A Mathematician and His Mathematical Work

The following sections are included:

• Foreword

• Contents

The following sections are included:

• Introduction

• Early life

• 1926-30, Nankai University

• 1930-34, Qing Hua graduate school

• 1934-36, Hamburg University

• 1936-37, Paris

• 1937-43, Kunming and The Southwest University Consortium

• Chern's family

• 1943-45, Institute for Advanced Study at Princeton

• 1946-48, Academia Sinica

• 1949-60, Chicago University

• Chern and C.N. Yang

• 1960-79, UC Berkeley

• 1981-present. The Three Institutes

• Honors and awards

• An overview of Chern's research

• Geometric structures and their equivalence problems

• Integral geometry

• Euclidean differential geometry

• The Generalized Gauss-Bonnet Theorem

• Characteristic classes

• "Retirement"

• References

The following sections are included:

• Early Education in China

• European Student Life

• Mathematical Isolation

• Princeton Sunshine

• Mathematics at the Midway

• Settelment on the West Coast

• Something to Play With in My Eighties

The following sections are included:

• Projective Differential Geometry

• Euclidean Differential Geometry

• Geometrical Structures and Their Intrinsic Connections

• Integral Geometry

• Characteristic Classes

• Holomorphic Mappings

• Minimal Submanifolds

• Webs

• Exterior Differential Systems and Partial Differential Equations

The friends and colleagues of S. S. Chern who have planned this volume in his honor have asked me for a contribution. Such an invitation is also an honor and could not easily be declined. At the same time, while I have no doubt that future historians of differential geometry will rank Chern as the worthy successor of Elie Cartan in that field, I do not feel competent to give an assessment of his work, nor called upon to do so, since the best part of it, or at least a very representative selection from it, is reproduced in this volume and speaks for itself. All I can do on this occasion is to evoke memories from a friendship of long standing—a friendship which has been among the most valuable ones, personally and scientifically, that I have been privileged to experience …

It is a special honor to have been asked to offer some comments and observations on Chern's mathematical work. Because of his own mathematical summaries and Weil's discussion of Chern's life and contributions provided from the perspective of a contemporary, perhaps this would be a good opportunity to explore how a few of Chern's papers and his view of mathematics appear to me, coming as I did a generation following the publication of his earliest work …

I met Shiing-Shen Chern for the first time in Hamburg sometime in the fall of 1934. We were at that time both students at the Hamburg University; Chern was at that time studying with Blaschke, while I was attending the lectures of Artin. I purposely use the rather vague expression "attending the lectures of Artin" in order to indicate the tentative nature of my sojourn in Hamburg at that time; more correctly, I should say that I was at that time strictly speaking a student at the University of Leipzig, but I decided to stay in Hamburg for personal reasons, as I shall explain later, and I took the opportunity to learn something by attending the lectures of Artin. In order to explain this rather strange lack of definiteness in my mathematical aspirations and also in order to understand the importance of Chern's influence at a critical juncture later in my life, let me say a few words about my rather unusual educational background …

The following sections are included:

• Der allgemeine Fall

• Die Ausnahmefälle

• Beispiele für Gewebe höchsten Ranges

The following sections are included:

• Some Fundamental Notions in Klein Spaces

• Definition of the Incidence of Elements of Different Fields

• The Generalized Crofton's Formula and Cauchy's Formula

The following sections are included:

• Introduction

• Résumé of some fundamental formulas in Riemannian Geometry

• The space of unit vectors and a formula for Ω

• Proof of the Gauss-Bonnet formula

• REFERENCES

The following sections are included:

• Introduction

• Definition of the Intrinsic Differential Form in M^2n-1

• Remarks on the Formula of Allendoerfer-Weil

• A New Integral Formula

• Fields of Normal Vectors

• REFERENCES

The following sections are included:

• INTRODUCTION

• COMPLEX SPHERE BUNDLES AND THEIR IMBEDDING

• STUDY OF THE COCYCLES ON A COMPLEX GRASSMANN MANIFOLD

• THE BASIC CHARACTERISTIC CLASSES ON A COMPLEX MANIFOLD

• HERMITIAN MANIFOLDS

• APPLICATIONS TO ELLIPTIC HERMITIAN GEOMETRY

• BIBLIOGRAPHY

On donnE dans cette Note unc propriété caractéristique d'une classe de variétés dans l'espace projectif à n dimensions, étudiées auparavant par Cartan. Ces variétés fournissent une généralisation des surfaces ayant un réseau. Il existe une transformation entre ces variétés généralisant la transformation bien connue de Laplace.

The following sections are included:

• Introduction

• Review of some known results on sphere bundles

• Plücker coordinates in G

• The Theorem

• Remarks

The following sections are included:

• Introduction

• Measures in spaces associated with a Riemann manifold

• Differential geometry of a submanifold in Euclidean space

• A formula on densities

• Total curvature and Euler characteristic

• Proof of the kinematic formula

• REFERENCES

The following sections are included:

• Introduction

• G-connection in a tangent bundle; torsion tensor

• Proof of the decomposition theorem

• The case of the unitary group

• Riemann manifolds with a field of parallel oriented linear spaces

• REFERENCES

The following sections are included:

• Introduction

• Moving frames

• The total curvature

• Proof of Theorem 1

• Proof of Theorem 2

• Proof of Theorem 3

• A further theorem

• REFERENCES

The following sections are included:

• Introduction

• Algebraic properties of the index of a martix

• Poincaré rings

• Proof of the theorem

The following sections are included:

• TOTAL CURVATURE AND THE SUM OF BETTI NUMBERS

• IMMERSIONS WITH MINIMAL TOTAL CURVATURE

• HYPERSURFACES WITH NONNEGATIVE GAUSS-KRONECKER CURVATURE

• REFERENCES

The following sections are included:

• Introduction

• Integral formulas

• The uniqueness theorem of Alexandroff-Fenchel-Jessen

• Characterization of the hypersphere

• REFERENCES

The object of this note is to prove the following theorem:

Let Σ, Σ′ be two closed, strictly convex, C²-differentiable hypersurfaces in a Euclidean space of dimension n + 1 (≧ 3). Let f: Σ → Σ′ be a diffeomorphism such that Σ and Σ′ have parallel outward normals at p ε Σ and p′ = f(p) respectively. Denote by P_l(p) (respectively P_l(p′)) the l^th elementary symmetric function of the principal radii of curvature of Σ (resp. Σ′) at p (resp. p′). If, for a fixed l, 2 ≦ l ≦ n, we have

for all points p Υ Σ, then f is a translation …

The following sections are included:

• Introduction

• The hypersphere in the real and complex euclidean space

• An integral formula

• The first main theorem

• The integrated form of the first main theorem

• REFERENCES

The following sections are included:

• Introduction

• Curvature and characteristic classes

• Hermitian vector-bundles

• The duality formula

• Remarks on Positivity. The proof of Theorem I completed

• The relative Gauss Bonnet theorem

• The second integration; definition of the order function

• Equidistribution in measure

• The proof of Proposition 8.6

• Some remarks on integral geometry. The proof of Proposition 8.7

• The Nevanlinna Theorem

• References

The following sections are included:

• Introduction

• Preliminaries

• A differential formula

• Higher sectional curvatures and mixed curvatures of a pair of riemannian manifolds

• A Meusnier-Euler type theorem

• Proof of the main formula

• Determination of the constants

• Integration over linear space; reproductivity of the invariants

• REFERENCES

The following sections are included:

• Introduction

• A lemma on bounded plurisubharmonic functions in C_n

• Semi-norms and their properties

• Comparision of semi-norms

• Some examples

• Intrinsic pseudo-metrics

• Remarks on the differential equations

• REFERENCES

The following sections are included:

• Introduction

• Multilinear Algebra Over a Complex Vector Space

• Affine Connection

• Hermitian Structure

• The Recursive Formulas

• Proof of the Theorem

• Bibliography

The following sections are included:

• Introduction

• Review of connection theory*

• The forms TP(θ)

• Conformal invariance

• Conformal immersions

• Applications to 3-manifolds

• REFERENCES

The following sections are included:

• Introduction

• The real hyperquadrics

• Construction of a normal form

• Existence theorems

• Solution of an equivalence problem

• The connection

• Actual computation for real hypersurfaces

• Appendix. Bianchi Identities

• References

It is well-known that there is a correspondence between solutions of the Sine-Gordon equation (SGE)

and the surfaces of constant curvature -1 in R3 (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.

The following sections are included:

• Geometry

• Triangles

• Curves in the plane; rotation index and regular homotopy

• Euclidean three-space

• From coordinate spaces to manifolds

• Manifolds; local tools

• Homology

• Vector fields and generalizations

• Elliptic differential equations

• Euler characteristic as a source of global invariants

• Gauge field theory

• Concluding remarks

• References

The following sections are included:

• Introduction

• KdV equations

• MKdV equations

• Miura transformation

• References

The following sections are included:

• Introduction and Statement of Results

• Formulation of Problem

• Connection From Associated to a Coframe

• Surfaces of Non-Constant Mean Curvature

• References

The following sections are included:

• Introduction

• Lie Sphere Geometry

• Legendre Submanifolds

• Cyclides of Dupin

• Dupin submanifolds for n = 4

• REFERENCES

The following sections are included:

• Review of Surface Theory

• Contributions of Gauss, Bonnet, etc.

• Finsler Surfaces and Landsberg Surfaces

• REFERENCES

SHIING-SHEN CHERN. Born October 26, 1911 in Kashing, Chekiang Province, China, Dr. Chern was educated at Nankai University, Tsinghua University, and the University of Hamburg, where he received his doctorate in 1936. He has been Professor at UC Berkeley since 1960, Emeritus since 1979, and is now Director Emeritus of MSRI in Berkeley as well as Director of the Nankai Institute of Mathematics in Tianjin, China. He is a member of the National Academy of Sciences and of numerous other Academies, domestic and foreign. He has received the National Medal of Science of the USA, and many other prizes, distinctions and honors.

The following sections are included:

• Dupin submanifolds and Lie sphere geometry

• Taut imbedding

• A generalization of Lie sphere geometry and an equivalence problem

• References

The following sections are included:

• Introduction

• Formulation as a problem of equivalance

• A notable pseudo-group of contact transformations

• An invariant for ∞ⁿ⁺¹ hypersurfaces

• REFERENCES

The following sections are included:

• ISOMETRY

• APPLICABLE SURFACES

• W-SURFACES

• W-CONGRUENCES

• TRANSFORMATION OF SURFACES

• REFERENCES

The following sections are included:

• Introduction and Review

• Differentiable Bundles

• Holomorphic Bundles

• References

The following sections are included:

• Stiefel–Whitney and Pontrjagin Classes

• Characteristic Classes in Terms of Curvature

• Transgression

• Holomorphic Line Bundles and the Nevanlinna Theory

• REFERENCES

A new connection is introduced in a Finsler space, solving at the same time the problem of equivalence. In terms of this connection the curvature is divided into two parts, to be called the Minkowski and the Riemannian curvature respectively. A formula for the second variation of the arc length is derived, which involves only the Riemannian curvature.

The following sections are included:

• SIXTY-PLUS YEARS OF FRIENDSHIP

• YANG-MILLS EQUATIONS

• YANG-BAXTER EQUATIONS*

• A POEM

• REFERENCES

This paper explores the merits of a special connection in Finsler geometry which solves the equivalence problem. This connection is torsionfree and 'almost' compatible with the induced metric. We develop the calculus of geodesics and Jacobi fields. We also show that the expression for the second variation of arc length, when expressed in terms of this connection and its curvature tensor, is formally identical to that in Riemannian geometry. This fact then effects a straightforward generalization of the concept of sectional curvature, as well as a number of comparison theorems, to Finsler geometry.

The following sections are included:

• Riemannian geometry and Erlanger program

• Connections after Elie Cartan

• Global differential geometry

• BIBLIOGRAPHY

This is the 150th anniversary of Sophus Lic (1812-99). I think he was a great mathematician even without Lic groups. In the following I shall discuss tow of his contributions to differential geometry which have developed and promised to have a future.

The following sections are included:

• Introduction

• Segre family

• Local criterion for non-degeneracy

• Hachtroudi connection

• A new proof of Lemma 1

• Proof of Theorem 2

• Proof of Theorem 1

• References

The following sections are included:

• Shiing-Shen Chern

• List of Ph.D. Theses Written Under the Supervision of S.S. Chern

• Bibliography of the Publications of S.S. Chern