Shiing-Shen Chern

Shiing-Shen Chern was a Chinese-American mathematician whose work shaped modern differential geometry, especially through the Chern classes and the Chern–Weil theory of characteristic classes.

Differential geometry: The study of smooth spaces using calculus, geometric structures, and differential equations. Chern made this field a central setting for studying global geometric structure.
Chern class: A sequence of cohomology classes that captures topological information about a complex vector bundle. These classes are Chern’s best-known contribution and a foundation of his geometric legacy.
Élie Cartan: A French mathematician who developed influential methods in differential geometry and Lie theory. Cartan’s work on moving frames and differential forms shaped the tradition Chern advanced.
Chern–Gauss–Bonnet theorem: A theorem expressing the Euler characteristic of an even-dimensional Riemannian manifold as an integral of curvature. Chern proved a higher-dimensional form of Gauss–Bonnet using characteristic forms.
Fiber bundle: A space built from local products of a base space and a typical fiber, joined by transition maps. Bundles provide the geometric setting for characteristic classes developed in Chern’s work.
Chern–Weil homomorphism: A construction that turns invariant polynomials of bundle curvature into cohomology classes. It gives a differential-geometric method for representing characteristic classes.
André Weil: A French mathematician whose work connected geometry, number theory, and topology. Weil and Chern jointly established the Chern–Weil approach to characteristic classes.
Chern number: An integer obtained by evaluating products of Chern classes on a manifold’s fundamental class. Chern numbers turn his characteristic classes into computable topological invariants.
Connection (mathematics): A structure on a bundle that defines how to differentiate sections and compare fibers. Connections let Chern translate local differential data into global invariants.
Chern form: A differential form built from curvature whose cohomology class represents a Chern class. Chern forms realize bundle invariants through local curvature calculations.

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