Celebratio Mathematica

article S.-S. Chern: “On integral geometry in Klein spaces,” Ann. Math. (2) 43 : 1 (January 1942), pp. 178–189. MR 0006075 Zbl 0147.22303

article S.-S. Chern: “The geometry of isotropic surfaces,” Ann. Math. (2) 43 : 3 (July 1942), pp. 545–559. MR 0006477 Zbl 0060.39207

The classical results in integral geometry found by Crofton, Poincaré, Cartan, and recently developed by Blaschke [1936, 1937] and his school, are mostly restricted to Eulidean spaces. It is the object of this paper to give the fundamental concepts of integral geometry in a general space of Klein, by which we mean a number space of \( n \) dimensions with a transitive \( r \)-parameter group \( G_r \) of transformations. The discussion is mainly based on Cartain’s theory of Lie’s groups [1937].

The paper is divided into three sections. In §1 we give a brief summary of Cartan’s theory of Lie’s groups with some results concerning the measures of geometrical elements. In §2 we define the incidence of the geometrical elements of different fields, which plays a fundamental rôle in all subsequent discussions. As applications of the general notions we give in the last section two formulas which are respectively generalizations of the well-known formulas of Crofton and Cauchy.

article S.-S. Chern: “A simple intrinsic proof of the Gauss–Bonnet formula for closed Riemannian manifolds,” Ann. Math. (2) 45 : 4 (October 1944), pp. 747–752. MR 0011027 Zbl 0060.38103

article S.-S. Chern: “Some new viewpoints in differential geometry in the large,” Bull. Am. Math. Soc. 52 : 1 (1946), pp. 1–30. MR 0021706 Zbl 0063.00834

article S.-S. Chern: “Characteristic classes of Hermitian manifolds,” Ann. Math. (2) 47 : 1 (January 1946), pp. 85–121. MR 0015793 Zbl 0060.41416

article S.-S. Chern: “The integrated form of the first main theorem for complex analytic mappings in several complex variables,” Ann. Math. (2) 71 : 3 (May 1960), pp. 536–551. MR 0125979 Zbl 0142.04802

The object of this paper is to make a beginning of the study of complex analytic mappings of the complex euclidean space of \( m \) dimensions into the complex projective space of the same dimension in the direction of a generalization of the classical theory of Picard–Borel for the one-dimensional case. The main geometrial conclusion of the Picard–Borel theory can be interpreted as a statement on the “size” of the image set. It is well-known that similar statements are not true in the case of higher dimensions. In fact, there are examples due to Bieberbach and Fatou in which the image set omits open subsets of the complex projective space [Bochner and Martin 1948, p. 45]. A proper approach to the theory should therefore begin with introducing the suitable concepts and draw conclusions on them. In a problem of this nature it is the general dimension of the original set which creates the difference from the classical theory. We take it to be the complex euclidean space. When the situation in this particular case becomes clear, various generalizations are possible.

R. Bott and S. S. Chern: “Hermitian vector bundles and the equidistribution of the zeroes of their holomorphic sections,” Acta Math. 114 : 1 (1965), pp. 71–112. A Russian translation was published in Matematika 14:2 (1970). MR 0185607 Zbl 0148.31906 article

article S.-S. Chern and J. K. Moser: “Real hypersurfaces in complex manifolds,” Acta Math. 133 : 1 (1974), pp. 219–271. MR 0425155 Zbl 0302.32015

S.-S. Chern and J. Simons: “Characteristic forms and geometric invariants,” Ann. Math. (2) 99 : 1 (January 1974), pp. 48–69. MR 0353327 Zbl 0283.53036 article

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Shiing-Shen Chern

Shiing-shen Chern: 1911–2004

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