Chern's work in geometry (Q2373592)

This paper contains the author's remarks made in his speech at the Harvard Memorial Conference for S. S. Chern. The author starts with a list of ``the major events in the glorious history of geometry'' (up to about 1900) defined by names from Pythagoras to Felix Klein, and he quotes A. Weil from the preface of selected papers of S. S. Chern that mathematics in the 20th century was to a great extent determined by the geometric insights of Élie Cartan, Heinz Hopf and S. S. Chern. The author gives credit to Élie Cartan for the birth of modern (i.e., 20th century) differential geometry by Élie Cartan's continuation of the pioneering work of Gauss and Riemann through systematic use of Lie group theory and invariant theory of differential systems. The author recalls Heinz Hopf's thesis of 1925 on the Gauss-Bonnet formula in the hypersurface case which led to the development of differential topology. Chern is called the father of global intrinsic differential geometry who inspired the Allendörfer-Weil-Chern treatment of the Gauss-Bonnet formula. Successful use of fiber bundles is mentioned as it played an important role in the theory of Stiefel-Whitney classes for differentiable manifolds (involving the orthogonal group) and the Chern classes in the complex (unitary) case (in \textit{S. S. Chern's} seminal paper ``Characteristic classes of Hermitian manifolds'' [Ann. Math. (2) 47, 85--121 (1946; Zbl 0060.41416)]). Chern's life and work in mathematics, teaching and administration is described.