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Shiing-Shen Chern (陳省身 October 26, 1911 – December 3, 2004) was a Chinese American mathematician who won the 2004 Shaw prize.
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- It is well known that in three-dimensional elliptic or spherical geometry the so-called Clifford's parallelism or parataxy has many interesting properties. A group-theoretical reason for the most important of these properties is the fact that the universal covering group of the proper orthogonal group in four variables is the direct product of the universal covering groups of two proper orthogonal groups in three variables. This last-mentioned property has no analogue for orthogonal groups in n (>4) variables. On the other hand, a knowledge of three-dimensional elliptic or spherical geometry is useful for the study of orientable Riemannian manifols of four dimensions, because their tangent spaces possess a geometry of this kind.
- (1945). "On Riemannian manifolds of four dimensions". Bulletin of the American Mathematical Society 51 (12): 964–971.
- In 1917 Levi-Civita discovered his celebrated parallelism which is an infinitesimal transportation of tangent vectors preserving the scalar product and is the first example of a connection. The salient fact about the Levi-Civita parallelism is the result that it is the parallelism, and not the Riemannian metric, which accounts for most of the properties concerning curvature.
- Differential Manifolds (Classroom Notes) Math 352A, Spring 1952. Department of Mathematics, University of Chicago.
- Integral geometry, started by the English geometer M. W. Crofton, has received recently important developments through the works of W. Blaschke, L. A. Santaló, and others. Generally speaking, its principal aim is to study the relations between the measures which can be attached to a given variety.
- "Differential geometry and integral geometry". Proc. Int. Congr. Math. Edinburgh. 1958. pp. 411–449.
- Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan.
- (1966). "The geometry of 𝐺-structures". Bulletin of the American Mathematical Society 72 (2): 167–219.
- The main object of study in differential geometry is, at least for the moment, the differential manifolds, structures on the manifolds (Riemannian, complex, or other), and their admissible mappings. On a manifold the coordinates are valid only locally and do not have a geometric meaning themselves.
- "Differential geometry, its past and its future". Actes, Congrès inter. math. 1970. pp. 41–53.
- The treatises of Darboux (1842–1917) and Bianchi (1856–1928) on surface theory are among the great works in the mathematical literature. They are: G. Darboux, Théorie générale des surfaces, Tome 1 (1887), 2 (1888), 3 (1894), 4 (1896), and later editions and reprints. L. Bianchi. Lezioni di Geometria Differenziale, Pisa 1894; German translation by Lukat, Lehrbuch der Differentialgeometrie, 1899. The subject is basically local surface theory.
- "Surface Theory with Darboux and Bianchi". Miscellanea Mathematica. Springer. 1991. pp. 59–69.
Quotes about Chern
- I have no doubt that future historians of differential geometry will rank Chern as the worthly successor of Elie Cartan in that field.
- André Weil as quoted in A mathematician and his mathematical work: selected papers of S. S. Chern. 1996. p. ix.
- Recently, having refreshed my understanding of the mathematics of relativity theory, I called one of my old Berkeley professors to ask him some questions about the geometry of general relativity. S. S. Chern is arguably the greatest living geometer. We spoke on the phone for a long time, and he patiently answered all my questions. When I told him I was contemplating writing a book about relativity, cosmology, and geometry and how they interconnect to explain the universe, he said, "It's a wonderful idea for a book, but writing it will surely take too many years of your life ... I wouldn't do it." Then he hung up.