*Editors*: H. Amann (Zürich), G. Auchmuty (Houston), D. Bao
(Houston), H. Brezis (Paris), J. Damon (Chapel Hill), K. Davidson (Waterloo), C.
Hagopian (Sacramento), R. M. Hardt (Rice), J. Hausen (Houston), J. A. Johnson
(Houston), J. Nagata (Osaka), B. H. Neumann (Canberra), V. I. Paulsen (Houston),
G. Pisier (College Station and Paris), S. W. Semmes (Rice)
*Managing Editor*: K. Kaiser (Houston)

Houston Journal of Mathematics

*Editorial*

It gives us distinct pleasure to dedicate this issue to
Professor Shiing-Shen Chern. Prof. Chern had been an editor of the Houston
Journal of Mathematics from 1992 to 2000. His association with the Department of
Mathematics involved yearly visits and stimulating lectures that began in
1988. He has collaborated with members of the department
in publications and in organizing conferences. He is
currently a *Distinguished Visiting Professor Emeritus*
*of the** University of Houston.*
Professor Chern is also *Professor Emeritus of Mathematics** **
at the University of California at Berkeley*, and *
Director Emeritus of the Mathematical Sciences Research* *
Institute (M.S.R.I.).* He is the founding father of three renowned
institutes: the *Institute of Mathematics of the** **
Academia Sinica, the M.S.R.I.,* and the *Nankai Institute** **
of Mathematics*.

Professor Chern has rather broad but impeccable tastes in
Mathematics. We have taken the liberty of singling out some
areas in which he has made profound contributions: real and
complex differential geometry, several complex variables,
and Finsler geometry. The collection of papers that we
have assembled in this issue comes from a *small*
cross-section of mathematicians in whose careers Professor Chern
has played a unique role. We are delighted, but not
surprised, that the authors in question submitted their
manuscripts with much enthusiasm. We hope this special
issue will become a source of inspiration for many years
to come.

**Editors of the special issue for S.S. Chern, Volume 28, No. 2**

David Bao

Shanju Ji

Klaus Kaiser (Managing Editor)

Min Ru

Subscribers might wish to order an additional copy of this issue Vol. 28(2)
which is also available to individuals.

The price is **$25**, plus **$5 **for shipping and handling.

*Contents*

**Robert L. Bryant, ** Duke University Mathematics Department, P.O. Box
90320 Durham, NC 27708-0320 (bryant@math.duke.edu) .

Some Remarks on Finsler Manifolds with
Constant Flag Curvature, pp. 221-262.

ABSTRACT.
Abstract: This article is an exposition of four loosely related remarks on the
geometry of Finsler manifolds with constant positive flag curvature.

The first remark is that there is a canonical Kähler structure on the space of
geodesics of such a manifold.

The second remark is that there is a natural way to construct a (not necessarily
complete) Finsler *n*-manifold of constant positive flag curvature out of a
hypersurface in suitably general position in **CP ^{n}**.

The third remark is that there is a description of the Finsler metrics of constant curvature on

The fourth remark concerns the generality of the space of (local) Finsler metrics of constant positive flag curvature in dimension

**Patrick Foulon, ** Institut de Recherche Mathèmatique Avancèe, Universitè
Louis Pasteur Strasbourg, France.

Curvature and Global Rigidity in Finsler
Manifolds, pp. 263-292.

ABSTRACT.
We present some strong global rigidity results for reversible Finsler manifolds.
Following É Cartan's definition (1926), a locally symmetric Finsler metric is
one whose curvature is parallel. These spaces strictly contain the spaces such
that the geodesic reflections are local isometries and also constant curvature
manifolds. In the case of negative curvature, we prove that the locally
symmetric Finsler metrics on compact manifolds are Riemannian and this,
therefore, extends A. Zadeh's rigidity result. Our approach uses dynamical
properties of the flag curvature. We also give a full generalization of the
Ossermann Sarnak minoration of the metric entropy of the geodesic flow. In
positive curvature, we just announce some partial results and remarks concerning
Finsler metrics of curvature +1 on the 2-sphere. We show that in the reversible
case the geodesic flow is conjugate to the standard one. We also observe that a
condition of integral geometry (of Radon type) forces such a metric to be
Riemannian. This indicates a deep link with (exotic) projective structures

**Daniel S. Freed, ** Department of Mathematics, University of Texas,
Austin, TX 78712 (dafr@math.utexas.edu).

Classical Chern-Simons Theory, Part 2,
pp. 293-310.

ABSTRACT.
We first recall the construction of Chern-Weil and Chern-Simons forms, and then
their synthesis in Cheeger-Simons differential characters. Then given a family
of connections parametrized by a manifold *T*, we integrate these
invariants to obtain differential forms and differential characters on *T*.
There are special constructions of geometric representatives in low degrees, but
the general case requires a suitable "cochain theory" for differential
characters. This is provided by recent work of Hopkins-Singer. It applies in
particular to the classical three-dimensional topological field theory known as
Chern-Simons theory.

**Peter B. Gilkey,** Mathematics Department, University of Oregon, Eugene
Or 97403 USA (gilkey@darkwing.uoregon.edu) and **Tan Zhan, ** Department of
Mathematics and Statistics, Murray State University, Murray, KY 42071-0009
(tan.zhang@murraystate.edu).

Algebraic Curvature Tensors for Indefinite
Metrics whose Skew-Symmetric Curvature Operator has Constant Jordan Normal Form,
pp. 311-328.

ABSTRACT.
We classify the connected pseudo-Riemannian manifolds of signature *(p,q)*
with *q*≥5 so that at each point of *M* the skew-symmetric curvature
operator has constant rank 2 and constant Jordan normal form on the set of
spacelike 2 planes and so that the skew-symmetric curvature operator is not
nilpotent for at least one point of *M*.

**Mark L. Green,** Department of Mathematics, UCLA, Los Angeles, CA 9009
(mlg@math.ucla.edu) and **Phillip A. Griffiths,** Institute for Advanced
Study, Einstein Drive, Princeton, NJ 08540 (pg@ias.edu).

Abel's Differential Equations, pp. 329-351.

ABSTRACT.
Abel's differential equations govern the rational motion of algebraic cycles on
an algebraic variety. In this paper, we will discuss Abel's differential
equations, first from a historical and classical perspective. Then we will
discuss their modern form, which is definitely non-classical in that arithmetic
considerations enter in an essential way. Finally, we will discuss the
integration of Abel's differential equations, which may be accomplished only by
assuming an important conjecture of Bloch/Beilinson.

**Wolf Kohn, ** **Vladimir Brayman, ** Hynomics Corporation,Kirkland, WA
98033-7921 (wk@hynomics.com) (vbrayman@hynomics.com) and **Anil Nerode,**
Department of Mathematics, Cornell University, Ithaca, New York 14853
(anil@math.cornell.edu).

Control Synthesis in Hybrid Systems with Finsler Dynamics, pp. 353-375.

ABSTRACT.
This paper is concerned with a symbolic-based synthesis of feedback control
policies for hybrid and continuous dynamic systems. A key step in our synthesis
procedure is a new method to solve a dynamic optimization problem in which the
continuous dynamics generates trajectories on a smooth manifold in which a
Finsler metric has been defined. The proposed method can be generalized for
finding explicitly the control laws for a wide variety of problems by
"Finslerizing" their formulation. The feedback control law generated is of the
form of the Cartan connection. The coefficients of this control law (generalized
Christoffel symbols) can be determined analytically. This paper illustrates the
computation by establishing a direct connection between optimality in a Finsler
domain (geodesic trajectories) and Dynamic Programming.

**Bernard Shiffman,** Department of Mathematics, Johns Hopkins University,
Baltimore, MD 21218, USA (shiffman@math.jhu.edu) and **Mikhail Zaidenberg, **
Université Grenoble I, Institut Fourier, UMR 5582 CNRS-UJF, BP 74, 38402 St.
Martin d'Hères Cédex, France (zaidenbe@ujf-grenoble.fr) .

Constructing Low Degree Hyperbolic
Surfaces in **P **^{3}
, pp. 377-388.

ABSTRACT.
We describe a new method of constructing Kobayashi-hyperbolic surfaces in
complex projective 3-space based on deforming surfaces with a ``hyperbolic
non-percolation'' property. We use this method to show that general small
deformations of certain singular abelian surfaces of degree 8 are hyperbolic. We
also show that a union of 15 planes in general position in projective 3-space
admits hyperbolic deformations.

**Yum-Tong Siu, **Department of Mathematics, Harvard University,
Cambridge, Massachusetts 02138} (siu@math.harvard.edu).

A New Bound for the Effective Matsusaka Big
Theorem , pp. 389- 409.

ABSTRACT.
Matsusaka's Big Theorem gives the very ampleness of a multiple of an ample line
bundle over an *n*-dimensional compact complex manifold with the factor
depending ineffectively on the top Chern class of the line bundle and the
product of the second top Chern class of the line bundle and the canonical
class. An earlier result of the author gives an effective bound on the factor
which is of the order of the sum of the absolute values of the above two Chern
numbers raised to the power 4^{n}. Demailly reduced the power to
3^{n} by reducing the twisting required for the existence of
nontrivial global holomorphic sections for anticanonical sheaves of subvarieties
which occur in the verification of the numerical effectiveness of the sum of the
anticanonical line bundle and an effective multiple of the ample line bundle.
Twisted sections of anticanonical sheaves are needed to offset the addition of
the canonical sheaf in vanishing theorems. We introduce here a technique to get
a new bound with power 2^{n}. The technique avoids the use of
sections of twisted anticanonical sheaves of subvarieties by transferring the
use of vanishing theorems on subvarieties of the ambient space to the ambient
space itself and is more in line with techniques for Fujita conjecture type
results.

**Gang Tian, ** Department of Mathematics, Massachusetts Institute of
Technology, Cambridge, MA 02139 (tian@math.mit.edu).

Extremal Metrics and Geometric Stability, pp. 411-432.

ABSTRACT.
This paper grew out of my lectures at Nankai Institute as well as a few other
conferences in the last few years. The purpose of this paper is to describe some
of my works on extremal Kähler metrics in the last fifteen years in a more
unified way.

At the beginning of the 90's, the author developed a method of relating certain
stability of underlying manifolds to Kähler-Einstein metrics (cf. [Ti4], [Ti2]).
An necessary and new condition was derived in terms of the stability for a
Kähler manifold to admit Kähler-Einstein metrics with positive scalar curvature.
It was clear then that similar results should also hold for general extremal
Kähler metrics. Extremal Kähler metrics were introduced by Calabi [Ca]. Extremal
Kähler metrics are critical points of the K-energy introduced by T. Mabuchi.
Most extremal metrics are Kähler metrics of constant scalar curvature. It was
conjectured by the author before that the existence of Kähler metrics with
constant scalar curvature is equivalent to the properness of the K-energy. This
has been verified for the case of Kähler-Einstein metrics ([Ti2]).

We will explain how extremal metrics are related to the stability of the
underlying manifolds and compare it with the standard picture from symplectic
geometry. We will outline the proof of the Calabi's conjecture for complex
surfaces. We will also list a few problems and indicate the difficulties in
solving them.

*References*

[Ca] : Calabi, E.: Extremal Kähler metrics. Seminar on Diff. Geom., Ann. of
math. Stud., **102**, Princeton Univ. Press, 1982.

[Ti2] : Tian, G: Kähler-Einstein metrics with positive scalar curvature. Invent.
Math., **130 **(1997), 1-39.

[Ti4]: Tian, G: The K-energy on Hypersurfaces and Stability. Communications in
Geometry and Analysis, **2 ** (1994), 239-265.

** S. M. Webster , **University of Chicago (webster@math.uchicago.edu).

A remark on the Chern-Moser tensor, pp. 433-435.

ABSTRACT.
We compute the fourth order Chern-Moser tensor for real hypersurfaces of
revolution in complex Euclidean space.