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Reading Seminar:
Group theory meets differential geometry: Crystallographic groups and flat manifolds


together with Jacques Audibert

Time: Thursdays, 15:15-16:45 (Start: October 16, no class on December 25 and January 1)
Place: SG 3-12
Reference: Bieberbach groups and flat manifolds by Leonard S. Charlap


Prerequisites: The prerequisites for this seminar are group theory and smooth manifolds. Apart from that everything will be defined from scratch. You do not need to know yet what Riemannian geometry is, but it would be good to know what a fundamental group is.

Organization: Every week students will be given a few number of pages of the book to read at home. We expect students to solve the exercises throughout the book. The exercises, as well as necessary background, will be discussed during the weekly in presence meeting. 		Charlap's book
Goal: The goal of this reading seminar is to read the first three chapters of the book "Bieberbach groups and flat manifolds" by Leonard Charlap. This book presents the correspondence between crystallographic groups and flat manifolds, which illustrates the strong interplay between group theory and Riemannian geometry.
The first chapter presents the three famous theorems of Bieberbach on crystallographic groups, that describe their algebraic structure. A crystallographic group is a subgroup G of the isometries of ℝ n , that is discrete and for which ℝ n /G is compact. Once we learn the basics from Riemannian geometry in the second chapter, we see that the manifold ℝ n /G is "flat". Thus crystallographic groups provide numerous examples of interesting Riemannian manifolds. This point of view also allows for a geometric reformulation of Bieberbach's theorems.
The third chapter, after introducing group cohomology, investigates the classification of crystallographic groups.

Last updated: September 2025