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