Celebratio Mathematica — Chern

[1] article H. S. Chen: “Pairs of plane curves with points in one-to-one correspondence,” Sci. Rep. Nat. Tsing Hua Univ. (A) 1 (1932), pp. 145–153. JFM 58.1239.03 Zbl 0005.41004

[2] article S.-S. Chern: “Triads of rectilinear congruences with generators in correspondence,” Tohoku Math. J. 40 (1935), pp. 179–188. JFM 61.1436.03 Zbl 0010.41704

[3] article S.-S. Chern: “Associate quadratic complexes of a rectilinear congruence,” Tohoku Math. J. 40 (1935), pp. 293–316. JFM 61.1437.01 Zbl 0010.41801

[4] article S.-S. Chern: “Eine Invariantentheorie der Dreigewebe aus \( r \)-dimensionalen Mannigfaltigkeiten im \( \mathbb{R}_{2r} \)” [An invariant theory of three-webs on \( r \)-dimensional manifolds in \( \mathbb{R}_{2r} \)], Abh. Math. Semin. Hamb. Univ. 11 : 1 (1935–1936), pp. 333–358. Topologische Fragen der Differentialgeometrie 62. JFM 62.0829.03 Zbl 0013.41802

[5] article S.-S. Chern: “Abzählungen für Gewebe” [Enumeration of webs], Abh. Math. Semin. Hamb. Univ. 11 : 1 (1935–1936), pp. 163–170. Topologische Fragen der Differentialgeometrie 60. See also the author’s PhD thesis (1936). Zbl 0011.13202

[6] phdthesis S.-S. Chern: Abzählungen für Gewebe [Enumeration of webs]. Ph.D. thesis, Hamburg University, 1936. Advised by W. J. E. Blaschke. See also Abh. Math. Semin. Hamb. Univ. 11:1 (1935–1936).

[7] 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

[8] article S.-S. Chern: “Sur la possibilité de plonger un espace à connexion projective donné dans un espace projectif” [On the possibility of immersing a projective connection space in a projective space], Bull. Sci. Math., II. Sér. 61 (1937), pp. 234–243. JFM 63.0692.03 Zbl 0017.22804

[9] article S.-S. Chern: “On two affine connections,” J. Yunnan Nation. Univ., Nat. Sci. 1 (1938), pp. 1–18.

[10] article S.-S. Chern: “On projective normal coördinates,” Ann. Math. (2) 39 : 1 (January 1938), pp. 165–171. MR 1503394 Zbl 0018.23604

[11] 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

[12] article S.-S. Chern: “Sur les invariants intégraux en géométrie” [On integral invariants in geometry], Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1940), pp. 85–95. MR 0004527 Zbl 0024.18201

[13] article S.-S. Chern: “Generalization of a formula of Crofton,” Wuhan Univ. J. Sci. 7 (1940), pp. 1–16. See also C. R. Acad. Sci., Paris 210 (1940).

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

[15] 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

[16] article S.-S. Chern: “Sur une généralisation d’une formule de Crofton” [On a generalization of a formula of Crofton], C. R. Acad. Sci., Paris 210 (1940), pp. 757–758. See also Wuhan Univ. J. Sci. 7 (1940). MR 0006076 Zbl 0024.35601

[17] article S.-S. Chern and C.-t. Yien: “Sulla formula principale cinematica dello spazio ad \( n \) dimensioni” [The main kinematic formula for space of \( n \) dimensions], Boll. Unione Mat. Ital., II. Ser. 2 (1940), pp. 434–437. See also Am. J. Math. 74 (1952). MR 0004983 Zbl 0024.08402

[18] article S.-S. Chern: “Sur les invariants de contact en géométrie projective différentielle” [On invariants of contact in projective differential geometry], Acta 5 (1941), pp. 123–140. MR 0025786 Zbl 0028.08901

[19] 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

[20] article S.-S. Chern: “On the invariants of contact of curves in a projective space of \( n \) dimensions and their geometrical interpretation,” Sci. Rec. 1 (1942), pp. 11–15. MR 0007636 Zbl 0063.00830

[21] 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

[22] 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.

[23] 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

[24] article S.-S. Chern: “On the Euclidean connections in a Finsler space,” Proc. Natl. Acad. Sci. U. S. A. 29 : 1 (January 1943), pp. 33–37. MR 0008191 Zbl 0060.39210

[25] article S.-S. Chern: “Integral formulas for the characteristic classes of sphere bundles,” Proc. Nat. Acad. Sci. U.S.A. 30 : 9 (September 1944), pp. 269–273. MR 0011028 Zbl 0060.38105

The notion of sphere bundles of H. Whitney [1941], [Stiefel 1936] has proved to be very fruitful in the applications of topology. So far in this theory an important rôle has been played by a class of topological invariants in the base space, called the characteristic cohomology classes and which can be regarded as generalizations of the Euler–Poincare characteristic. By assuming the base space to be a differentiable manifold with a Riemannian metric, on which the bundles of vectors and \( p \)-vectors are defined in a natural way, we shall give in this note various relations which exist between these topological invariants and certain differential invariants of the Riemannian manifold. In fact, the relations are such that the former are expressed as integrals of the latter. In the formulae the generalized Gauss–Bonnet formula due to Allendoerfer and Weil [1943] is included as a particular case.

[26] article S.-S. Chern: “Laplace transforms of a class of higher dimensional varieties in a projective space of \( n \) dimensions,” Proc. Natl. Acad. Sci. U.S.A. 30 : 4 (April 1944), pp. 95–97. MR 0009882 Zbl 0063.00831

[27] article S.-S. Chern: “On a theorem of algebra and its geometrical application,” J. Indian Math. Soc., New Ser. 8 (1944), pp. 29–36. MR 0011786 Zbl 0060.38201

[28] 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

[29] article S.-S. Chern: “On Riemannian manifolds of four dimensions,” Bull. Am. Math. Soc. 51 : 12 (1945), pp. 964–971. MR 0013910 Zbl 0060.41405

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-theoretic 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 manifolds of four dimensions, because their tangent spaces possess a geometry of this kind. It is the purpose of this note to give a study of a compact orientable Riemannian manifold of four dimensions at each point of which is attached a three-dimensional spherical space. This necessitates a more careful study of spherical geometry than hitherto given in the literature, except, so far as the writer is aware, in a paper by E. Study [1907]. 0ur main result consists of two formulas, which express two topological invariants of a compact orientable differentiable manifold of four dimensions as integrals over the manifold of differential invariants constructed from a Riemannian metric previously given on the manifold. These two topological invariants have a linear combination which is the Euler–Poincaré characteristic.

[30] article S.-S. Chern: “On Grassmann and differential rings and their relations to the theory of multiple integrals,” Sankhyā 7 : 1 (August 1945), pp. 2–8. MR 0013752 Zbl 0063.00832

[31] article S.-S. Chern: “On the curvatura integra in a Riemannian manifold,” Ann. Math. (2) 46 : 4 (October 1945), pp. 674–684. MR 0014760 Zbl 0060.38104

In a previous paper [1944] we have given an intrinsic proof of the formula of Allendoerfer–Weil which generalizes to Riemannian manifolds of \( n \) dimensions the classical formula of Gauss–Bonnet for \( n=2 \). The main idea of the proof is to draw into consideration the manifold of unit tangent vectors which is intrinsically associated to the Riemannian manifold. Denoting by \( R^n \) the Riemannian manifold of dimension \( n \) and by \( M^{2n-1} \) the manifold of dimension \( 2n-1 \) of its unit tangent vectors, our proof has led, in the case that \( n \) is even, to an intrinsic differential form of degree \( n-1 \) (which we denoted by \( \Pi \)) in \( M^{2n-1} \). We shall introduce in this paper a differential form of the same nature for both even and odd dimensional Riemannian manifolds. We find that this differential form bears a close relation to the “Curvatura Integra” of a submanifold in a Riemannian manifold, because it will be proved that its integral over a closed submanifold of \( R^n \) is equal to the Euler–Poincaré characteristic of the submanifold. The method can be carried over to deduce relations between relative topological invariants of a submanifold of the manifold and differential invariants derived from the imbedding, and some remarks are to be added to this effect.

[32] article S.-S. Chern: “Some new characterizations of the Euclidean sphere,” Duke Math. J. 12 : 2 (1945), pp. 279–290. MR 0012492 Zbl 0063.00833

[33] 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

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

[35] article S.-S. Chern and H.-c. Wang: “Differential geometry in symplectic space, I,” Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1947), pp. 453–477. MR 0025788

[36] article S.-S. Chern: “Sur une classe remarquable de variétés dans l’espace projectif à \( n \) dimensions” [On a remarkable class of varieties in projective space of \( n \) dimensions], Sci. Rep. Nat. Tsing Hua Univ. (A) 4 (1947), pp. 328–336. MR 0025787

[37] article S.-S. Chern: “Note on affinely connected manifolds,” Bull. Am. Math. Soc. 53 : 8 (1947), pp. 820–823. A correction was published in Bull. Am. Math. Soc. 54:10 (1948). MR 0021443 Zbl 0032.31004

[38] article S.-S. Chern: “On the characteristic ring of a differentiable manifold,” Sci. Rec. 2 (1947), pp. 1–5. MR 0023053

[39] article S.-S. Chern: “On the characteristic classes of Riemannian manifolds,” Proc. Nat. Acad. Sci. U. S. A. 33 : 4 (April 1947), pp. 78–82. MR 0019980 Zbl 0034.25002

[40] article S.-S. Chern: “Note on projective differential line geometry,” Sci. Rec. 2 : 2 (1948), pp. 137–139. MR 0026848 Zbl 0200.00005

[41] article S.-S. Chern: “On the multiplication in the characteristic ring of a sphere bundle,” Ann. Math. (2) 49 : 2 (April 1948), pp. 362–372. MR 0024127 Zbl 0033.40202

[42] article S.-S. Chern and Y.-l. Jou: “On the orientability of differentiable manifolds,” Sci. Rep. Nat. Tsing Hua Univ. (A) 5 (1948), pp. 13–17. MR 0025153 Zbl 0200.00006

[43] article S.-S. Chern: “Correction: ‘Note on affinely connected manifolds’,” Bull. Am. Math. Soc. 54 : 10 (1948), pp. 985–986. Correction to Bull. Am. Math. Soc. 53:8 (1947). MR 0026849 Zbl 0032.31005

[44] article S.-S. Chern: “Local equivalence and Euclidean connections in Finsler spaces,” Sci. Rep. Nat. Tsing Hua Univ. (A) 5 (1948), pp. 95–121. MR 0031812 Zbl 0200.00004

[45] article S.-S. Chern and S.-T. Hu: “Parallelisability of principal fibre bundles,” Trans. Am. Math. Soc. 67 (1949), pp. 304–309. MR 0032997 Zbl 0037.10301

[46] 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

[47] article S.-S. Chern and E. Spanier: “The homology structure of sphere bundles,” Proc. Natl. Acad. Sci. U.S.A. 36 : 4 (April 1950), pp. 248–255. MR 0035994 Zbl 0035.38803

[48] book S.-S. Chern: Topics in differential geometry. Institute for Advanced Study (Princeton, NJ), 1951. MR 0090080 Zbl 0054.06801

[49] article S.-S. Chern and E. Spanier: “A theorem on orientable surfaces in four-dimensional space,” Comment. Math. Helv. 25 (1951), pp. 205–209. MR 0044883 Zbl 0043.38403

Let \( M \) be a closed oriented surface differentiably imbedded in a Euclidean space \( E \) of four dimensions. Let \( G \) denote the Grassmann manifold of oriented planes through a fixed point \( O \) of \( E \). It is well known that \( G \) is homeomorphic to the topological product \( S_1\times S_2 \) of two 2-spheres. By mapping each point \( P \) of \( M \) into the oriented plane through \( O \) parallel to the oriented tangent plane to \( M \) at \( P \), we define a mapping \( t:M\to G \). If \( M \), \( S_1 \), \( S_2 \) denote also the fundamental cycles of the respective manifolds and \( t_* \) denotes the homomorphism induced by \( t \), we have \[ t_*(M) \sim u_1S_1 + u_2S_2. \] In a recent paper [1949] Blaschke studied the situation described above by methods of differential geometry and proved that the sum \( u_1 + u_2 \) equals the Euler charateristic of \( M \). He also asserted that \( u_1 = u_2 \). The object of this note is to give a proof of this assertion, as well as a new proof of the theorem on \( u_1+u_2 \).

[50] article S.-S. Chern: “La géométrie des sous-variétés d’un espace euclidien à plusieurs dimensions” [The geometry of submanifolds of Euclidean space in several dimensions], Enseign. Math. 40 (1951–1954), pp. 26–46. MR 0068280 Zbl 0064.17504

[51] article S.-S. Chern and N. H. Kuiper: “Some theorems on the isometric imbedding of compact Riemann manifolds in Euclidean space,” Ann. Math. (2) 56 : 3 (November 1952), pp. 422–430. MR 0050962 Zbl 0052.17601

This paper will be concerned with some estimates on the lower bound of the dimension of the Euclidean space in which a compact Riemann manifold can be imbedded isometrically, if its curvatures satisfy certain conditions. Our basic geometrical idea is a very simple one. Denote by \( M \) a compact Riemann manifold of dimension \( n \) in a Euclidean space \( E \) of dimension \( n+N \), the Riemann metric on \( M \) being induced by the imbedding. Let \( O \) be a fixed point of \( E \). The distance \( OP \), \( p\in M \), is a continuous function in \( M \) and attains a maximum at a point \( P_0\in M \), since \( M \) is compact. It is intuitively clear that \( M \) will be “concave toward \( O \)” at \( P_0 \), so that there will be some restrictions on the Riemann curvature of \( M \) at \( P_0 \). If \( M \) is given abstractly, the imbedding will not be possible, if these restrictions are not fulfilled by the given Riemann metric at an of the points of \( M \). Actually, however, if the difference \( N \) of the dimensions of \( M \) and \( E \) is greater than one, the implication of this geometrical fact on the Riemann curvature of \( M \) is not very simple. The question leads to algebraic problems which probably do not have simple answers. We propose to give in this paper a few conclusions which can be drawn. It should be mentioned that the above geometrical idea has been used by Tompkins [1939] to prove that a locally flat compact Riemann manifold of dimension \( n \) cannot be isometrically imbedded in an Euclidean space of dimension \( 2n-1 \). Among other things this theorem will be generalized and the invariants entering in the problem geometrically interpreted. As for the differentiability assumptions we suppose our manifold and the imbedding to be of class \( \geq 4 \).

[52] article S.-S. Chern and C. Chevalley: “Élie Cartan and his mathematical work,” Bull. Am. Math. Soc. 58 (1952), pp. 217–250. MR 0046972 Zbl 0046.00304

Élie Cartan	Related
Claude Chevalley	Related

[53] incollection S.-S. Chern: “Differential geometry of fiber bundles,” pp. 397–411 in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 30 August–6 September 1950), vol. 2. American Mathematical Society (Providence, RI), 1952. MR 0045432 Zbl 0049.24201

[54] 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

[55] article S.-S. Chern: “Some formulas in the theory of surfaces,” Bol. Soc. Mat. Mex. 10 : 1–2 (1953), pp. 30–40. MR 0058267 Zbl 0053.29903

[56] incollection S.-S. Chern: “Pseudo-groupes continus infinis” [Infinite continuous pseudogroups], pp. 119–136 in Géométrie différentielle [Differential geometry] (Strasbourg, 26 May–1 June 1953). Edited by C. Ehresmann and A. Lichnerowicz. Colloques Internationaux 52. CNRS (Paris), 1953. MR 0063377 Zbl 0053.01604

Charles Ehresmann	Related
André Lichnerowicz	Related

[57] book S.-S. Chern: Differentiable manifolds. University of Chicago, 1953. Notes for Spring 1952 course Math 354A. See also Differentiable manifolds (1959) and Differentiable manifolds (1960).

[58] article S.-S. Chern: “Book review: ‘Sur les espaces fibrés et les variétés feuilletées’, by W.-T. Wu and G. Reeb,” Bull. Am. Math. Soc. 59 : 3 (1953), pp. 258–263. MR 1565488

Wen-tsün Wu	Related
Georges Reeb	Related

[59] article S.-S. Chern: “Relations between Riemannian and Hermitian geometries,” Duke Math. J. 20 : 4 (1953), pp. 575–587. MR 0060885 Zbl 0052.17604

[60] article S.-S. Chern: “On the characteristic classes of complex sphere bundles and algebraic varieties,” Am. J. Math. 75 : 3 (July 1953), pp. 565–597. MR 0056966 Zbl 0051.14301

[61] article S.-S. Chern, P. Hartman, and A. Wintner: “On isothermic coordinates,” Comment. Math. Helv. 28 (1954), pp. 301–309. Dedicated to Professor H. Hopf on his 60th birthday. MR 0066718 Zbl 0056.40206

Philip Hartman	Related
Aurel Wintner	Related
Heinz Hopf	Related

[62] article S.-S. Chern: “An elementary proof of the existence of isothermal parameters on a surface,” Proc. Am. Math. Soc. 6 (1955), pp. 771–782. MR 0074856 Zbl 0066.15402

[63] article S. S. Chern: “Book review: ‘Introduction to integral geometry’, by L. Santaló,” Bull. Am. Math. Soc. 61 : 4 (1955), pp. 358–359. MR 1565701

[64] article S.-S. Chern: “On special \( W \)-surfaces,” Proc. Am. Math. Soc. 6 : 5 (October 1955), pp. 783–786. MR 0074857 Zbl 0067.13801

[65] article S.-S. Chern: “On curvature and characteristic classes of a Riemann manifold,” Abh. Math. Semin. Univ. Hamb. 20 : 1–2 (1955), pp. 117–126. Dedicated to Wilhelm Blaschke on his seventieth birthday. MR 0075647 Zbl 0066.17003

[66] book S.-S. Chern: Complex manifolds. University of Chicago, 1956. See also Complex manifold (1959). Russian translation appeared as Kompleksnye mnogoobrazija (1961).

[67] article S.-S. Chern: “Scientific report on the Second Summer Institute, part II: Complex manifolds,” Bull. Am. Math. Soc. 62 (1956), pp. 101–117. University of Colorado, 21 June–31 July 1954. Zbl 0074.30301

[68] S. S. Chern, F. Hirzebruch, and J.-P. Serre: “On the index of a fibered manifold,” Proc. Am. Math. Soc. 8 : 3 (1957), pp. 587–596. MR 87943 Zbl 0083.17801 article

F. Hirzebruch	Related	Volume
Jean-Pierre Serre	Related

[69] 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

[70] article S. S. Chern: “Book review: ‘Théorie globale des connexions et des groupes d’holonomie’, by A. Lichnerowicz,” Bull. Am. Math. Soc. 63 : 1 (1957), pp. 57–59. MR 1565836

[71] article S.-S. Chern: “A proof of the uniqueness of Minkowski’s problem for convex surfaces,” Am. J. Math. 79 : 4 (October 1957), pp. 949–950. MR 0096285 Zbl 0086.14603

[72] incollection S.-S. Chern: “On a generalization of Kähler geometry,” pp. 103–121 in Algebraic geometry and topology: A symposium in honor of S. Lefschetz (Princeton, NJ, 8–10 April 1954). Edited by R. H. Fox, D. C. Spencer, and A. W. Tucker. Princeton Mathematical Series 12. Princeton University Press, 1957. MR 0087172 Zbl 0078.14103

Ralph Hartzler Fox	Related
Solomon Lefschetz	Related
Albert William Tucker	Related
Donald Clayton Spencer	Related

[73] article S. Chern: “Book review: ‘The theory of Lie derivatives and its applications’, by K. Yano,” Am. Math. Mon. 65 : 4 (April 1958), pp. 294–295. MR 1529919

[74] incollection S.-S. Chern: “Geometry of submanifolds in a complex projective space,” pp. 87–96 in Symposium internacional de topología algebraica [International symposium on algebraic topology] (Mexico City, August 1956). Universidad Nacional Autónoma de México and UNESCO (Mexico City and Paris), 1958. MR 0100288 Zbl 0178.55501

[75] 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

[76] article S. S. Chern: “Book review: ‘Neue topologische Methoden in der algebraischen Geometrie’, by F. Hirzebruch,” Bull. Am. Math. Soc. 64 : 3 (1958), pp. 119–123. MR 1565916

[77] book S.-S. Chern: Complex manifolds. Textos de Matemática 5. Universidade do Recife, 1959. See also Complex manifolds (1956). MR 0111055 Zbl 0088.37801

[78] book S.-S. Chern: Differentiable manifolds. Textos de Matemática 4. Universidade do Recife, 1959. See also Differentiable manifolds (1953) and Differentiable manifolds (1960). MR 0133741 Zbl 0099.37402

[79] article S.-S. Chern: “Integral formulas for hypersurfaces in Euclidean space and their applications to uniqueness theorems,” J. Math. Mech. 8 : 6 (1959), pp. 947–955. MR 0114170 Zbl 0090.12802

[80] book S.-S. Chern: Differentiable manifolds. University of Chicago, 1960. Notes for Winter 1959 course Math 243. See also Differentiable manifolds (1959) and Differentiable manifolds (1953).

[81] booklet S.-S. Chern: Geometrical structures on manifolds (East Lansing, MI, 30 August–2 September, 1960). American Mathematical Society (Providence, RI), 1960.

[82] article S.-S. Chern: “Complex analytic mappings of Riemann surfaces, I,” Am. J. Math. 82 : 2 (April 1960), pp. 323–337. Dedicated to Professor Marston Morse. MR 0115183 Zbl 0103.30104

The geometrical nature of the theory of functions in one complex variable is a well-known fact and has been particularly emphasized by L. Ahlfors [1937]. It is also the most natural viewpoint, beause complex function thoery should be regarded as the first chapter of the theory of complex analytic mappings of complex manifolds and the classical study of value distributions is the study of the “size” of the image of a complex analytic mapping. We give in this paper a treatment, from a purely differential-geometric viewpoint, of complex analytic mappings of a Riemann surface (= one dimensional complex analytic manifold) into a compact Riemann surface. In the case when the first Riemann surface is a compact one with a finite number of points deleted, we derive defect relations which generalize the classical relations of Nevanlinna–Ahlfors. In a subsequent paper we will consider the case when the first Riemann surface is a compact one with a finite number of points and a finite number of disks deleted. The paper is written for differential geometers, so that concepts currently in use in differential geometry are freely used and a minimum of function theory will be required. The explicit models of the Gaussian plane or the unit disk are avoided.

[83] 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.

[84] incollection S.-S. Chern: “Differential geometry and integral geometry,” pp. 441–449 in Proceedings of the International Congress of Mathematicians (Edinburgh, 14–21 August 1958). Edited by J. A. Todd. Cambridge University Press, 1960. A German translation was published in Differentialgeometrie und Integralgeometrie (1972). MR 0120601 Zbl 0119.38003

[85] article S. S. Chern, J. Hano, and C. C. Hsiung: “A uniqueness theorem on closed convex hypersurfaces in Euclidean space,” J. Math. Mech. 9 : 1 (1960), pp. 85–88. MR 0116289 Zbl 0101.13903

Jun-Ichi Hano	Related
Chuan-Chih Hsiung	Related

[86] article S.-S. Chern: “Holomorphic mappings of complex manifolds,” Enseign. Math., II. Sér. 7 (1961), pp. 179–187. MR 0136768 Zbl 0113.06402

[87] book S.-S. Chern: Kompleksnye mnogoobrazija [Complex manifolds]. Publisher of Foreign Literature (Moscow), 1961. In Russian. Translation of Complex manifolds (1956). Zbl 0098.35201

[88] incollection S.-S. Chern: “Sur les métriques riemanniennes compatibles avec une reduction du groupe structural” [On Riemannian metrics compatible with a reduction of the structure group], pp. 1–2 in Séminaire Ehresmann: Topologie et géométrie différentielle, vol. 2: 1958–1960. Secrétariat Mathématique (Paris), 1961. Exposé no. 8. Zbl 0129.36402

[89] article S.-S. Chern: “Geometry of a quadratic differential form,” J. Soc. Ind. Appl. Math. 10 : 4 (1962), pp. 751–755. MR 0154230 Zbl 0106.15102

[90] article S.-S. Chern and C.-c. Hsiung: “On the isometry of compact submanifolds in euclidean space,” Math. Ann. 149 (1963), pp. 278–285. MR 0148011 Zbl 0124.37501

[91] article S.-S. Chern: “Pseudo-Riemannian geometry and the Gauss–Bonnet formula,” Anais Acad. Brasil. Ci. 35 (1963), pp. 17–26. MR 0155261 Zbl 0113.37001

[92]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

[93] article S.-S. Chern: “On the curvatures of a piece of hypersurface in euclidean space,” Abh. Math. Semin. Univ. Hamb. 29 : 1–2 (December 1965), pp. 77–91. Dedicated to Professor Emanuel Sperner on his sixtieth birthday. MR 0188949 Zbl 0147.20901

Let \( z=z(x,y) \) be a \( C^2 \)-surface in euclidean three-space, defined over the disk \( x^2+y^2 < R^2 \) in the \( (x,y) \)-plane. Let \( H \) and \( K \) denote respectively its mean and Gaussian curvatures. In [1955] Heinz proved the folowing theorem:

If \( |H|\geq c > 0 \), then \( R\leq 1/c \). If \( K\geq c > 0 \), then \( R\leq 1/\sqrt{c} \). If \( K\leq -c < 0 \), then \( R\leq e(3/x)^{1/2} \). (\( c= \) const. in all cases).

The purpose of this paper is to extend this theorem to a hypersurface in an euclidean space of dimension \( m+1 \). Having the global problem in mind, we will consider an immersed hypersurface and establish, in so far as possible, the intermediary results in this general setting. In this sense some of our formulations are more general even in the classical case \( m=2 \).

[94] 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

[95] article S. S. Chern: “The geometry of \( G \)-structures,” Bull. Am. Math. Soc. 72 (1966), pp. 167–219. MR 0192436 Zbl 0136.17804

[96] incollection S.-S. Chern: “Geometric structures on manifolds and submanifolds,” pp. 127–135 in Belfer Graduate School of Science annual science conference proceedings: Some recent advances in the basic sciences (Belfer Graduate School of Science, Yeshiva University, New York, 1962–1964), vol. 1. Edited by A. Gelbart. Yeshiva University Press (New York), 1966. MR 0208511

[97] incollection S.-S. Chern: “On the differential geometry of a piece of submanifold in Euclidean space,” pp. 17–21 in Proceedings of the United States–Japan seminar in differential geometry (Kyoto, 14–19 June 1965). Edited by K. Daigaku and S. K. Kenkyūjo. Nippon Hyoronsha (Tokyo), 1966. MR 0216435 Zbl 0151.27902

Sūri Kaiseki Kenkyūjo	Related
Kyōto Daigaku	Related

[98] book Studies in global geometry and analysis. Edited by S.-S. Chern. MAA Studies in Mathematics 4. Mathematical Association of America (Washington, DC), 1967. 2nd edition was no. 27 in this series, titled Global differential geometry (1989). MR 0210537 Zbl 0175.48102

[99] article S.-S. Chern: “Einstein hypersurfaces in a Kählerian manifold of constant holomorphic curvature,” J. Diff. Geom. 1 : 1 (1967), pp. 21–31. MR 0219013 Zbl 0168.19505

In his dissertation Brian Smyth studied the complete hypersurfaces in a complex space-form whose induced metric is einsteinian and proved that these are either totally geodesic or certain hyperquadrics of the complex projective space. We wish to show in this note that the corresponding local theorem is true:

Let \( V \) be a kählerian manifold of dimension \( \geq 3 \) with constant holomorphic sectional curvature \( K \). Let \( f:M\to V \) be a holomorphically immersed hypersurface such that the induced metric is einsteinian. Then, if \( K\leq 0 \), \( M \) is totally geodesic. If \( K > 0 \) and \( V \) is identified with the complex projective space, \( M \) is either totally geodesic or a hypersphere.

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

[101] misc S.-S. Chern: Lectures on integral geometry, 1967. Typescript notes from 1965 Summer Science Seminar, Academia Sinica, Taipei. Seminar sponsored by National Tsinghua University, Academia Sinica and National Taiwan University. Zbl 0142.20801

[102] incollection S.-S. Chern: “Curves and surfaces in Euclidean space,” pp. 16–56 in Studies in global geometry and analysis. Edited by S.-S. Chern. MAA Studies in Mathematics 4. Mathematical Association of America (Washington, DC), 1967. This later appeared in Curves and surfaces in Euclidean space (1989) . MR 0212744

[103] 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

[104] book S. S. Chern: Complex manifolds without potential theory. Van Nostrand Mathematical Studies 15. D. Van Nostrand (Princeton, NJ), 1967. MR 0225346 Zbl 0158.33002

[105] book Entire functions and related parts of analysis (La Jolla, CA, 27 June–22 July 1966), vol. 11. Edited by J. Korevaar, S.-S. Chern, L. Ehrenpreis, W. H. J. Fuchs, and L. A. Rubel. Proceedings of Symposia in Pure Mathematics. American Mathematical Society (Providence, RI), 1968. MR 0232935 Zbl 0177.29306

Leon Ehrenpreis	Related
Jacob Korevaar	Related
Wolfgang Heinrich Johannes Fuchs	Related
Lee Albert Rubel	Related

[106] incollection S.-S. Chern: “On holomorphic mappings of hermitian manifolds of the same dimension,” pp. 157–170 in Entire functions and related parts of analysis (La Jolla, CA, 27 June–22 July 1966). Edited by J. Korevaar, S.-S. Chern, L. Ehrenpreis, W. H. J. Fuchs, and L. A. Rubel. Proceedings of Symposia in Pure Mathematics 11. American Mathematical Society (Providence, RI), 1968. MR 0234397 Zbl 0184.31202

Leon Ehrenpreis	Related
Jacob Korevaar	Related
Wolfgang Heinrich Johannes Fuchs	Related
Lee Albert Rubel	Related

[107] techreport S.-S. Chern: Minimal submanifolds in a Riemannian manifold. Technical report 19, University of Kansas, Lawrence, 1968. MR 0248648

[108] article S.-S. Chern: “Simple proofs of two theorems on minimal surfaces,” Enseign. Math. (2) 15 (1969), pp. 53–61. MR 0246212 Zbl 0175.18603

[109] incollection S. S. Chern, H. I. Levine, and L. Nirenberg: “Intrinsic norms on a complex manifold,” pp. 119–139 in Global analysis: Papers in Honor of K. Kodaira. Edited by D. C. Spencer. University Tokyo Press, 1969. MR 0254877 Zbl 0202.11603

We propose to define in this paper certain norms (or more precisely, semi-norms) on the homology groups of a complex manifold. They will be invariants of the complex strucgture and do not increase under holomorphic mappings. Their definitions depend on the bounded plurisubharmonic functions on the manifold and are modelled after the notion of harmonic length introduced by H. Landau and R. Osserman [1959–1960] for Riemann surfaces (= one-dimensional complex manifolds). It is possible to extend the definition to certain families of chains. In particular we get in this way an intrinsic pseudo-metric on the manifold which is closely related to that of Carathéodory [1927] and to one recently introduced by S. Kobayashi [1967].

Kunihiko Kodaira	Related
Harold I. Levine	Related
Louis Nirenberg	Related
Donald Clayton Spencer	Related

[110]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

[111] incollection S. S. Chern, M. do Carmo, and S. Kobayashi: “Minimal submanifolds of a sphere with second fundamental form of constant length,” pp. 59–75 in Functional analysis and related fields: Proceedings of a conference in honor of Professor M. Stone (University of Chicago, May 1968). Edited by F. E. Browder. Springer (New York), 1970. MR 0273546 Zbl 0216.44001

Let \( M \) be an \( n \)-dimensional manifold which is minimally immersed in a unit sphere \( S^{n+p} \) of dimension \( n+p \). Let \( h \) be the second fundamental form of this immersion; it is a certain symmetric bilinear mapping \[ T_x\times T_x\to T_x^{\perp} \] for \( x\in M \), where \( T_x \) is the tangent space of \( M \) at \( x \) and \( T_x^{\perp} \) is the normal space to \( M \) at \( x \). We denote by \( S \) the square of the length of \( h \). It is known that if \( M \) is moreover compact, then \[ \int_M\Bigl(\Bigl(2-\frac{1}{p}\Bigr)S - n\Bigr)S\cdot *1\geq 0, \] where \( *1 \) denotes the volume element of \( M \) (Simons [1968]; a slightly more general formula will be proved in §§2 and 3). It follows that if \( S\leq n/(2-1/p) \) everywhere on \( M \), then either

\( S=0 \) (i.e., \( M \) is totally geodesic), or
\( S = n/(2-1/p) \).

The purpose of the present paper is to determine all minimal submanifolds \( M \) of \( S^{n+p} \) satisfying \( S = n/(2-1/p) \). The proof depends upon the results presented by the first named author in his lectures on minimal submanifolds in Berkeley in the Winter of 1968, in which an exposition of the work of Simons [1968] was made by the use of moving frames. To describe our result, we begin with examples of minimal submanifolds.

Marshall Harvey Stone	Related	Volume
Manfredo P. do Carmo	Related
Shōshichi Kobayashi	Related	Volume
Felix E. Browder	Related

[112]M. Atiyah and S. S. Chern: “Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections,” Matematika, Moskva 14 : 2 (1970), pp. 117–154. Russian translation of an article in Acta Math. 114:1 (1965). Zbl 0208.35101 article

[113] article S. S. Chern: “Book review: ‘Differential geometry’, by J. J. Stoker,” Am. Math. Mon. 77 : 3 (March 1970), pp. 317. MR 1535844

[114] incollection S.-S. Chern: “Holomorphic curves and minimal surfaces,” pp. 1–28 in Proceedings of the Carolina conference on holomorphic mappings and minimal surfaces (Chapel Hill, NC, 30 March 30–3 April 1970). Edited by D. A. Eisenman and L. E. Taylor. University of North Carolina (Chapel Hill, NC), 1970. MR 0273522 Zbl 0222.53005

Laird Edwin Taylor	Related
Donald Alfred Eisenman	Related

[115] 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

[116] book Global analysis (Berkeley, CA, 1–26 July 1968). Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 14. American Mathematical Society (Providence, RI), 1970. MR 0263081 Zbl 0204.07601

[117] book Global analysis. Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 16. American Mathematical Society (Providence, RI), 1970. Zbl 0204.07603

[118] book Global analysis. Edited by S.-S. Chern and S. Smale. Proceedings of Symposia in Pure Mathematics 15. American Mathematical Society (Providence, RI), 1970. Zbl 0204.07602

[119] incollection S.-S. Chern: “On minimal spheres in the four-sphere,” pp. 137–150 in Studies and essays. National Taiwan University (Taipei), 1970. Presented to Yu-why Chen on his 60th birthday, April 1, 1970. MR 0278205 Zbl 0212.26402

[120] incollection S.-S. Chern: “Differential geometry: Its past and its future,” pp. 41–53 in Actes du Congrès International des Mathématiciens 1970 (Nice, France, 1–10 September 1970), vol. 1. Gauthier-Villars (Paris), 1971. MR 0428217 Zbl 0232.53001

[121] incollection S. S. Chern: “Brief survey of minimal submanifolds,” pp. 43–60 in Differentialgeometrie im Grossen [Differential geometry in the large] (Oberwolfach, Germany, 13–19 July 1969). Edited by W. Klingenberg. Berichte, Mathematisches Forschungsinstitut Oberwolfach 4. Bibliographisches Institut (Mannheim), 1971. MR 0358634 Zbl 0218.53070

[122] S.-S. Chern and J. Simons: “Some cohomology classes in principal fiber bundles and their application to Riemannian geometry,” Proc. Natl. Acad. Sci. U.S.A. 68 : 4 (April 1971), pp. 791–794. MR 0279732 Zbl 0209.25401 article

[123] incollection S.-S. Chern: “Differentialgeometrie und Integralgeometrie” [Differential geometry and integral geometry], pp. 306–317 in Geometrie. Edited by K. Strubecker. Wege der Forschung 177. Wissenschaftliche Buchgesellschaft (Darmstadt), 1972. In German. Translation from Proceedings of the International Congress of Mathematicians (1960), pp. 441–449.

[124] incollection S.-S. Chern: “Holomorphic curves in the plane,” pp. 73–94 in Differential geometry: In honor of Kentaro Yano. Edited by S. Kobayashi, M. Obata, and T. Takahashi. Kinokuniya (Tokyo), 1972. MR 0422693 Zbl 0249.32016

Kentaro Yano	Related
Morio Obata	Related
Shōshichi Kobayashi	Related	Volume
Tsunerō Takahashi	Related

[125] incollection S.-S. Chern: “Geometry of characteristic classes,” pp. 1–40 in Proceedings of the thirteenth biennial seminar of the Canadian Mathematical Congress (Dalhousie University, Halifax, NS, 16 August–3 September 1971), vol. 1. Edited by J. R. Vanstone. Canadian Mathematical Congress (Montreal), 1972. A Russian translation was published in Uspehi Mat. Nauk 28:3(171) (1973). MR 0370613 Zbl 0269.57013

[126] book Collection of articles dedicated to S. S. Chern and D. C. Spencer on their sixtieth birthdays, published as J. Diff. Geom. 6 : 4. Lehigh University (Bethlehem, PA), 1972. MR 0317858

[127] article S.-S. Chern: “The mathematical works of Wilhelm Blaschke,” Abh. Math. Sem. Univ. Hamburg 39 : 1 (1973), pp. 1–9. See also Gesammelte Werke, vol. 5 (1985). MR 0331273 Zbl 0264.01021

[128] article S. S. Čern: “Geometry of characteristic classes,” Uspehi Mat. Nauk 28 : 3(171) (1973), pp. 121–160. In Russian. Translation from Proceedings of the thirteenth biennial seminar of the Canadian Mathematical Congress, vol. 1 (1972), pp. 1–40. MR 0385881 Zbl 0292.57016

[129] article S. S. Chern: “Meromorphic vector fields and characteristic numbers,” Scripta Math. 29 : 3–4 (1973), pp. 243–251. Special issue dedicated to the memory of Abraham Adrian Albert. MR 0412486 Zbl 0265.32013

[130] incollection S. S. Chern and M. J. Cowen: “Frenet frames along holomorphic curves,” pp. 191–203 in Value distribution theory (Tulane University, New Orleans, 1972–1973), part A. Edited by R. O. Kujala and A. L. Vitter, III. Monographs and textbooks in pure and applied mathematics 25. Dekker (New York), 1974. MR 0361170 Zbl 0287.53044

Albert Leopold Vitter, III	Related
Robert O. Kujala	Related
Michael Jeffrey Cowen	Related

[131] 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

[132] 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

[133] book Differential geometry (Stanford, CA, 30 July–17 August 1973), part 2. Edited by S. S. Chern and R. Osserman. Proceedings of Symposia in Pure Mathematics XXVII. American Mathematical Society (Providence, RI), 1975. MR 0370637 Zbl 0304.00009

[134] book Differential geometry (Stanford, CA, 30 July–17 August 1973), part 1. Edited by S. S. Chern and R. Osserman. Proceedings of Symposia in Pure Mathematics XXVII. American Mathematical Society (Providence, RI), 1975. MR 0370636

[135] 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

[136] article S. S. Chern: “On the projective structure of a real hypersurface in \( C_{n+1} \),” Math. Scand. 36 (1975), pp. 74–82. Special volume dedicated to Werner Fenchel on his 70th birthday. MR 0379910 Zbl 0305.53019

[137] article S. S. Chern and J. White: “Duality properties of characteristic forms,” Invent. Math. 35 : 1 (1976), pp. 285–297. Dedicated to Jean-Pierre Serre. MR 0431193 Zbl 0343.53015

This paper proves various statements and propositions concerning fiber bundles with a Lie group, connections, curvature forms, and characteristic classes, and in particular concerning the recent work of Chern–Simons on secondary characteristic classes [1974] and of Cheeger–Simons on differential characters [1973]. Our main interest in this work is not so much in the characteristic classes but in the differential forms which represent them. Indeed, the aim of this paper is to study the duality properties of the characteristic forms, both the primary ones and the secondary ones. We begin by reviewing some basic notions. For general references, the reader is referred to [Chern 1972; Kobayashi and Nomizu 1963, 1969; Koszul 1973; Milnor and Stasheff 1974].

Jean-Pierre Serre	Related
James White	Related

[138] article S. S. Chern: “Book review: ‘Integral geometry and geometric probability’, by L. Santaló,” Bull. Am. Math. Soc. 83 : 6 (1977), pp. 1289–1290. MR 1567008

[139] incollection S.-S. Chern: “Circle bundles,” pp. 114–131 in Geometry and topology: Proceedings of the third Latin American school of mathematics (Rio de Janeiro, July 1976). Edited by J. Palis and M. P. Do Carmo. Lecture Notes in Mathematics 597. Springer (Berlin), 1977. MR 0461353 Zbl 0356.55005

Manfredo P. do Carmo	Related
Jacob Palis, Jr.	Related

[140] article S.-S. Chern and P. A. Griffiths: “An inequality for the rank of a web and webs of maximum rank,” Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 5 : 3 (1978), pp. 539–557. Dedicated to Hans Lewy. MR 507000 Zbl 0402.57001

Hans Lewy	Related
Phillip A. Griffiths	Related

[141] article S. S. Chern and P. A. Griffiths: “Abel’s theorem and webs,” Jahresber. Deutsch. Math.-Ver. 80 : 1–2 (1978), pp. 13–110. MR 494957 Zbl 0386.14002

[142] 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

[143] incollection S.-S. Chern and P. A. Griffiths: “Linearization of webs of codimension one and maximum rank,” pp. 85–91 in Proceedings of the international symposium on algebraic geometry (Kyoto, 10–14 January 1977). Edited by M. Nagata. Kinokuniya Book Store (Tokyo), 1978. MR 578854 Zbl 0406.14003

Masayoshi Nagata	Related
Phillip A. Griffiths	Related

[144] incollection S.-S. Chern: “On projective connections and projective relativity,” pp. 225–232 in The Ta-You Wu Festschrift: Science of matter. Edited by S. Fujita. Gordon & Breach (New York), 1978. MR 547033

Ta-You Wu	Related
Shigeji Fujita	Related

[145] incollection S.-S. Chern: “Affine minimal hypersurfaces,” pp. 17–30 in Minimal submanifolds and geodesics: Proceedings of the Japan–United States seminar (Tokyo, 1977). Edited by M. Obata. North-Holland (Amsterdam), 1979. MR 574250 Zbl 0439.53008

[146] book S. S. Chern: Complex manifolds without potential theory, 2nd edition. Universitext. Springer (New York), 1979. With an appendix on the geometry of characteristic classes. A revised printing of this edition appeared in 1995. MR 533884 Zbl 0444.32004

[147] book G. Herglotz: Gesammelte Schriften [Collected works]. Edited by H. Schwerdtfeger. Vandenhoeck & Ruprecht (Göttingen), 1979. With a foreword by the editor and introductory articles by Peter Bergmann, S. S. Chern, Ronald B. Guenther, Claus Müller, Theodor Schneider and H. Wittich. MR 526569 Zbl 0402.01010

Peter Gabriel Bergmann	Related
Ronald Bernard Guenther	Related
Theodor Schneider	Related
Hans Wilhelm Eduard Schwerdtfeger	Related
H. Wittich	Related
Gustav Herglotz	Related
Claus Müller	Related

[148] article S. S. Chern: “From triangles to manifolds,” Am. Math. Mon. 86 : 5 (May 1979), pp. 339–349. MR 528789 Zbl 0425.53002

[149] incollection S.-S. Chern: “Geometrie,” pp. xx–xxi in G. Herglotz: Gesammelte Schriften [Collected writings]. Edited by H. Schwerdtfeger. Vandenhoeck und Ruprecht (Göttingen), 1979. In German.

Gustav Herglotz	Related
Hans Wilhelm Eduard Schwerdtfeger	Related

[150] article S.-S. Chern and C. K. Peng: “Lie groups and KdV equations,” Manuscr. Math. 28 : 1–3 (1979), pp. 207–217. MR 535702 Zbl 0408.35074

In recent years there have been extensive studies of evolution equations with soliton solutions, among which the most important ones are the Korteweg-deVries and sine-Gordon equations. We will show that the algebraic basis of these mathematical phenomena lies in Lie groups and their structure equations; their explicit solutions with special properties often give the evolution equations. The process is thus similar to the introdution of a “potential”. In fact, from \( \textrm{SL}(2;\mathbf{R}) \), the special linear group of all \( (2\times 2) \)-real unimodular matrices, one is led naturally to the KdV and MKdV (= modified Korteweg-deVries) equations of higher order. A Miura transformation exists between them. Following H. H. Chen [1976], this leads to the Bäcklund transformations of the KdV.

[151] article R. Wells: “Chern retires from Berkeley, almost,” Math. Intelligencer 2 : 2 (1979–1980), pp. 78–81. Includes an after-dinner speech given by Chern at the international symposium held in his honor, U.C. Berkeley, 25–29 June 1979. MR 577556

[152] article W. M. Boothby, S. S. Chern, and S. P. Wang: “The mathematical work of H. C. Wang,” Bull. Inst. Math. Acad. Sinica 8 : 2–3, part 1 (1980), pp. xiii–xxiv. MR 595525 Zbl 0446.01026

S. P. Wang	Related
William Munger Boothby	Related
Hsien-chung Wang	Related

[153] incollection A. Beauville: “Géométrie des tissus,” pp. 103–119 in Séminaire Bourbaki, 1978/79. Lecture Notes in Mathematics 770. Springer (Berlin), 1980. After S. S. Chern and P. A. Griffiths. Exposé no. 531. In French. MR 572420 Zbl 0436.57008

Phillip A. Griffiths	Related
Arnaud Beauville	Related

[154] incollection S.-S. Chern: “General relativity and differential geometry,” pp. 271–287 in Some strangeness in the proportion: A centennial symposium to celebrate the achievements of Albert Einstein, 2nd edition. Edited by H. Woolf. Addison-Wesley (Reading, MA), 1980.

Harry Woolf	Related
Albert Einstein	Related

[155] incollection S. S. Chern: “Geometry and physics,” pp. 1–5 in Proceedings of the mathematical seminar (Singapore, 1980). Edited by K. N. Cheng and C. T. Chong. National University of Singapore (Singapore), 1980. MR 622216

C. T. Chong	Related
K. N. Cheng	Related

[156] article S.-S. Chern and C. L. Terng: “An analogue of Bäcklund’s theorem in affine geometry,” Rocky Mountain J. Math. 10 : 1 (1980), pp. 105–124. MR 573866 Zbl 0407.53002

It is well known that there is a correspondence between solutions of the Sine-Gordon equation (SGE) \[ \frac{\partial^2\phi}{\partial x^2} - \frac{\partial^2\phi}{\partial t^2} = \sin\phi \] and the surfaces of constant curvature \( -1 \) in \( \mathbf{R}^3 \) (see below). The classical Bäcklund transformation of such surfaces furnishes a way to generate new solutions of the SGE from a given solution. This has received much attention in recent studies of the soliton solutions of the SGE, and the technique has been used successfully in the study of other non-linear evolution equations. In the first section of this paper we present a simple derivation of the classical Bäcklund theorem and its applications by using the method of moving frames.

Our main result concerns affine minimal surfaces. They arise as the solution of the variation problem for affine area. The corresponding Euler–Lagrange equation is a fourth order partial differential equation. In §2, we develop the basic properties of affine minimal surfaces. In §3 we study the transformation of affine surfaces by realizing them as the focal surfaces of a line congruence. The natural conditions that the congruence be a \( W \)-congruence and that the affine normals at corresponding points be parallel lead to the conclusion that both surfaces are affine minimal. This is the content of Theorem 4, the main result of our paper. As in the classical case, the Theorem leads to the construction of new affine minimal surfaces from a given one by the solution of a completely integrable system of first order partial differential equations.

[157] book B. Su: Lectures on differential geometry. World Scientific (Singapore), 1980. With a foreword by S.-S. Chern. MR 664600 Zbl 0517.53006

[158] article S.-S. Chern: “Geometry and physics,” Math. Medley 8 : 1 (1980), pp. 1–6. MR 604751 Zbl 0455.53052

[159] article S. S. Chern: “Geometrical interpretation of the \( \sinh \)-Gordon equation,” Ann. Polon. Math. 39 (1981), pp. 63–69. MR 617450 Zbl 0497.53056

[160] article S. S. Chern and C. K. Peng: “On the Bäcklund transformations of KdV equations and modified KdV equations,” J. China Univ. Sci. Tech. 11 : 3 (1981), pp. 1–6. MR 701775

[161] incollection S. S. Chern and R. Osserman: “Remarks on the Riemannian metric of a minimal submanifold,” pp. 49–90 in Geometry symposium (Utrecht, 27–29 August 1980). Edited by E. Looijenga, D. Siersma, and F. Takens. Lecture Notes in Mathematics 894. Springer (Berlin), 1981. MR 655419 Zbl 0477.53056

Robert Osserman	Related
Dirk Siersma	Related
Eduard J. N. Looijenga	Related
Floris Takens	Related

[162] incollection S.-S. Chern and J. G. Wolfson: “A simple proof of Frobenius theorem,” pp. 67–69 in Manifolds and Lie groups: Papers in honor of Yozô Matsushima. Edited by J.-I. Hano. Progress in Mathematics 14. Birkhäuser (Boston, MA), 1981. MR 642852 Zbl 0481.58007

Jun-Ichi Hano	Related
Jon Gordon Wolfson	Related
Yozô Matsushima	Related

[163] article S. S. Chern and P. Griffiths: “Corrections and addenda: ‘Abel’s theorem and webs’,” Jahresber. Deutsch. Math.-Verein. 83 : 2 (1981), pp. 78–83. MR 612412 Zbl 0474.14003

[164] article S. S. Chern and K. Tenenblat: “Foliations on a surface of constant curvature and the modified Korteweg–de Vries equations,” J. Differential Geom. 16 : 3 (1981), pp. 347–349. Dedicated to Professor Buchin Su on his 80th birthday. MR 654630 Zbl 0483.53019

Keti Tenenblat	Related
Buchin Su	Related

[165] article S. S. Chern: “Web geometry,” Bull. Am. Math. Soc. (N.S.) 6 : 1 (1982), pp. 1–8. A Chinese translation was published in Adv. Math. 14:4 (1985). See also The mathematical heritage of Henri Poincaré, part 1 (1983), pp. 3–10. MR 634430 Zbl 0483.53013

[166] article S. S. Chern: “Projective geometry, contact transformations, and CR-structures,” Arch. Math. (Basel) 38 : 1 (1982), pp. 1–5. MR 646314 Zbl 0456.53023

In a recent paper [1980] M. Kuranishi gave a new proof of a theorem of Kashiwara–Kawai–Sato and Boutet de Monvel that any non-degenerate \( CR \)-system on a conic neighborhood of its characteristic is conjugate under a canonical transformation to the \( CR \)-sstem derived from teh Heisenberg structure. The theorem is clearly related to the classical theorem of Sophus Lie that a second-order differential equation \[ z^{\prime\prime} = F(x,z,z^{\prime}) \] in the \( (x,z) \)-plane has no local invariant under contact transformations. In this paper we shall give a generalization of Lie’s theorem to high dimensions and apply it to \( CR \) stuctures. Various notions of equivalence of \( CR \)-structures will be discussed.

[167] book Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (Beijing, 18 August–21 September 1980), vol. 2. Edited by S. S. Chern and W. T. Wu. Science Press (Beijing), 1982.

[168] book W. Blaschke: Gesammelte Werke [Collected works], Band 1. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1982. With commentary by K. Strubecker and with introductory material by Walter Benz and Emanuel Sperner. MR 741181 Zbl 0541.01012

Hans Robert Müller	Related
Walter Benz	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Emanuel Sperner	Related
Karl Strubecker	Related
Udo Simon	Related

[169] book Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (Beijing, 18 August–21 September 1980). Edited by S. S. Chern and W. T. Wu. Science Press (Beijing), 1982. In 3 volumes. MR 714332 Zbl 0509.00015

[170] book K. Yano: Selected papers of Kentaro Yano. Edited by M. Obata. North-Holland Mathematics Studies 70. North-Holland (Amsterdam), 1982. With an introduction by Shiing-Shen Chern. MR 676813 Zbl 0492.01016

Kentaro Yano	Related
Morio Obata	Related

[171] incollection R. L. Bryant, S.-S. Chern, and P. A. Griffiths: “Exterior differential systems,” pp. 219–338 in Proceedings of the 1980 Beijing Symposium on differential geometry and differential equations (Beijing, 18 August–21 September 1980), vol. 1. Edited by S. S. Chern and W. T. Wu. Science Press (Beijing), 1982. MR 714337 Zbl 0516.58003

Wen-tsün Wu	Related
Robert Leamon Bryant	Related
Phillip A. Griffiths	Related

[172] book Proceedings of the 1980 Beijing symposium on differential geometry and differential equations (Beijing, 18 August–21 September 1980), vol. 3. Edited by S. S. Chern and W. T. Wu. Science Press (Beijing), 1982.

[173] article S.-S. Chern and J. K. Moser: “Erratum: ‘Real hypersurfaces in complex manifolds’,” Acta Math. 150 : 1 (July 1983), pp. 297. MR 0709144

[174] article S.-S. Chern and J. K. Moser: “Real hypersurfaces in complex manifolds,” Usp. Mat. Nauk 38 : 2(230) (1983), pp. 149–193. In Russian. Translation from Acta Math. 133:1 (1974), pp. 219–271. Zbl 0519.32014

[175] book S.-S. Chern and W.-h. Chen: Differential geometry notes. Beijing University Press, 1983. In Chinese. English translation published as Lectures on differential geometry (1999).

[176] article S. S. Chern and J. K. Moser: “Erratum: ‘Real hypersurfaces in complex manifolds’,” Acta Math. 150 : 1 (1983), pp. 297.

[177] article S. S. Chern and J. G. Wolfson: “Minimal surfaces by moving frames,” Am. J. Math. 105 : 1 (February 1983), pp. 59–83. MR 692106 Zbl 0521.53050

In this paper we will develop, by moving frames, the basic formulas and some results on minimal surfaces, both in a Riemannian and in a Kählerian manifold. When the ambient Riemannian manifold is of constant curvature, we will show that a quartic form defined in a natural way from the second fundamental form is holomorphic, relative to the underlying complex structure of the surface. This will be applied to the study of minimal surfaces on the unit 4-sphere \( S^4(1) \). We will show how to derive by this approach a theorem of R. Bryant that a compact Riemann surface can be conformally immersed as a minimal surface in \( S^4(1) \).

When the ambient space is a Kählerian manifold, the first-order jet of the surface defines a scalar invariant, which is the Kähler form divided by the element of area of the surface. This will be used to normalize the frames attached to the surface. When the surface is minimal, an invariant cubic form appears naturally. We will show that it is holomorphic, if the ambient Kählerian manifold is of constant holomorphic sectional curvature. This will be applied to the complex projective plane \( P_2(\mathbf{C}) \) (with the Fubini–Study metric), to give a family of minimal surfaces in \( P_2(\mathbf{C}) \), and all its minimal two-spheres. A determination of all the minimal 2-spheres in \( P_n(\mathbf{C}) \) method has been carried out by the second author [1982]. This latter problem was first studied by Din–Zakrewski [1980] and a mathematician’s treatment was given by Eels–Wood [1983].

[178] book H. Hopf: Differential geometry in the large: Seminar lectures (New York University, 1946, and Stanford University, 1956). Lecture Notes in Mathematics 1000. Springer (Berlin), 1983. Notes taken by Peter Lax and John Gray. With a preface by S. S. Chern. A 2nd edition was published in 1989. MR 707850 Zbl 0526.53002

John Walker Gray	Related
Peter D. Lax	Related
Heinz Hopf	Related

[179] article “1983 Steele Prizes awarded in Albany,” Notices Am. Math. Soc. 30 : 6 (1983), pp. 574–579. MR 718773

Lars Valerian Ahlfors	Related
Stephen Cole Kleene	Related

[180] incollection S. S. Chern: “On surfaces of constant mean curvature in a three-dimensional space of constant curvature,” pp. 104–108 in Geometric dynamics (Rio de Janeiro, July–August 1981). Edited by J. Palis, Jr. Lecture Notes in Mathematics 1007. Springer (Berlin), 1983. MR 730266 Zbl 0521.53006

[181] techreport S.-S. Chern: Deformation of surfaces preserving principal curvatures. Technical report 010-83, Mathematical Sciences Research Institute, February 1983. Later published in Differential geometry and complex analysis (1985).

[182] incollection S.-S. Chern: “Web geometry,” pp. 3–10 in The mathematical heritage of Henri Poincaré (Bloomington, IN 1980), part 1. Edited by F. E. Browder. Proceedings of Symposia in Pure Mathematics 39. American Mathematical Society (Providence, RI), 1983. See also Bull. Am. Math. Soc. 6:1 (1982). Zbl 0519.53010

Felix E. Browder	Related
Jules Henri Poincaré	Related

[183] book É. Cartan: Œuvres complètes. Partie III [Complete works. Part III], 2nd edition, vol. 2: Géométrie différentielle. Divers. Centre National de la Recherche Scientifique (Paris), 1984. In French. With biographical material by Shiing-Shen Chern, Claude Chevalley and J. H. C. Whitehead. MR 753093

Claude Chevalley	Related
Élie Cartan	Related
J. H. C. Whitehead	Related

[184] book G. de Rham: Differentiable manifolds: Forms, currents, harmonic forms. Grundlehren der Mathematischen Wissenschaften 266. Springer (Berlin), 1984. With an introduction by S. S. Chern. MR 760450 Zbl 0534.58003

[185] book Seminar on nonlinear partial differential equations (Berkeley, CA, 9 May 1983). Edited by S.-S. Chern. MSRI Publications 2. Springer (New York), 1984. MR 765224 Zbl 0542.00009

[186] article S. S. Chern and J. Wolfson: “Harmonic maps of \( S^2 \) into a complex Grassmann manifold,” Proc. Nat. Acad. Sci. U.S.A. 82 : 8 (April 1985), pp. 2217–2219. Part II was published in Ann. Math. 125:2 (1987). MR 788858 Zbl 0601.58023

Let \( G(k,n) \) be the Grassmann manifold of all \( C_k \) in \( C_n \), the complex spaces of dimensions \( k \) and \( n \), respectively, or, what is the same, the manifold of all projective spaces \( P_{k-1} \) in \( P_{n-1} \), so that \( G(1,n) \) is the complex projective space \( P_{n-1} \) itself. We study harmonic maps of the two-dimensional sphere \( S^2 \) into \( G(k,n) \). The case \( k=1 \) has been the subject of investigation by several authors [see, for example, Din, A. M. & Zakrzewski, W. J. (1980) Nucl. Phys. B 174, 397–406; Eells, J. & Wood, J. C. (1983) Adv. Math. 49, 217–263; and Wolfson, J. G. Trans. Am. Math. Soc., in press]. The harmonic maps \( S^2\to G(2,4) \) have been studied by Ramanathan [Ramanathan, J. (1984) J. Differ. Geom. 19, 207–219]. We shall describe all harmonic maps \( S^2\to G(2,n) \). The method is based on several geometrical constructions, which lead from a given harmonic map to new harmonic maps, in which the image projective spaces are related by “fundamental collineations.” The key result is the degeneracy of some fundamental collineations, which is a global consequence, following from the fact that the domain manifold is \( S^2 \). The method extends to \( G(k,n) \).

[187] 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

[188] book W. Blaschke: Gesammelte Werke, Band 2: Kinematik und Integralgeometrie. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1985. With commentaries by Hans Robert Müller and Luis A. Santaló and with introductory material by Erwin Kruppa. MR 909486 Zbl 0646.53002

Hans Robert Müller	Related
Erwin Kruppa	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Karl Strubecker	Related
Udo Simon	Related

[189] incollection S.-S. Chern: “Wilhelm Blaschke and web geometry,” pp. 25–27 in W. Blaschke: Gesammelte Werke [Collected works], vol. 5: Geometrie der Gewebe. Edited by W. Burau. Thales-Verlag (Essen), 1985.

Werner Burau	Related
Wilhelm J. E. Blaschke	Related

[190] incollection S.-S. Chern: “Deformation of surfaces preserving principal curvatures,” pp. 155–163 in Differential geometry and complex analysis: A volume dedicated to Harry Ernest Rauch. Edited by I. Chavel and H. M. Farkas. Springer (Berlin), 1985. Originally published as MSRI technical report 010-83 (1983). MR 780041 Zbl 0566.53002

Isaac Chavel	Related
Hershel Mark Farkas	Related
Harry Ernest Rauch	Related

[191] incollection S.-S. Chern: “The mathematical works of Wilhelm Blaschke: An update,” pp. 21–23 in W. Blaschke: Gesammelte Werke [Collected works], vol. 5: Geometrie der Gewebe. Edited by W. Burau. Thales-Verlag (Essen), 1985. See also Abh. Math. Sem. Univ. Hamburg 39:1 (1973).

Werner Burau	Related
Wilhelm J. E. Blaschke	Related

[192] book S. Sasaki: Selected papers. Edited by S.-i. Tachibana. Kinokuniya (Tokyo), 1985. With a foreword by Shiing-Shen Chern. MR 834384 Zbl 0578.53001

Shun-ichi Tachibana	Related
Shigeo Sasaki	Related

[193] book W. Blaschke: Gesammelte Werke [Collected works], Band 5: Geometrie der Gewebe. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1985. With commentaries in English by S. S. Chern and with introductory material by Otto Haupt, Werner Burau, Enrico Bompiani, Hans Reichardt and I. M. Yaglom. MR 909489 Zbl 0656.53003

Hans Robert Müller	Related
Hans Reichardt	Related
Enrico Bompiani	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Otto Haupt	Related
Karl Strubecker	Related
Isaak Moiseivich Yaglom	Related
Udo Simon	Related

[194] incollection S. S. Chern and R. S. Hamilton: “On Riemannian metrics adapted to three-dimensional contact manifolds,” pp. 279–308 in Arbeitstagung Bonn 1984 [Workshop Bonn 1984] (Max-Planck-Institut für Mathematik, Bonn, 15–24 June 1984). Edited by F. Hirzebruch, J. Schwermer, and S. Suter. Lecture Notes in Mathematics 1111. Springer (Berlin), 1985. With an appendix by Alan Weinstein. MR 797427 Zbl 0561.53039

F. Hirzebruch	Related	Volume
Silke Suter	Related
Richard Streit Hamilton	Related
Joachim Schwermer	Related
Alan Weinstein	Related

[195] article S. S. Chern: “Web geometry,” Adv. Math. (Beijing) 14 : 4 (1985), pp. 301–307. In Chinese. Translation from Bull. Am. Math. Soc. 6:1 (1982). MR 848829 Zbl 0593.53005

[196] book W. Blaschke: Gesammelte Werke [Collected works], Band 3: Konvexgeometrie. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1985. With commentary by Kurt Leichtweiß and with introductory material by Hans Reichardt. MR 909487 Zbl 0655.52001

Hans Robert Müller	Related
Hans Reichardt	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Karl Strubecker	Related
Udo Simon	Related

[197] book W. Blaschke: Gesammelte Werke [Collected works], Band 4: Affine Differentialgeometrie. Differentialgeometrie der Kreis- und Kugelgruppen. Edited by W. Burau, S.-S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1985. With commentaries by Werner Burau and Udo Simon. MR 909488 Zbl 0656.53002

Hans Robert Müller	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Karl Strubecker	Related
Udo Simon	Related

[198] article S. S. Chern and K. Tenenblat: “Pseudospherical surfaces and evolution equations,” Stud. Appl. Math. 74 : 1 (1986), pp. 55–83. MR 827492 Zbl 0605.35080

[199] incollection S. S. Chern: “On a conformal invariant of three-dimensional manifolds,” pp. 245–252 in Aspects of mathematics and its applications: Collection of papers written in honor of Leopoldo Nachbin. Edited by J. A. Barroso. North-Holland Mathematics Library 34. North-Holland (Amsterdam), 1986. MR 849555 Zbl 0589.53011

Jorge Alberto Barroso	Related
Leopoldo Nachbin	Related

[200] incollection S.-S. Chern and P. A. Griffiths: “Pfaffian systems in involution,” pp. 233–256 in Proceedings of the 1982 Changchun symposium on differential geometry and differential equations (Changchun, China, 23 August–16 September 1982). Edited by S.-S. Chern, R.-H. Wang, and M.-Y. Chi. Science Press (Beijing), 1986. Zbl 0677.58005

Min-you Chi	Related
Phillip A. Griffiths	Related
Rou-Huai Wang	Related

[201] book W. Blaschke: Gesammelte Werke [Collected works], Band 6: Selecta. Edited by W. Burau, S. S. Chern, K. Leichtweiß, H. R. Müller, L. A. Santaló, U. Simon, and K. Strubecker. Thales-Verlag (Essen), 1986. With commentaries by Walter Benz, Karl Strubecker, Kurt Leichtweiß, Werner Burau, Helmut Grunsky and Dietrich Kölzow. MR 909490 Zbl 0656.53004

Hans Robert Müller	Related
Walter Benz	Related
Werner Burau	Related
Wilhelm J. E. Blaschke	Related
Helmut Grunsky	Related
Kurt Leichtweiß	Related
Luis Antoni Santaló	Related
Karl Strubecker	Related
Udo Simon	Related
Dietrich Kölzow	Related

[202] book Proceedings of the 1983 Beijing symposium on differential geometry and differential equations (Beijing, 29 August–10 September 1983). Edited by S. T. Liao and S.-S. Chern. Science Press (Beijing), 1986. MR 880868 Zbl 0632.00008

[203] book Proceedings of the 1982 Changchun symposium on differential geometry and differential equations (Changchun, China, 23 August–16 September 1982). Edited by S.-S. Chern, R.-H. Wang, and M.-y. Chi. Science Press (Beijing), 1986. Zbl 0646.00007

Min-you Chi	Related
Rou-Huai Wang	Related

[204] article S.-S. Chern and J. G. Wolfson: “Harmonic maps of the two-sphere into a complex Grassmann manifold, II,” Ann. Math. (2) 125 : 2 (March 1987), pp. 301–335. Part I was published in Proc. Nat. Acad. Sci. U.S.A. 82:8 (1985). MR 881271 Zbl 0627.58017

[205] 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

[206] article S. S. Chern: “Book review: ‘Prescribing the curvature of a Riemannian manifold’, by J. L. Kazdan,” Bull. Am. Math. Soc. (N.S.) 16 : 1 (1987), pp. 143–144. MR 1567600

[207] incollection S. S. Chern: “American differential geometry: Some personal notes,” pp. 375–377 in A century of mathematics in America, part I. Edited by P. Duren, R. A. Askey, and U. C. Merzbach. History of Mathematics 1. American Mathematical Society (Providence, RI), 1988. MR 1003183 Zbl 0695.01012

Peter L. Duren	Related
Uta Caecilia Merzbach	Related
Richard Allen Askey	Related	Volume

[208] book Partial differential equations (Tianjin, China, 23 June–5 July 1986). Edited by S.-S. Chern. Lecture Notes in Mathematics 1306. Springer (Berlin), 1988. Proceedings of the 7th symposium on differential geometry and differential equations. MR 1032766 Zbl 0631.00004

[209] misc R. M. Schoen: New developments in the theory of geometric partial differential equations, 1988. 60 minute videocassette, American Mathematical Society ICM Series. A plenary address presented at the International Congress of Mathematicians, Berkeley, CA, August 1986, introduced by Shiing-Shen Chern. MR 1059458

[210] book H. Hopf: Differential geometry in the large: Seminar lectures (New York University, 1946, and Stanford University, 1956), 2nd edition. Lecture Notes in Mathematics 1000. Springer (Berlin), 1989. Notes taken by Peter Lax and John W. Gray, with prefaces by S. S. Chern and K. Voss. MR 1013786 Zbl 0669.53001

Konrad Voss	Related
John Walker Gray	Related
Peter D. Lax	Related
Heinz Hopf	Related

[211] incollection S.-S. Chern: “Vector bundles with a connection,” pp. 1–26 in Global differential geometry. Edited by S.-S. Chern. MAA Studies in Mathematics 27. Mathematical Association of America (Washington, DC), 1989. MR 1013808 Zbl 0683.53026

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

[213] incollection T. E. Cecil and S. S. Chern: “Dupin submanifolds in Lie sphere geometry,” pp. 1–48 in Differential geometry and topology (Tianjin, China, 1986–1987). Edited by B. Jiang, C.-K. Peng, and Z. Hou. Lecture Notes in Mathematics 1369. Springer (Berlin), 1989. Proceedings of the special year at Nankai Institute of Mathematics. MR 1001178 Zbl 0678.53003

Chia-Kuei Peng	Related
Thomas Edward Cecil	Related
Zixin Hou	Related
Boju Jiang	Related

[214] incollection S.-S. Chern: “Curves and surfaces in Euclidean space,” pp. 99–139 in Global differential geometry. Edited by S.-S. Chern. MAA Studies in Mathematics 27. Mathematical Association of America (Washington, DC), 1989. This earlier appeared in Studies in global geometry and analysis (1967). Zbl 0683.53002

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

[216] article “Shiing Shen Chern,” C. R. Acad. Sci. Paris, Sér. Gén., Vie Sci. 6 : 6 (1989), pp. 488–489. In French. MR 1063960

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

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

[219] book Global differential geometry, 2nd edition. Edited by S. S. Chern. MAA Studies in Mathematics 27. Mathematical Association of America (Washington, DC), 1989. Original was published as no. 4 in this series, as Studies in global geometry and analysis (1967). MR 1013807 Zbl 0683.53001

[220] article S.-S. Chern: “What is geometry?,” Am. Math. Mon. 97 : 8 (1990), pp. 679–686. MR 1072811 Zbl 0717.51002

[221] incollection S.-S. Chern: “Historical remarks on Gauss–Bonnet,” pp. 209–217 in Analysis, et cetera: Research papers in honor of J. Moser’s 60th birthday. Edited by P. H. Rabinowitz and E. Zehnder. Academic Press (Boston, MA), 1990. MR 1039345 Zbl 0701.53039

Paul H. Rabinowitz	Related
Eduard J. Zehnder	Related
Jürgen K. Moser	Related

[222] book Physics and mathematics of anyons: Proceedings of the TCSUH workshop (Houston, TX, 1–2 February 1991), published as Int. J. Mod. Phys. B 5 : 10. Issue edited by S. S. Chern, C. W. Chu, and C. S. Ting. 1991. MR 1118209

Paul C. W. Chu	Related
C. S. Ting	Related

[223] incollection S. S. Chern: “Characteristic classes and characteristic forms,” pp. 169–177 in Paul Halmos: Celebrating 50 years of mathematics. Edited by J. H. Ewing and F. W. Gehring. Springer (New York), 1991. MR 1113271 Zbl 0853.57025

Paul Richard Halmos	Related
Frederick W. Gehring	Related
John H. Ewing	Related

[224] article S. S. Chern: “Transgression in associated bundles,” Int. J. Math. 2 : 4 (1991), pp. 383–393. In memory of Guang-Lei Wu. MR 1113567 Zbl 0768.53014

[225] incollection S. S. Chern: “An introduction to Dupin submanifolds,” pp. 95–102 in Differential geometry: A symposium in honour of Manfredo do Carmo (Rio de Janeiro, 1988). Edited by H. B. L. Lawson and K. Tenenblat. Pitman Monographs and Surveys in Pure and Applied Mathematics 52. Longman Scientific & Technical (Harlow, UK), 1991. MR 1173035 Zbl 0721.53052

Manfredo P. do Carmo	Related
H. Blaine Lawson, Jr.	Related
Keti Tenenblat	Related

[226] incollection S.-S. Chern: “Surface theory with Darboux and Bianchi,” pp. 59–69 in Miscellanea mathematica. Edited by M. F. Atiyah, P. J. Hilton, F. Hirzebruch, and R. Remmert. Springer (Berlin), 1991. Festschrift for Heinz Götze. MR 1131117 Zbl 0734.53008

F. Hirzebruch	Related	Volume
M. F. Atiyah	Related	Volume
Reinhold Remmert	Related
Heinz Götze	Related
Peter John Hilton	Related

[227] article X. S. Chen: “How to make China a mathematically advanced country,” Adv. Math. (China) 20 : 2 (1991), pp. 129–134. In Chinese. MR 1121551

[228] article J.-P. Bourguignon: “Entretien avec un optimiste, S. S. Chern” [Interview with an optimist, S. S. Chern], Gaz. Math. 48 (1991), pp. 5–10. An English translation was published in S. S. Chern: A great geometer of the twentieth century (1992). MR 1116889

[229] book R. L. Bryant, S. S. Chern, R. B. Gardner, H. L. Goldschmidt, and P. A. Griffiths: Exterior differential systems. MSRI Publications 18. Springer (New York), 1991. MR 1083148 Zbl 0726.58002

Hubert Leopold Goldschmidt	Related
Robert Leamon Bryant	Related
Robert Brown Gardner	Related
Phillip A. Griffiths	Related

[230] incollection S. S. Chern: “Families of hypersurfaces under contact transformations in \( \mathbb{R}^n \),” pp. 49–56 in International symposium on number theory and analysis in memory of Hua Loo Keng (Beijing, 1–7 August 1988), vol. II: Analysis. Edited by S. Gong, Q.-k. Lu, Y. Wang, and L. Yang. Series in pure and applied mathematics 21. Springer (Berlin), 1991. MR 1135825 Zbl 0819.53009

Hua Loo Keng	Related
Sheng Gong	Related
Yuaan Wang	Related
Qi-keng Lu	Related
Lo Yang	Related

[231]R. Bott: “For the Chern volume,” pp. 106–108 in S. S. Chern: A great geometer of the twentieth century (Los Angeles, 1990). Edited by S.-T. Yau. International Press (Cambridge, MA), 1992. MR 1201350 incollection

Raoul H. Bott	Related	Volume
Shing-Tung Yau	Related

[232] I. M. Singer: “S. S. Chern at Chicago,” pp. 88–90 in Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. International Press (Hong Kong), 1992. MR 1201344 incollection

Isadore M. Singer	Related	Volume
Shing-Tung Yau	Related

[233] incollection I. Kaplansky: “Shiing Shen Chern with admiration as he approaches his 80th birthday,” pp. 95–97 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201346

Irving Kaplansky	Related	Volume
Shing-Tung Yau	Related

[234] incollection S.-S. Chern: “My mathematical education,” pp. 1–17 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. Polish translation was published in Rocz. Pol. Tow. Mat., Ser. II, Wiadom. Mat. 42 (2006). Also appeared in Chin. Ann. Math., Ser. B 27:1 (2006). MR 1201339

[235] article S. S. Chern: “On Finsler geometry,” C. R. Acad. Sci., Paris, Sér. 314 : 10 (1992), pp. 757–761. MR 1163872 Zbl 0755.53044

[236] incollection R. Lashof: “Personal recollection of Chern at Chicago,” pp. 104–105 in S. S. Chern: A great geometer of the twentieth century. Edited by S.-T. Yau. Monographs in geometry and topology. International Press (Hong Kong), 1992. MR 1201349

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