Learning seminar on G-structures

Seminar Description

The purpose of this seminar is to study basic notions of G-structures. 

G-structures provide a framework for studying several geometric structures (such as Riemannian, symplectic, complex, etc.) in an unified way, where the symmetries relevant for each geometry play a central role. One point of view is that they let us study a geometric structure by studying adapted frames: for example, knowing a Riemannian structure is the same as knowing all orthonormal frames on a manifold.

We will have as a guiding example the study of differential geometry on quaternionic manifolds, which have some formal similarities with complex manifolds, but also crucial differences. For example, quaternionic manifolds can be distinguished locally by curvature - like Riemannian manifolds, and unlike complex manifolds.  

Seminar organization

The format is that of a learning seminar - the material will be presented by participants of the seminar; although the speaker of each session prepares the material in advance, the idea is to keep the environment welcoming (particularly to students), having in mind that the speaker and the audience are learning together.

The first session is scheduled October 21, 2025. The following sessions (which will will happen weekly) and their topics will be organized by the participants. The schedule of the seminar will be posted and updated on this webpage.

Schedule

Where: Room 2.5

1. October 21, 2025: Introduction and planning ; bundles and connections - Pedro Pessoa

2. October 28, 2025: Bundles and connections - Part II - Pedro Pessoa

3. November 4, 2025: G-structures - Pedro Pessoa

4. To be decided... 

Bibliography

Bundles and Connections

S. Morita, Geometry of Differential Forms, AMS Translations of Mathematical Monographs, 201 (2001)

L. W. Tu, Differential Geometry: Connections, Curvature and Characteristic Classes, Springer Graduate Texts in Mathematics, 275 (2017)

G-structures

M. Crainic - Lecture notes for the Mastermath course "Differential Geometry 2015/2016"

A Čap, Lecture notes on G-structures, Fall Term 2021/22

S. M. Salamon, Differential geometry of quaternionic manifolds. Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 19 (1986)

Organisers

João Nuno Mestre, Pedro Pessoa, Ivan Yudin