Notes sur les vari\'et\'es diff\'erentiables, structures complexes et quaternioniques et applications

TL;DR

This paper provides lecture notes on differential geometry, focusing on complex and quaternionic structures, and explores their applications in theoretical physics, including Penrose transformations and formulations of Yang-Mills and Einstein equations.

Contribution

It offers a comprehensive overview of complex and quaternionic structures with novel applications to physics, including unusual formulations of fundamental equations.

Findings

Analysis of Penrose transformation in Riemannian context

Various formulations of Yang-Mills equations

Applications to Einstein equations

Abstract

These are notes of lectures given at the Third School of Theoretical Physics in Jijel (Algeria, September 2009). The subject of these notes is differential geometry, complex and quaternionic structures with applications to theoretical physics. Concerning the physical applications, they contain several aspects of Penrose transformation in the Riemannian context (Euclidean signature) and various formulations of the Yang-Mills and Einstein equations among which several are unusual ones.