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哲人已遠，典型猶存
幾何學一代宗師陳省身
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作 者 丘成桐
作者簡介 丘成桐為哈佛大學數學與物理教授，費爾茲獎、克拉福得獎、沃爾夫獎、馬賽爾‧ 格羅斯曼獎得主，中央研究院院士。科普著作有《丘成桐談空間的內在形狀》，並為《數理人文》主編。
譯 者 夏木清
譯者簡介 夏木清是香港專業數學科普譯者。
本文出處 哈佛大學數學科學及應用中心新近開辦了《數理科學文獻》講座，旨在介紹現代數學各分枝之主要發展，使聽眾對數學有一大域的認識。作者給出了這系列講座的第一講，本文即此演講整理而成。
延伸閱讀
參考資料
[1] Carl B. Allendoerfer and Andr? Weil, The Gauss-Bonnet theorem for
Riemannian polyhedra, Transactions of the American Mathematical Society
53 (1943), 101–129. Note.
[2] Raoul Bott and Shiing-Shen Chern, Hermitian vector bundles and the
equidistribution of the zeroes of their holomorphic sections, Acta
Mathematica 114 (1965), 71–112.
[3] ,
Some formulas related to complex transgression, Essays on Topology and
Related Topics (M?moires d?di?s ? Georges de Rham), 1970, pp. 48–57.
[4] Élie Cartan, Les groupes d’holonomie des espaces généralisés, Acta Math. 48 (1926), no. 1-2, 1–42.
[5] , Sur les invariants integraux de certains espaces homogenes clos et les proprietes topologiques de ces espaces, Imprimerie de l’Univesite, 1929.
[6] , Leçons sur la géométrie projective complexe, Cahiers scientifiques, Gauthier-Villars, 1931.
[7] , Sur les variétés
à connexion affine et la théorie de la relativité
généralisée (première partie), Annales Scientifiques de l’École Normale
Supérieure. Troisième Série 40 (1923), 325–412.
[8] , Sur les variétés
à connexion affine, et la théorie de la relativité
généralisée (première partie)
(Suite),
Annales Scientifiques de l’École Normale
Supérieure. Troisième Série 41 (1924), 1–25.
[9] , Sur les nombres de betti des espaces de groupes clos, C. R. Acad.Sci. Paris (1928), 196–198.
[10] , L’extension du calcul tensoriel aux géométries non-affines, Annals of Mathematics. Second Series 38 (1937), no. 1, 1–13.
[11] Élie Cartan and J.A. Schouten, On riemannian geometries admitting an absolute parallelism.
[12] Shiing-Shen Chern, On integral geometry in Klein spaces, Annals of Mathematics. Second Series 43 (1942), 178–189.
[13] , On the curvatura integra in a Riemannian manifold, Annals of Mathematics. Second Series 46 (1945), 674–684.
[14] , Characteristic classes of Hermitian manifolds,
Annals of Mathematics. Second Series 47 (1946), 85–121. In the
introduction of this paper, Chern said: “The present paper will be
devoted to a study of the fiber bundle of the complex tangent vectors
of complex manifolds and their characteristic classes in the sense of
Pontrjagin. ...Roughly speaking, the difficulty in the real case lies
in the existence of finite homotopy group of certain real manifolds,
namely the manifolds formed by the ordered sets of linearly independent
vectors of a finite dimensional vector space. ...We are therefore led
to the study of the cocyles or cycles on a complex Grassmannian
manifold, a problem treated exhaustively by Ehresmann.”
[15] , On the multiplication in the characteristic ring of a sphere bundle, Annals of Mathematics. Second Series 49 (1948), 362–372.
[16] , Topics in differential geometry.
Lectures given in Institute for Advanced Study in Veblen seminar in
1949 spring semester and continue in university of Chicago finished in
April, 1951.
[17] , Differential geometry of fiber bundles, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, 1952, pp. 397–411.
[18] , On the characteristic classes of complex sphere bundles and algebraic varieties,
American Journal of Mathematics 75 (1953), 565–597. In the preface of
this paper, he said “In a recent paper, Hodge studied the question of
identifying, for non singular algebraic varieties over the complex
field, the characteristic classes of complex manifolds with the
canonical systems introduced by M. Eger and J.A. Todd. He proved that
they are identical up to a sign, when the variety is the complete
intersections of nonsingular hyper surfaces in a projective space. His
method does not seem to extend to a general algebraic variety. ...We
shall give in this paper a more direct treatment of the problem, by
proving that there is an equivalent definition of the characteristic
classes, which is valid for algebraic varieties.”
[19] , Review: A. Lichnerowicz, Théorie globale des connexions
et des groupes d’holonomie, Bulletin of the American Mathematical Society 63 (1957), no. 1, 57–59.
[20] , Holomorphic mappings of complex manifolds, L’Enseignement Math?matique. Revue Internationale. 2e S?rie 7 (1961), 179–187 (1962).
[21] , On holomorphic mappings of hermitian manifolds of the same dimension, Entire Functions and Related Parts of Analysis (Proc. Sympos. Pure Math., La Jolla, Calif., 1966), 1968, pp. 157– 170.
[22] , Selected papers,
Springer-Verlag, New York-Heidelberg, 1978. With a foreword by H. Wu
and introductory articles by Andr? Weil and Phillip A. Griffiths. In
the preface, Weil said: “I was able to tell Chern about the ‘canonical
classes’ in algebraic geometry, as introduced in the work of Todd and
Eger. Their resemblance with the Stiefel–Whitney classes was apparent,
while they were free from the defect (if it was one) of being defined
modulo 2.”; their status, however, was somewhat uncertain, since that
work had been done in the spirit of Italian geometry and still rested
on some unproved assumptions.
[23] , Affine minimal hypersurfaces, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977), 1979, pp. 17–30.
[24] , General relativity and differential geometry,
Some strangeness in the proportion: a centennial symposium to celebrate
the achievements of Albert Einstein, 1980. In this paper, Chern said:
“a definitive treatment of affine connections, together with a
generalization to connections with torsion, was given by Élie Cartan in his fundamental paper ’ Sur les variétés
à connexion affine, et la théorie
de la relativité généralisée’,
published in 1923-24. The paper did not receive the attention it
deserved for the simple reason it was ahead of its time. For it is more
than a theory of affine connections, its idea can be easily generalized
to give a theory of connections in a fiber bundle with any Lie group,
for whose treatment the Ricci calculus is no longer adequate.
...Specifically the following contributions could be singled out: 1.
the introduction of the equations of structure and the interpretation
of the so called Bianchi identities as the result of exterior
differentiation of the equations of structure. 2. the recognition of
curvature as a tensor valued exterior quadratic differential form.”
[25] , Deformation of surfaces preserving principal curvatures, Differential geometry and complex analysis, 1985, pp. 155–163.
[26] , Selected papers. Vol. II,
Springer-Verlag, New York, 1989. Chern said: “The general case of a
principle bundle with an arbitrary Lie group as structure group, of
which my work above is a special case concerning the unitary group, was
carried out by Weil in 1949 in an unpublished manuscript. Part of
Weil’s result was presented in the Bourbaki seminar. The main
conclusion is the so called Weil homomorphism which identifies the
characteristic classes (through the curvature forms) with the invariant
polynomials under the action of the adjoint group.”
[27] , Selected papers. Vol. III, Springer-Verlag, New York, 1989.
[28] , Selected papers. Vol. IV, Springer-Verlag, New York, 1989.
[29] Shiing-Shen Chern and Claude Chevalley, Obituary: Elie Cartan and his mathematical work,
Bulletin of the American Mathematical Society 58 (1952), 217–250. In
the preface of this paper, the authors mentioned “a generalized space
(space generalise) in the sense of Cartan is a space of tangent spaces
such that two infinitely near tangent spaces are related by an
infinitesimal transformation of a given Lie group. Such a structure is
known as a connection. The tangent space may not be the spaces of
tangent vectors. This generality, which is absolutely necessary, gave
rise to misinterpretation among differential geometers. It is now
possible to express these concepts in a more satisfactory way, by
making use of the modern notion of fiber bundles.” On page 243 of this
paper, the authors said “without the notion and terminology of fiber
bundles, it was difficult to explain these concepts in a satisfactory
way. the situation was further complicated by the fact that Cartan
called tangent space which is now known as fiber bundle while the base
space X, being a differentiable manifold, has a tangent space from its
differentiable structure,But he saw clearly that the geometrical
situation demands the introduction of fiber bundles with rather general
fibers.” The above reference [5] of Cartan showed that Cartan knew the
theory of connection, which is called non-abelian gauge theory, back in
1926.
[30] Shiing-Shen Chern, Harold I. Levine, and Louis Nirenberg, Intrinsic norms on a complex manifold, Global Analysis (Papers in Honor of K. Kodaira), 1969, pp. 119–139.
[31] Shiing-Shen Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Mathematica 133 (1974), 219–271.
[32] Shiing-Shen Chern and James Simons, Characteristic forms and geometric invariants, Annals of Mathematics. Second Series 99 (1974), 48–69.
[33] Shiing-Shen Chern and E. Spanier, The homology structure of sphere bundles,
Proceedings of the National Academy of Sciences of the United States of
America 36 (1950), 248–255. In this paper, Chern and Spanier proved the
Thom isomorphism which is fundamental in differential topology of fiber
bundles. Thom’s paper appeared in C.R. Paris in the same year.
[34] Shiing-Shen Chern and Jon Wolfson, Harmonic maps of S2 into a complex Grassmann manifold, Proceedings of the National Academy of Sciences of the United States of America 82 (1985), no. 8, 2217–2219.
[35] Georges de Rham, Sur l’analysis situs
des variétés à n dimensions, Ph.D.Thesis, 1931.
[36] Max Eger, Les systèmes canoniques d’une variété algébrique à plusieurs dimensions, Annales scientifiques de l’École Normale
Supérieure 3e série,, 60 (1943), 143–172 (fr).
[37] Charles Ehresmann, Sur la topologie de certains espaces homogènes, Annals of Mathematics. Second Series 35 (1934), no. 2, 396–443.
[38] , Sur la topologie des groupes simple clos, C. R. Acad. Sci. Paris 208 (1939), 1263–1265.
[39] , Various notes on fiber spaces, C. R. Acad. Sci. Paris 213, 214, 216 (1941, 1942, 1943), 762–764, 144–147, 628–630.
[40] , Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie (espaces fibrés), Bruxelles, 1950, 1951, pp. 29–55.
[41] W. V. D. Hodge, The characteristic classes on algebraic varieties, Proceedings of the London Mathematical Society. Third Series 1 (1951), 138–151.
[42] W. V. D. Hodge and D. Pedoe, Methods of Algebraic Geometry. Vol. I, Cambridge, at the University Press; New York, The Macmillan Company, 1947.
[43] , Methods of algebraic geometry. Vol. II. Book III: General theory of algebraic varieties in projective space. Book IV: Quadrics and Grassmann varieties, Cambridge, at the University Press, 1952.
[44] , Methods of algebraic geometry. Vol. III. Book V: Birational geometry, Cambridge, at the University Press, 1954.
[45] Erich Kähler, Über eine bemerkenswerte Hermitesche Metrik, Abhandlungen aus dem Mathema- tischen Seminar der Universität Hamburg 9 (1933), no. 1, 173–186.
[46] Felix Klein, Vergleichende betrachtungen über neuere geometrische forschungen, Verlag von Andreas Deichert, Erlangen, 1872.
[47] Solomon Lefschetz, Topology, 1930.
[48] André Lichnerowicz, Théorie globale des connexions
et des groupes d’holonomie, Edizioni Cre- monese, Roma, 1957.
[49] L. Pontrjagin, Characteristic cycles on manifolds, C. R. (Doklady) Acad. Sci. URSS (N.S.) 35 (1942), 34–37.
[50] , On some topologic invariants of Riemannian manifolds, C. R. (Doklady) Acad. Sci. URSS (N. S.) 43 (1944), 91–94.
[51] E. Stiefel, Richtungsfelder und Fernparallelismus in n-dimensionalen Mannigfaltigkeiten, Com- mentarii Mathematici Helvetici 8 (1935), no. 1, 305–353.
[52] J. A. Todd, The Geometrical Invariants of Algebraic Loci, Proceedings of the London Mathematical Society. Second Series 43 (1937), no. 2, 127–138.
[53] , The geometrical invariants of algebraic loci. (Second paper.), Proceedings of the London Mathematical Society. Third Series 45 (1939), 410–424.
[54] André Weil, Geometrie differentielle des espaces fibres, unpublished (1949). Appears in Vol. 1, pp. 422–436, of his Collected Papers.
[55] Hermann Weyl, Die idee der riemannschen fläche, Die idee der riemannschen fl?che, B. G. Teubner, 1913.
[56] , On the Volume of Tubes, American Journal of Mathematics 61 (1939), no. 2, 461–472.
[57] Hassler Whitney, On the topology of differentiable manifolds, Lectures in Topology, 1941, pp. 101–141.