Shiing-Shen Chern - Research

Research

Chern's work extends over all the classic fields of differential geometry. It includes areas currently fashionable (the Chern-Simons theory arising from a 1974 paper written jointly with Jim Simons), perennial (the Chern-Weil theory linking curvature invariants to characteristic classes from 1944, after the Allendoerfer-Weil paper of 1943 on the Gauss-Bonnet theorem), the foundational (Chern classes), and some areas such as projective differential geometry and webs that have a lower profile. He published results in integral geometry, value distribution theory of holomorphic functions, and minimal submanifolds.

He was a follower of Élie Cartan, working on the 'theory of equivalence' in his time in China from 1937 to 1943, in relative isolation. In 1954 he published his own treatment of the pseudogroup problem that is in effect the touchstone of Cartan's geometric theory. He used the moving frame method with success only matched by its inventor; he preferred in complex manifold theory to stay with the geometry, rather than follow the potential theory. Indeed, one of his books is entitled, "Complex Manifolds without Potential Theory". In the last years of his life, he advocated the study of Finsler geometry, writing several books and articles on the subject.