This is a reading seminar aimed at students that are interested in differential geometry.

Talks will be prepared and given by the participants, in close coordination with the organisers. Along with their talk speakers will also provide a handout with homework exercises for all participants, to be approved by the organisers. The speaker is responsible for grading this homework.

The aim of the course is twofold: students will learn about principal bundles and the general theory of geometric structures on manifolds on the one hand, and improve their ability to present mathematics on the other.

**
Organisers**

Francesco Cattafi, Marius Crainic and Maarten Mol

**Day, time and place
**

Mondays 13.15-15.00 in room 610 of the Hans Freudenthal building, weekly from 11 February 2019 until 17 June 2019.

**Prerequisites
**

Students should be familiar with the contents of a first course in differentiable manifolds, covering differential forms and vector bundles. Knowledge of connections on vector bundles is useful but is not necessary. No prerequisite on Lie groups and principal bundles is required.

**Plan of the talks**

• 11 February - Lecture 0: introduction to the course (Francesco & Maarten)

18 February - Lecture 1: Basics on Lie groups and Lie algebras

25 February - Lecture 2: Principal bundles

4 March - Lecture 3: Principal connections and Ambrose-Singer

11 March - Lecture 4: Frame bundles, G-structures and first examples

18 March - Lecture 5: Riemannian structures

25 March - Lecture 6: Symplectic structures

1 April - Lecture 7: Integrability

8 April - Lecture 8:

15 April - Lecture 9:

22 April: *no lecture (Easter Monday)*

29 April - Lecture 10:

6 May - Lecture 11:

13 May - Lecture 12:

20 May - Lecture 13:

27 May - Lecture 14:

3 June - not needed?

10 June: *no lecture (Pentecost)*

17 June - not needed?

**References**

• Marius Crainic, Lecture notes on Differential Geometry

• Shlomo Sternberg, Lectures on differential geometry, Chelsea Publishing Co., New York, 1983

• Shoshich Kobayashi, Katsumi Nomizu, Foundations of differential geometry, Vol. I and II, John Wiley & Sons, Inc., New York, 1996

• Victor Guillemin, The integrability problem for G-sructures, Trans. Amer. Math. Soc. 116, 1965

**Evaluation
**

Participants are expected to give two seminar talks (each of which is a 2x45 min presentation), possibly more or less depending on the number of participants. They will study the material beforehand, hold a blackboard presentation about it, and distribute a hand-in exercise to the other seminar participants (to be approved beforehand by the seminar organizers and to be handed in by the students at the next lecture). The speaker is responsible for grading this hand-in exercise. In case of discussion about the solutions, the seminar organizers decide. Participants are expected to attend every seminar meeting. The final grade for the seminar is based on your talks and handouts (40%) and on your homework formulation and grades (60%).

**ADDRESS**

Mathematical Institute

Budapestlaan 6

Utrecht, 3584 CD

The Netherlands

**CONTACTS**

Email: f.cattafi[at]uu.nl

Phone: +31 30 253 4557