# Differential Geometry (Mastermath course, Fall 2015)

This is the functional web-page for the master math course Differential Geometry; here we will make all the announcements regarding the content of the lectures, material used, changes in the schedule, regulations, etc etc.
Here ARE the final marks. They are based on the homework marks and the mark for the take-home exam. Since the number of homeworks was only 3 (and not 4 as originally planned), the weight of the homework in computing the final mark is 40%. Note also that the rounding of the marks (up or down), whenever there was some doubt, it was based on a careful look at take home exam.

If you want to improve your mark, please let me know so that we can arrange an oral examination.

Here is the take home exam (a small typo was corrected in part a of exercise 1; the last $u$ should have been a $v$). As we discussed:

- I will try to correct each exam as soon as I get it.

- you are expected to send the solutions to me by the end of January (of course, earlier is fine as well!)

- after correcting the exam, I will let you know if there is the need for an extra-discussion at the blackboard.

- if you decide to discuss the problems with some of your colleagues, please do write the solution yourself, in your own style, with your own way of understanding it (otherwise it is not acceptable).

Here is the entire set of lecture notes, (including the last two chapters on $G$-structures).

Here is the first part of Chepter 3 (linear G-structures on vector spaces).

Announcements (please keep an eye on this field for important announcements during the semester):
• Reminder: the lecture from week 45 (november 4) will take place in BBG 201 and the one from week 51 (December 16) in KBG ATLAS (1.38).
week 45: BBG 201
• Last homework (October 28th): show, using only material from the lecture notes (i.e. the defintion of $Ad$ and $ad$ from teh lecture notes), that $ad_{\alpha}(\beta)= [\alpha, \beta]$ (for any Lie grouyp $G$). I.e. do Exercise 42; hint: look at Exercise 43. .
• October 28th: the teaching assistants will not be available and the I will use the entire 3 hours for the lectures (we will look at Lie groups).
• FIRST HOMEWORK (given on October 7; to be handed in by October 14): Exercise 17+ 23 + generalize to arbitray dimensions (but be careful: to integrate on CP^n, which is 2n-dimensional, you need a differential form/cohomology class of degree 2n, while the first Chern class is of degree 2; so you better start taking some powers). Please hand it in directly to the teaching assistants (i.e. do not send emails with the homework to the lecturer, since they may get lost).
• Here is the first part of the new lecture notes: on Vector bundles and connections (version of October 6, 2015; after this version I will not make any big changes anymore, just correcting typos).
• The first lecture takes place on September 16. The lecturer for the first hour will be Ori Yudilevich. After the brake, M. Crainic will take over and we will also discuss some practical aspects of the course (e.g. regarding examination).

• Location and time:
• Time: Wednesdays, from 10:00 to 12:45 (the first two hours are for the lectures, the last one is for the exercise classes, sometime the order is reversed)
• Location: room Ruppert A, De Uithof, Utrecht, except for week 45 (november 4) which will be in BBG 201 and week 51 (December 16) in KBG ATLAS (1.38).

• Teachers:
• lecturer: Marius Crainic (UU)
• teaching assistant: Francesco Cattafi.

• Exam: Hand in exercises (about 4 homework sets) + final exam (written or oral, depending on number of students). Each of these two items count for 50 percent of the grade; more details will be discussed during the lectures. The hand-in problems will be posted on this page, as a separate sheet.

Prerequisites:
• A good knowledge of multi-variable calculus.
• Some basic knowledge of topology (such as compactness).
• The standard basic notion that are tought in the first course on Differential Geometry, such as: the notion of manifold, smooth maps, immersions and submersions, tangent vectors, Lie derivatives along vector fields, the flow of a tangent vector, the tangent space (and bundle), the definition of differential forms, DeRham operator (and hopefully the definition of DeRham cohomology).

• Aim/content of the course:
The aim of this course is to provide an introduction to the differential geometry of vector bundles and principal bundles (connections, curvature, parallel transport) and then to the general concept of a G-structure, which includes several significant geometric structures on differentiable manifolds (for instance, Riemannian or symplectic structures).
The course will start with a discussion of vector/principal bundles, the will move to the discussion of "geometric structures" on vector spaces and on manifolds. The last part of the course will focus on topics such as equivalence and integrability of G-structures and discuss their interpretation in the some specific examples. Some of the key-words are: bundles, connections, curvature, Riemannian metrics, distributions, foliations, symplectic structures, almost complex and complex structures, torsion, integrability.

Literature:
We will occasionally use lecture notes (see above). During the semester, for the various parts of the course, we will provide extra-literature as well. For instance, on book that you may want to consult from time to time is:
• S. Sternberg, "Lectures on differential geometry", Prentice-Hall, First (1964) or Second (1983) edition.

• Lecture notes: Lecture notes will be made available during the semester. Here is the first part of the new lecture notes: on Vector bundles and connections . And here is Chapter 2: Principal bundles , as well as the updated table of contents .
In principle, the new set of lecture notes will be a revision of the old lecture notes; they will be updated during the semester and they will be made available on this page. And here is a very short reminder that covers some of the basic notions of differential geometry (mainly to fix the notations).

The schedule week by week (here we will try to add, after each lecture, a description of what was discussed in the lectures + the exercises):
• Lecture 1 (September 16): Reminder on vector bundles (look at the lecture notes, up to page 15).
• Lecture 2 (September 23): Connections.
• Lecture 3 (September 30): Parallel transport etc; the curvature of a connection.
• Lecture 4 (October 7): Interpretations of the curvature of a connection; the first Chern class. FIRST HOMEWORK: Exercise 17+ 23 + generalize to arbitray dimensions (but be careful: to integrate on CP^n, which is 2n-dimensional, you need a differential form/cohomology class of degree 2n, while the first Chern class is of degree 2; so you better start taking some powers). )
• Lecture 5 (October 14):
• Lecture 6 (October 21):
• Lecture 7 (October 28): Show, using only material from the lecture notes (i.e. the defintion of $Ad$ and $ad$ from teh lecture notes), that $ad_{\alpha}(\beta)= [\alpha, \beta]$ (for any Lie grouyp $G$). I.e. do Exercise 42; hint: look at Exercise 43.
• Lecture 8 (November 4): Free and proper actions.
• Lecture 9 (November 11):

Exercises for the class: 53, 54, 55, 56, 57, 58.

Homework: do 48+49+59 as one single exercise.

• Lecture 10 (November 18):
• Lecture 11 (November 25):
• Lecture 12 (December 2):
• Lecture 13 (December 9):
• Lecture 14 (December 16):
• Lecture 15 (December 23):

• Last update: 30-1-2014