# Differential Geometry 2

- Symmetric Spaces

Winter Semester 2020-2021

### Schedule

For the whole month of November the lecture will be held online on zoom (you should have got the link and password through Müsli, if you didn't, or you lost it, write me an email). I will post the recordings on MaMpf.

It is in your interest to come to class and use the recorded videos only as a last resource: this way you can ask questions if you don't understand something, and it will be easier to keep up the pace, and slow down the lecturer if necessary. As with a normal lecture, in case of a no-show no lecture will happen, and reading material will be assigned instead.

**Main Lecture**

- Tuesday, 9:15-10:50, Online
- Thursday, 9:15-10:50, Online

First lecture Tuesday, November 3rd.

**Exercise Session**

- Wednesday, 2:15-3:45, Online

Starting from Wednesday, November 4th.

**Office Hour**

- Wednesday, 16:00-17:00, online

Starting from Wednesday, November 4th.

### Müsli

There is a page for this course on MÜSLI. Registration is open, please, register on MÜSLI.

There is a page for this course on Mampf.

### Lectures

Week 1 | Introduction to Symmetric spaces Summary of Differential Geometry 1 |
[DC] |

Week 2 | Fundamental groups and covering spaces Jacobi Fields relation with sectional curvature |
[Ha, Ch 2] [DC, Ch 5] [DC, Ch 7] |

Week 3 | Hadamard theorem Riemannian characterization of locally symmetric spaces Integrating linear isometries |
[DC, Ch 7] [Ma, Sec 3] |

Week 4 | Coverings of locally symmetric spaces Transvections Compact open topology on isometry groups |
[He p.62-63] [Io. 2.3] [He IV.2] |

### Exercise sheets

Exercise sheets will be published on Tuesday on the MaMpf page.

You should try to solve the exercises on your own before the exercise class the following week, so that you can ask what was not clear to the tutors, and volunteer to present solutions on the board.

When a graded exercise sheet appears, you will have one week to solve the exercises, write the solutions and hand them in, by Tuesday at 4pm. The grades of every exercise sheet will be written on MÜSLI by Giulio Belletti. To be admitted to the final exam, you need to obtain at least 50% of the available points of the graded sheets.

The first question of every exam will be taken from one of the non-graded exercise sheets. You will be allowed to bring to your oral exams your hand written solutions to the exercise sheets and have a quick look at it before solving it on the board.

### Final exam

During the last week of the semester we will organize oral exams. To attend the exam, it will be necessary to register on MÜSLI. We will communicate the date of the final exam as soon as possible. The oral exam will be half an hour long for every student, and the first question of every exam will be to explain the solution of an exercise from an exercise sheets. The final exam will be in English or German, depending on your preference.

### Contact

Dr. Giulio Belletti (Exercises)

### Office Hours

Time | Office | |
---|---|---|

JProf. Dr. Beatrice Pozzetti | Wed, 4-5 | INF 205, 05.229 and zoom |

### Description

In this lecture course we will discuss symmetric and locally symmetric spaces. Symmetric spaces are Riemannian manifolds in which the geodesic symmetry, at any point, is induced by an isometry. In particular the group of isometries acts transitively on the space. We will study the Riemannian geometry of symmetric spaces as well as their connection to the theory of semisimple Lie groups. A preliminary outline of the material covered in the lecture is the following:

- Generalities on symmetric spaces: locally and globally symmetric spaces, groups of isometries, examples.
- Relation with orthogonal symmetric Lie Algebras, decomposition of symmetric spaces in irreducible pieces, duality between compact and non-compact type, curvature computation.
- Symmetric spaces of non-compact type: flat subspaces and the notion of rank, roots and root space decomposition. Iwasawa decomposition, Weyl group, Cartan decomposition.
- Geometry at infinity: geometric boundary, Furstenberg boundary, Bruhat decomposition, visibility at infinity, Busemann functions.

### Prerequisites

This course is aimed at students who are interested in differential geometry. Students are expected to have a certain familiarity with Riemannian geometry, ideally they have followed Differential Geometry I or a similar course. The course will be taught in English.

### References

- [He] Helgason: Differential Geometry, Lie groups and Symmetric Spaces.
- [Ma] Maubon: Riemannian symmetric spaces of the non-compact type: differential geometry.
- [Io] Iozzi: Symmetric spaces.

Further reading

- [Ba] Ballmann: Symmetric spaces.
- [Pa] Paulin: Groupes et Geometries.
- [Eb] Eberlein: Geometry of non positively curved manifolds.
- [DC] Do Carmo: Riemannian geometry.
- [BH] Bridson, Haefliger: Metric spaces of non-positive curvature.
- [Ha] Hatcher: Algebraic topology.
- [Sc] Schroeder: Symmetrische Räume.
- [HI] Holland and Ion: Notes on symmetric spaces.
- [Bo] Borel: Semisimple Groups and Riemannian Symmetric Spaces.
- [KN] Kobayashi, Nomizu: Foundations of Differential Geometry vol. 1 and 2.
- [Lo] Loos: Symmetric Spaces, vol. 1 and 2.
- [Wo] Wolf: Spaces of constant curvature.
- [Pa] Paradan: Symmetric spaces of the non-compact type: Lie groups.