Shiing-Shen Chern

Characteristic Classes of Hermitian Manifolds

In recent years the works of Stiefel,1 Whitney,2 Pontrjagin,3 Steenrod,4 Feldbau,5 Ehresmann,6 etc. have added considerably to our knowledge of the topology of manifolds with a differentiable structure, by introducing the notion of so-called fibre bundles. The topological invariants thus introduced on a manifold, called the characteristic cohomology classes, are to a certain extent susceptible of characterization, at least in the case of Riemannian manifolds,7 by means of the local geometry. Of these characterizations the generalized Gauss-Bonnet formula of Allendoerfer-Weil8 is probably the most notable example. In the works quoted above, special emphasis has been laid on the sphere bundles, because they are the fibre bundles which arise naturally from manifolds with a differentiable structure. Of equal importance are the manifolds with a complex analytic structure which play an important role in the theory of analytic functions of several complex variables and in algebraic geometry. The present paper will be devoted to a study of the fibre bundles of the complex tangent vectors of complex manifolds and their characteristic classes in the sense of Pontrjagin. It will be shown that there are certain basic classes from which all the other characteristic classes can be obtained by operations of the cohomology ring. These basic classes are then identified with the classes obtained by generalizing Stiefel-Whitneys classes to complex vectors. In the sense of de Rham the cohomology classes can be expressed by exact exterior differential forms which are everywhere regular on the (real) manifold. It is then shown that, in case the manifold carries an Hermitian metric, these differential forms can be constructed from the metric in a simple way. This means that the characteristic classes are completely determined by the local structure of the Ilermitian metric. This result also includes the formula of Allendoerfer-Weil and can be regarded as a generalization of that formula. Concerning the relations between the characteristic classes of a complex manifold and an Hermitian metric defined on it, the problem is completely solved by the above results. It is to be remarked that corresponding questions for Rie-

What is geometry

Geometry is the visual study of shapes, sizes, patterns, and positions. It occurred in all cultures, through at least one of these five strands of human activities: 1. building/structures (building/repairing a house, laying out a garden, making a kite, ...) 2. machines/motion (using a pry-bar, riding a bike, sawing a board, swinging, ...) 3. navigating/star-gazing (How do I get from here to there?, using maps, ...) 4. art/patterns (designs, symmetries, representations, ...). 5. measurement (How big is it?, How far is it?, ...)

On the Riemann mapping theorem

We prove a generalization of the Riemann mapping theorem: if a bounded simply connected domain Q with connected smooth boundary has the spherical boundary, then it is biholomorphic to the unit ball.

On A Conformal Invariant of Three-Dimensional Manifolds

Publisher Summary This chapter focuses on a conformal invariant of 3-D manifolds. Φ ( M ) is a conformal invariant, that is, it remains unchanged under a conformal transformation of the metric. Φ (M) has a critical value at M if and only if M is locally conformally flat. The chapter gives direct proofs of the theorems, both because the 3-D case has special features and because the invariant Φ ( M ) has come up in various other connections. The chapter uses arbitrary frame fields to develop the Riemannian geometry on M . The chapter provides an equation that helps to compute Φ (M) through an arbitrary frame field. To the integral of the last term the Stokes theorem is applied to reduce it to an integral involving only the υ ij and not their derivatives.

Linear Differential Systems

The goal of this chapter is to develop the formalism of linear Pfaffian differential systems in a form that will facilitate the computation of examples.

On Riemannian manifolds of four dimensions

Introduction. It is well known that in three-dimensional elliptic or spherical geometry the so-called Cliffords 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. I t 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 [2] . 2 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-Poincare characteristic.

Contemporary trends in algebraic geometry and algebraic topology

Mathematics in the 20th Century (M Atiyah) The [PHI]4 of Minimal Gorenstein 3-Folds of General Type (M Chen) Morphisms of Curves and the Fundamental Group (M Cushman) Iterated Integrals and Algebraic Cycles: Examples and Prospects (R Hain) Chens Interated Integrals and Algebraic Cycles (B Harris) On Algebraic Fiber Spaces (Y Kawamata) Local Holomorphic Isometric Embeddings Arising from Correspondences in the Rank-1 Case (N Mok) Multiple Polylogarithms: Analytic Continuation, Monodromy, and Variations of Mixed Hodge Structures (J-Q Zhao) Deformation Types of Real and Complex Manifolds (F M E Catanese) The Life and Work of Kuo-Tsai Chen (R Hain & P Tondeur).

The Chern Connection

The Chern connection that we construct is a linear connection that acts on a distinguished vector bundle π*TM, sitting over the manifold TM \0 or SM. It is not a connection on the bundle TM over M. Nevertheless, it serves Finsler geometry in a manner that parallels what the Levi-Civita (Christoffel) connection does for Riemannian geometry. This connection is on equal footing with, but is different from, those due to Cartan, Berwald, and Hashiguchi (to name just a few).

A Simple Proof of Frobenius Theorem

Frobenius Theorem, as stated in Y. Matsushima, Differential Manifolds, Marcel Dekker, N.Y., 1972, p. 167, is the following: Let D be an r-dimensional differential system on an n-dimensional manifold M. Then D is completely integrabte if and only if for every local basis {X1,...,Xr} of D on any open set V of M , there are C∞ -functvons c ij k on V such that we have