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

In April 2003, Chern began a study of almost-complex structures on the six-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 S-6, he did prove a significant identity that resolves the question for an interesting class of almost-complex structures on S-6.