# Differential Geometry 2

- Symmetric Spaces

Winter Semester 2018-2019

### Schedule

**Main Lecture**

- Monday, 14:15-15:50, INF 205 / SR 5
- Thursday, 11:15-12:50, INF 205 / SR 5

**Exercise Session**

- Wednesday, 4:15-5:45, INF 205 / 02.104

Starting from Wednesday October 24.

### Müsli

There is a page for this course on MÜSLI.

### 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 |
[DC, Ch 7] [Ma, Sec 3] |

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

Week 5 | Transvections The isometry group of a symmetric space as a Lie group Lie groups and Lie algebras |
[Io. 2.3] [He, IV.3] [Sc., 2] |

Week 6 | Semisimple Lie groups Compact Lie groups |
[He, II.6] [He, Thm 6.9] |

Week 7 | Riemannian symmetric pairs Compact Lie groups as symmetric spaces |
[He. IV.3] [He. IV.6] |

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

Week 9 | Irreducible OSLAs Duality |
[He. VIII.5] [He. V.2] |

Week 10 | Curvature Summary of the first part of the lecture |
[He. V.3] |

Week 11 | Lie triple systems CAT(0) geomery |
[He. IV.7] [Ma, 5.2] |

Week 12 | Parallel sets Flat subspaces (Johannes Horn) |
[Ma, 5,2] [Io 3.2] |

Week 13 | Boundary at infinity Iwasawa decomposition (Mareike Pfeil) |
[Ma 5.2.3] [Eb, 2.2.17] |

Week 14 | Visibility Tits distance |
[Ma 6.5-6.6] [Eb, 3.6] |

### Exercise sheets

Exercise sheets will be published regularly here, usually on Thursday night.

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. The tutors will answer questions and give you feedback about your presentation but won't solve exercises for you.

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 solutions to the exercise sheets and have a quick look at it before solving it on the board.

### Final exam

During the last two weeks of the course 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.

### Contact

Dr. Daniele Alessandrini (Exercises)

Johannes Horn (tutor)

Mareike Pfeil (tutor)

### Office Hours

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

JProf. Dr. Beatrice Pozzetti | Wed 10:00-11:00 | INF 205, 03.312 |

### 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.