S.-S. Chern's study of almost-complex structures on the six-sphere

TL;DR

This paper reviews S.-S. Chern's investigation into almost-complex structures on the 6-sphere, highlighting his partial progress and a key identity related to a special class of these structures, despite the open problem remaining unsolved.

Contribution

It introduces a significant identity for a class of almost-complex structures on the 6-sphere, advancing understanding even without solving the main open problem.

Findings

Proved a key identity for certain almost-complex structures

Established the invariance under the group G_2

Progressed towards understanding the integrability question

Abstract

In 2003, S.-s. Chern began a study of almost-complex structures on the 6-sphere, with the idea of exploiting the special properties of its well-known almost-complex structure invariant under the exceptional group $G_2$. While he did not solve the (currently still open) problem of determining whether there exists an integrable almost-complex structure on the 6-sphere, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on the 6-sphere.