# Differential Geometry 2

- Symmetric Spaces

Winter Semester 2017-2018

### Schedule

**Main Lecture**

- Tuesday, 9:15-10:45, INF 205 / SR C
- Thursday, 9:15-10:45, INF 205 / SR C

Starting from Tuesday October 17.

**Exercise Session**

- Friday, 2:15-3:45, INF 205 / SR 4

### Müsli

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

### Lectures

Week 1 | Introduction to Symmetric spaces Models for symmetric spaces of rank one |
[BH Ch II.10] |

Week 2 | Totally geodesic subspaces of KH^{n}Summary of Differential Geometry 1 Jacobi fields |
[DC, Ch 5] |

Week 3 | Cartan Hadamard theorem Fundamental groups and covering spaces |
[DC, Ch 7] [Ha, Ch 2] |

Week 4 | Riemannian characterization of locally symmetric spaces | [Ma, Sec 3-4] |

Week 5 | The isometry group of a symmetric space Transvections Locally symmetric pairs |
[He, Ch. 4.2] [Io] |

Week 6 | Lie groups and Lie algebras Semisimple Lie groups and algebras |
[Io 2.10, He II.6] |

Week 7 | Compact Lie groups Riemannian Symmetric spaces |
[Sc, Ch. 2.1] [Io, 2.4] |

Week 8 | Orthogonal symmetric Lie algebras Compact/non-compact type |
[He, V.1] |

Week 9 | Decomposition into irreducible duality |
[Io, 2.8,2.11] |

Week 10 | Curvature Lie triple systems |
[Pa, 4.1.3] [Io, 2.6] |

Week 11 | Hadamard manifolds Flat subspaces |
[Ma, 5.2] [Io, 3.2] |

Week 12 | Transitivity on flat subspaces Visibility at infinity |
[Io 3.2] [Maubon 5.2.3, 6.1 and 6.5] |

Week 13 | Root spaces Iwasawa decomposition every two boundary points are in the boundary of a flat |
[Io 3.3] [Paulin 4.3.3] [Maubon 6.6] |

Week 14 | Discussion on previous topics | |

Week 15 | Gromov-Ballman-Schroeder rigidity theorem | [BGS Lecture IV] |

### Exercise sheets

Exercise sheets will be published regularly on this website, usually on Thursday night. Some exercise sheets will be marked as graded sheets, and the others will be marked as not graded.

When a graded exercise sheet appears, the students will have two weeks time to solve the exercises, write the solutions and hand them in, by Wednesday at 2pm. The grades of every exercise sheet will be written on MÜSLI by the tutors. To be admitted to the final exam, students need to obtain at least 50% of the available points of the graded sheets.

The exercises of the non-graded sheets will be used during the final exam, the first question of every exam will be taken from there.

19.10.2017 | Sheet 1: Grassmannians (not graded) |

26.10.2017 | Sheet 2: Hyperbolic spaces and Jacobi fields (graded) |

09.11.2017 | Sheet 3: Symmetric spaces (not graded) |

16.11.2017 | Sheet 4: Lie groups and Lie algebras (graded) |

30.11.2017 | Sheet 5: Compact symmetric spaces (not graded) |

07.12.2017 | Sheet 6: Riemannian symmetric pairs (graded) |

21.12.2017 | Sheet 7: Christmas exercise sheet (not graded, bonus) |

11.01.2018 | Sheet 8: Spaces of non-compact type (graded) |

25.01.2018 | Sheet 9: Decompositions into products (not graded) |

### Final exam

During the last two weeks of the course we will organize oral exams for the students who obtained at least 50% of the points in the graded exercise sheets. 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 an exercise from the non-graded exercise sheets. The final exam will be in English.

### Contact

### Office Hours

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

JProf. Dr. Beatrice Pozzetti | tba | INF 205, 03.312 |

Dr. Daniele Alessandrini | tba | INF 205, 03.311 |

### 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. An outline of the material covered in the lecture is the following:

- Jacobi Fields and Cartan Hadamard theorem.
- Generalities on symmetric spaces: locally and globally symmetric spaces, Riemannian characterization, topological groups of isometries, examples.
- Relations with Riemannian symmetric pairs and orthogonal symmetric Lie algebras.
- Decomposition in irreducible subspaces, type for irreducible subspaces, duality, curvature, totally geodesic submanifolds.
- Symmetric spaces of non-compact type: flat subspaces and the notion of rank. Iwasawa decomposition.
- Geometry at infinity: geometric boundary, 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.
- [Ba] Ballmann: Symmetric spaces.
- [Io] Iozzi: 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.
- [Ma] Maubon: Riemannian symmetric spaces of the non-compact type: differential geometry.
- [Sc] Schroeder: Symmetrische Räume.
- [BGS] Ballmann, Gromov, Schroeder: Manifolds of nonpositive curvature.

Further reading

- [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.

Zuletzt geändert: 25.01.2018