# Differential Geometry (Mastermath course, Fall 2016)

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.

The schedule for the oral exam:

• Monday, January 23:

10:00-11:00: Paulien Schets (RU)

11:00-12:00: Maarten Smit (RU)

12:00-13:00: Zoe Schroot (UvA)

13:00-14:00: Teun van Nuland (RU)

14:00-15:00: Jasper Bouman (UvA)

15:00-16:00: -

16:00-17:00: -

• Tuesday, January 24:

14:00-15:00: Luuk Verhoeven (RU)

15:00-16:00: Niek Lamoree (UvA)

16:00-17:00: Francisco Hernandez (VU)

17:00-18:00: Guillermo Jacob Bijkerk Vila (UU)

• Wednesday, January 25:

10:00-11:00: Nicola Morandi

11:00-12:00: -

12:00-13:00: Guillermo Arias ?

13:00-14:00: Eric Pap (RUG)

14:00-15:00: -

15:00-16:00: Floris Elzinga

16:00-17:00: Sebastian Lucic

17:00-18:00: Karandeep Singh (UvA)

• Thursday, January 26:

10:00-11:00: Riccardo Pengo

11:00-12:00: Daniel Spitz (UU)

12:00-13:00: -

13:00-14:00: Jeroen Dekker (UvA)

14:00-15:00: Jeremy vd Heijden (UvA)

15:00-16:00: Joao Crespo (VU)

16:00-17:00: Charly Beulenkamp (UU)

17:00-18:00: Wilfred Bondt (UU)

• Friday, January 27:

10:00-11:00: Maarten Mol (UU)

11:00-12:00: Bjarne Kosmeijer (UU)

12:00-13:00: Salvatore Tambasco (RU)

13:00-14:00: Mathijs Lip (UU)

14:00-15:00: Alex Ben Hassine

15:00-16:00: Chongchuo Li

16:00-17:00: Luka Zwaan

• The second homework (deadline: January 19. Please send the homework to Marius Crainic, either by email or in his maibox). Note: the 1-form $\omega$ in Exercise 2 is real-valued.
• A typo in the homework: in the second exercise, part 3, please read "connection matrix" instead of "curvature matrix" (I hope that was clear from the explanations there- the fact that it was a matrix of 1-forms ...).
• The first homework. There are quite a few questions, but I decided to go for "many but easier" instead of "few but difficult". The date by which this exercise has to be handed-in will be discussed in the class.
• Chapter 2 of the lecture notes.

Location and time:
• Time: Wednesdays, from 10:00 to 12:45 (in principle, each week there will be 2x45 minutes for the lectures, and 1x45 minutes for the exercise classes).
• Location: De Uithof, Utrecht. The room depends on the week:
• wk 38: Ruppert B
• wk 39-40: BBG 205
• wk 41-45: Ruppert B
• wk 46: BBG 165
• wk 47-51: Unnik 220

• Teachers:
• lecturer: Marius Crainic (UU)
• Teaching assistants: Francesco Cattafi, Lauran Toussaint, Panagiotis Christou.

• Exam: There will be two homeworks that will count for 10% of the final mark, a take home exam that will count for 30% of the final mark followed by an oral exam (60% of the final mark).

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.

Lecture notes: Lecture notes will be made available during the semester. They will be an updated version of the notes from the last year (but there will not be big changes). Here is Chapter 1 of the new lecture notes: Vector bundles and connections . And here is Chapter 2: Principal bundles.

Literature:
We will 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.

• The schedule week by week (this will be maintained only for as long as it is useful):
• Lecture 1, September 21, Ruppert B : vector bundles, operations, differential forms with coefficients in vector bundles.
• Lecture 2, September 28, BBG 205: connections on vector bundles.
• Lecture 3, October 5, BBG 205: parallel transport; the curvature of a connection; the curvature matrix in terms of the connection matrix.
• Lecture 4, October 12, Ruppert B:
• Lecture 5, October 19, Ruppert B: connections compatible with a metric; the torsion of a connection on $TM$; Riemannian manifolds and the existence of the Levi-Civita connection.
• Lecture 6, October 26, Ruppert B: using the Levi-Civita connection: geodesics.
• Lecture 7, November 2, Ruppert B: The tubular neighborhood theorem; start with Lie groups.
• Lecture 8, November 9, Ruppert B: Lie groups.
• Lecture 9, November 16, BBG 165: Recap on Lie groups, a bit differently than the previous time: now just the main properties that will be needed later, presented in a form that is "ready for use" (in case you had problems absorbing the material from the previous lecture). Then free and proper actions, then start with principal bundles.
• Lecture 10, November 23, Unnik 220: Principal bundles- different ways of looking at it; the 1-1 correspondence between rank r vector bundles and principal GL_r bundles.
• Lecture 11, November 30, Unnik 220: Connections on principal bundles.
• Lecture 12, December 7, Unnik 220:
• Lecture 13, December 14, Unnik 220:
• Lecture 14, December 21, Unnik 220: