Lecturer: Jose Figueroa-O'Farrill (University of Edinburgh)
The aim of these lectures is to explore what one may learn about some of the non-lorentzian geometries of current interest (Newton-Cartan, Carroll, Bargmann,…) by reformulating them in the language of G-structures and in particular the classification which results by studying their intrinsic torsion. In this way we will rederive the classification of Newton—Cartan geometries into NC, TTNC and TNC geometries, and hopefully say something novel about Carroll and Bargmann geometries.
The detailed lecture plan is the following:
1. The frame bundle of a smooth manifold and the soldering one-form
2. Connections on the frame bundle, their torsion and curvature
3. G-structures and intrinsic torsion
4. Non-lorentzian G-structures
4.1 Galilei (a.k.a., Newton-Cartan)
4.2 Carroll
4.3 Bargmann (if time allows)
The aim is to cover 1. and 2. in the first lecture, 3. in the second lecture and 4. in the last lecture, but this will become clear as the lectures progress.
The lectures will be held on July 14,15,17. All lectures will start at 13.30 CET and consist of two blocks of 40 minutes with a short break between them. Zoom link TBA.
Lecture 1:
The frame bundle of a smooth manifold, G-structures and the soldering one-form
Lecture 2:
Ehresmann connections on the frame bundle, their torsion, and intrinsic torsion
Lecture 3:
Intrinsic torsion of Galilean, Carrollian and Aristotelian G-structures
