Celebratio Mathematica

article S.-S. Chern: “Sur la géométrie d’une équation différentielle du troisième ordre” [On the geometry of a differential equation of third order], C. R. Acad. Sci., Paris 204 (1937), pp. 1227–1229. JFM 63.0419.01 Zbl 0016.16401

article S.-S. Chern: “Sur la géométrie d’un système d’équations différentielles du second ordre” [On the geometery of a system of second order differential equations], Bull. Sci. Math., II. Ser. 63 (1939), pp. 206–212. MR 0000889 Zbl 0023.07701

article S.-S. Chern: “The geometry of higher path-spaces,” J. Chin. Math. Soc. 2 (1940), pp. 247–276. MR 0004531 Zbl 0063.00828

article S.-S. Chern: “The geometry of the differential equation \( y^{\prime\prime\prime}=F(x,y,y^{\prime},y^{\prime\prime}) \),” Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), pp. 97–111. MR 0004538 Zbl 0024.19801

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: “On a Weyl geometry defined from an \( (n-1) \)-parameter family of hypersurfaces in a space of \( n \) dimensions,” Sci. Rec. 1 (1942), pp. 7–10. MR 0007637 Zbl 0060.39208

article S.-S. Chern: “A generalization of the projective geometry of linear spaces,” Proc. Natl. Acad. Sci. U. S. A. 29 : 1 (January 1943), pp. 38–43. MR 0008192 Zbl 0060.39209

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

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 and Y.-F. Sun: “The imbedding theorem for fibre bundles,” Trans. Am. Math. Soc. 67 (1949), pp. 286–303. MR 0032996 Zbl 0037.10203

article S.-S. Chern: “On the kinematic formula in the Euclidean space of \( n \) dimensions,” Am. J. Math. 74 (1952), pp. 227–236. See also Boll. Unione Mat. Ital. 2 (1940). MR 0047353 Zbl 0046.16101

article S.-S. Chern and R. K. Lashof: “On the total curvature of immersed manifolds,” Am. J. Math. 79 : 2 (April 1957), pp. 306–318. MR 0084811 Zbl 0078.13901

article S.-S. Chern and R. K. Lashof: “On the total curvature of immersed manifolds, II,” Mich. Math. J. 5 : 1 (1958), pp. 5–12. MR 0097834 Zbl 0095.35803

incollection S.-S. Chern: “Minimal surfaces in an Euclidean space of \( N \) dimensions,” pp. 187–198 in Differential and combinatorial topology (A symposium in honor of Marston Morse). Edited by S. S. Cairns. Princeton Mathematical Series 27. Princeton University Press, 1965. MR 0180926 Zbl 0136.16701

Norman Morse	Related
Stewart Scott Cairns	Related

article S.-S. Chern: “On the kinematic formula in integral geometry,” J. Math. Mech. 16 (1967), pp. 101–118. MR 0198406 Zbl 0142.20704

article S.-S. Chern and R. Osserman: “Complete minimal surfaces in Euclidean \( n \)-space,” J. Anal. Math. 19 : 1 (1967), pp. 15–34. MR 0226514 Zbl 0172.22802

R. Bott and S. S. Chern: “Some formulas related to complex transgression,” pp. 48–57 in Essays on topology and related topics: Mémoires dédiés à Georges de Rham [Essays on topology and related topics: Memoirs dedicated to Georges de Rham]. Edited by A. Haefliger and R. Narasimhan. Springer (New York), 1970. MR 0264715 Zbl 0203.54202 incollection

Let \( X \) be a complex manifold of complex dimension \( n \) and \( \pi:E\to X \) a holomorphic vector bundle whose fiber dimension is also \( n \). On \( E \) we introduce a positive definite hermitian norm \( N \) and denote by \[ B^*(E) = \{e\in E\mid 0 < N(e)\} \] the subset of non-zero vectors of \( E \). Let \( c_n(E) \) be the Chern form of the hermitian structure (to be described again below). In [1965] we proved the theorem: There exists a real-valued form \( \rho \) of type \( (n-1, \) \( n-1) \) on \( B^*(E) \) such that \begin{equation*}\tag{1} \pi^* c_n(E) = \frac{dd^c}{4\pi}\rho. \end{equation*} In this paper we wish to give a proof of (1) based on an explicit construction. We will follow and extend a formalism developed in the real case by H. Flanders [1953]. It is possible that this formalism will be useful for later purposes.

Georges de Rham	Related
Raghavan Narasimhan	Related
Raoul H. Bott	Related	Volume
André Haefliger	Related

incollection S.-S. Chern: “On the minimal immersions of the two-sphere in a space of constant curvature,” pp. 27–40 in Problems in analysis: A symposium in honor of Salomon Bochner (Princeton, NJ, 1–3 April 1969). Edited by R. C. Gunning. Princeton Mathematical Series 31. Princeton University Press, 1970. MR 0362151 Zbl 0217.47601

Robert Clifford Gunning	Related
Salomon Bochner	Related

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

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

article S.-S. Chern and S. I. Goldberg: “On the volume decreasing property of a class of real harmonic mappings,” Am. J. Math. 97 : 1 (1975), pp. 133–147. MR 0367860 Zbl 0303.53049

book S.-S. Chern: Selected papers, vol. 1. Springer (New York), 1978. With a foreword by Hung Hsi Wu and introductory articles by André Weil and Phillip A. Griffiths. MR 514211 Zbl 0403.01012

Hung-hsi Wu	Related
Phillip A. Griffiths	Related
André Weil	Related

incollection S.-S. Chern: “Moving frames,” pp. 67–77 in The mathematical heritage of Élie Cartan (Lyons, France, 25–29 June 1984). Astérisque. 1985. Numéro Hors Série. MR 837194 Zbl 0614.53038

article T. E. Cecil and S.-S. Chern: “Tautness and Lie sphere geometry,” Math. Ann. 278 : 1–4 (1987), pp. 381–399. Dedicated to Friedrich Hirzebruch on the occasion of his sixtieth birthday. MR 909233 Zbl 0635.53029

F. Hirzebruch	Related	Volume
Thomas Edward Cecil	Related

book S.-S. Chern: Selected essays by S. S. Chern. Science Press (Beijing), 1989. In Chinese.

book S. S. Chern: Selected papers, vol. II. Springer (New York), 1989. MR 1007138 Zbl 0682.01017

book S. S. Chern: Selected papers, vol. IV. Springer (New York), 1989. MR 1019829

book S. S. Chern: Selected papers, vol. III. Springer (New York), 1989. MR 1025503

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