# V4D4 - Advanced Geometry II

- Symmetric Spaces

Summer Semester 2018-2019

### Schedule

**Main Lecture**

- Wednesday, 16:10-17:45, MATH / Sem R 0.011
- Friday, 10:15-11:50, MATH / Sem R 0.011

**Exercise Session**

- Friday, 14:15-16:00, MATH / SemR 0.003

Starting from Friday April 12.

### Lectures

Week 1 | Introduction to Symmetric spaces | |

Week 2 | Summary of Differential Geometry 1 Jacobi Fields, Hadamard theorem |
[DC] [DC, Ch 5,7] |

Week 3 | Hadamard theorem Riemannian characterization of locally symmetric spaces |
[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] |

Week 5 | The isometry group of a symmetric space as a Lie group | [He, IV.3] |

Week 6 | Lie groups and Lie algebras Riemannian symmetric pairs |
[Sc, 2] [He, IV Thm 3.3, Prop 3.4] |

Week 7 | Simply connected Riemannian symmetric pairs Examples |
[Io, Remark 2.23] [He. IV.6] |

Week 8 | Cartan Decomposition Orthogonal symmetric Lie algebras: compact, non-compact, Euclidean type |
[He. IV.3] [He. V.1] |

Week 9 | Irreducible OSLAs Compact semisimple Lie groups |
[He. VIII.5] [He. II.6] |

Week 10 | Duality Curvature Lie triple systems |
[He. V.2] [Pa, 4.1.3] |

Week 11 | No lecture on 19.06! Flat subspaces |
[Io, 3.2] |

Week 12 | Root spaces and SL2 Symmetric spaces of non-compact type as Hadamard manifolds |
[Io, 3.3] [Ma, 5.2] |

Week 13 | Boundary at infinity Visibility (Only part of exam in August) |
[Ma 5.2.3] [Ma 6.5-6.6] |

Week 14 | Iwasawa decomposition (Only part of exam in August) No lecture on 12.07! |
[Eb, 2.2.17] |

### Exercise sheets

Exercise sheets will be published regularly here, at latest on Friday night.

You should try to solve the exercises on your own before the tutorials the following week, so that you can ask what was not clear to the tutor, and volunteer to present solutions on the board. The tutor will answer questions and give you feedback about your presentation but won't solve exercises for you.

When a graded exercise sheet appears, you will have ten days to solve the exercises, write the solutions and hand them in, in groups of at most three persons, by Wednesday at 6pm. You can either hand in your solutions at the end of the class, or email them to islegers(at)math.uni-bonn.de . To be admitted to the final exam, you need to obtain a passing grade in at least 4 exercise sheets out of 6.

The first question of every exam will be taken from one of the exercise sheets. You will be allowed to bring to your oral exams your hand written solution to the exercise sheets and have a quick look at it before presenting your solution.

05.04.2019 | Sheet 1: Grassmannians | Not to hand in |

12.04.2019 | Sheet 2: Revision | Hand in by 26.04 |

19.04.2019 | Sheet 3: Locally symmetric spaces | Not to hand in |

24.04.2019 | Sheet 4: Symmetric spaces | Hand in by 10.04 |

03.05.2019 | Sheet 5: Isometry groups | Not to hand in |

10.05.2019 | Sheet 6: Lie groups | Hand in by 25.05 |

17.05.2019 | Sheet 7: Compact Lie groups | Not to hand in |

24.05.2019 | Sheet 8: Riemannian symmetric pairs | Hand in by Wednesday 05.06 |

31.05.2019 | Sheet 9: Cartan Decompositions | Not to hand in |

07.06.2019 | Sheet 10: Curvature and duality | Hand in by Wednesday 19.05 |

21.06.2019 | Sheet 11: Lie triple systems and duality | (only hand in if in need of points) |

28.06.2019 | Sheet 12: Root spaces and parallel sets | (only hand in if in need of points) |

05.07.2019 | Sheet 13: Symmetric spaces of non-compact type | (only part of exam material in August) |

### Final exam

Exams will be oral.

As agreed in class, the first round of examinations will take place on Thursday, July 11, Friday, July 12 and Monday, July 15, while the second round will take place on Monday, August 12.

Each exam will last 30 minutes, and the first question will be to explain the solution of an exercise taken from a non graded exercise sheet.

### Contact

### Office Hours

Please send me an email a couple of days in advance.

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

JProf. Dr. Beatrice Pozzetti | Wed 14:00-15:00 | 2.009 |

### 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, geometric boundary, visibility at infinity.

### 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 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.