$S^6$ and the geometry of nearly K\"ahler $6$-manifolds

TL;DR

This paper reviews classical and modern results concerning the almost complex structure on the 6-sphere, focusing on its relation to nearly Kähler geometry and the ongoing debate about complex structures on $S^6$.

Contribution

It provides a comprehensive overview of the geometric properties and recent developments related to the almost complex structure on $S^6$, highlighting the interplay with nearly Kähler geometry.

Findings

Summarizes classical results on $S^6$ and its almost complex structure.

Discusses modern approaches and open problems in nearly Kähler geometry.

Highlights the non-existence of integrable complex structures on $S^6$.

Abstract

We review results on and around the almost complex structure on $S^6$, both from a classical and a modern point of view. These notes have been prepared for the Workshop "(Non)-existence of complex structures on $S^6$" (\emph{Erste Marburger Arbeitsgemeinschaft Mathematik -- MAM-1}), held in Marburg in March 2017.