MA24, 12.15-14 (check the location, green building.)|

11 January | Studieadministrasjonen has proposed another time for the second lecture of the week, namely on Tuesdays 12.15-14 (room decided later) instead of Fridays 14.15-16.Then there will be a minimal number of "collisions" for those who have signed up for TMA4190. There will, unfortunately, be overlapping with TMA4175 Complex analysis, for 3 registered students.This week, however, the second lecture will be on Friday 15 January, 14.15-16 , as originally scheduled to begin with. |
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7 januar | Første forelesning er mandag 11 januar. |

Det er viktig å forstå dei grunnleggande begrepa i generell topologi. Dei som ikkje har kjennskap til generell topologi (for eksempel frå kurset MA3002 Generell topologi), må lære seg det mest grunnleggande om topologi og topologiske rom så fort som mulig. Sjå f.eks. i læreboka til kurset MA3002 Generell topologi, eller sjå "Appendix A : Review of Topologi" i vår lærebok, som er boka John M. Lee: Introduction to Smooth Manifolds.|

7 January | The lectures will be in English only if foreign students (who dont speak Norwegian) want to participate. The first lecture is on Monday 11 January (as announced) |
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7 januar | I førstninga vil vi jobbe med Kap.1: Smooth manifolds (i boka til J.M. Lee). Meir informasjon om forkunnskap og grunnlaget for kurset blir diskutert på første forelesning. |

7 januar | Iflg katalogen er det forelesning på mandag 10-12 og fredag 14-16. Det vil bli gjort forsøk på å flytte forelesninga på fredag til ein anna dag. |
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