Topics in Geometry: In Memory of Joseph D’Atri

386 Pages·1996·11.838 MB·English

by  A. Bahri, H. Brezis (auth.), Simon Gindikin (eds.)

Progress in Nonlinear Differential Equations
and Their Applications
Volume 20
Editor
Haim Brezis
Universite Pierre et Marie Curie
Paris
and
Rutgers University
New Brunswick, N.J.
Editorial Board
Antonio Ambrosetti, Scuola Normale Superiore, Pisa
A. Bahri, Rutgers University, New Brunswick
Luis Cafarelli, Institute for Advanced Study, Princeton
L. C. Evans, University of California, Berkeley
Mariano Giaquinta, University of Florence
David Kinderlehrer, Carnegie-Mellon University, Pittsburgh
S. Klainerman, Princeton University
Robert Kohn, New York University
P. L. Lions, University of Paris IX
Jean Mawhin, Universite Catholique de Louvain
Louis Nirenberg, New York University
Lambertus Peletier, University of Leiden
Paul Rabinowitz, University of Wisconsin, Madison
John Toland, University of Bath
Topics in Geometry
In Memory of Joseph D'Atri
Simon Gindikin
Editor
Birkhauser
Boston • Basel • Berlin
Simon Gindikin
Department of Mathematics
Rutgers University
New Brunswick, NJ 08903
Library of Congress In-Publication Data
Topics in geometry : in memory of Joseph D'Atri I Simon
Gindikin, editor.
p. cm. -- (Progress in nonlinear differential equations and
their applications ; v. 20)
Includes bibliographical references.
ISBN-13: 978-1-4612-7534-3 e-ISBN-13: 978-1-4612-2432-7
DOl: 10.1007/978-1-4612-2432-7
1. Geometry, Differential. I. D'Atri, J. E., 1938-1993.
II. Gindikin, S. G. (Semen Grigor'evich) III. Series.
QA64I.T63 96-1703
516.3'6--dc20 CIP
Printe<i on acid-free paper
$
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Typeset and reformatted by TeXniques, Boston, MA
987 6 5 4 3 2 1
,
Joseph D' Atri
In Memory of Joseph D'Atri, our Friend and Colleague
Contents
Preface ............................................................... ix
Commentary .......................................................... xi
Summary of Research ................................................ xiii
Publications ......................................................... xvii
Curriculum Vitae ................................................... xix
Non-Linear Elliptic Equations on Riemannian Manifolds
with the Sobolev Critical Exponent
A. Bahri and H. Brezis ................................................ 1
Symmetric Cones
Josef Dorfrneister .................................................... 101
Pseudo-Hermitian Symmetric Spaces of Tube Type
Jacques Fa1uut and Simon Gindikin .................................. 123
Homogeneous Riemannian Manifolds
Whose Geodesics Are Orbits
Carolyn S. Gordon .................................................. 155
On the V-Module and Formal-Variable
Approaches to Vertex Algebras
Yi-Zhi Huang and James Lepowsky .................................. 175
The Lowest Eigenvalue for Congruence Groups
Henryk Iwaniec ...................................................... 203
Signatures of Roots and a New Characterization
of Causal Symmetric Spaces
Soji Kaneyuki ....................................................... 213
Admissible Limit Sets of Discrete Groups
on Symmetric Spaces of Rank One
Adam Koranyi ...................................................... 231
viii Contents
D'Atri Spaces
Oldrich Kowalski, Priedberl Priifer and Lieven Vanhecke ............ 241
Multiple Point Blowup Phenomenon in Scalar Curvature
Equations on Spheres of Dimension Greater Than Three
Yan Yan Li ......................................................... 285
The Harish-Chandra Realization for Non-Symmetric
Domains in en
R. Penney ........................................................... 295
How many Lorentz Surfaces Are There?
Robert Smyth and Tilla Weinstein .................................. 315
On a Theorem of Milnor and Thorn
Nolan R. Wallach ................................................... 331
Riemannian Exponential Maps and
Decompositions of Reductive Lie Groups
Joseph A. Wolf and Roger Zierau ................................... 349
Weakly Symmetric Spaces
Wolfgang Ziller ..................................................... 355
Preface
This collection of articles serves to commemorate the legacy of Joseph
D'Atri, who passed away on April 29, 1993, a few days after his 55th
birthday. Joe D' Atri is credited with several fundamental discoveries in ge
ometry. In the beginning of his mathematical career, Joe was interested in
the generalization of symmetrical spaces in the E. Carta n sense. Symmetric
spaces, differentiated from other homogeneous manifolds by their geomet
rical richness, allows the development of a deep analysis. Geometers have
been constantly interested and challenged by the problem of extending the
class of symmetric spaces so as to preserve their geometrical and analytical
abundance. The name of D'Atri is tied to one of the most successful gen
eralizations: Riemann manifolds in which (local) geodesic symmetries are
volume-preserving (up to sign). In time, it turned out that the majority of
interesting generalizations of symmetrical spaces are D'Atri spaces: natu
ral reductive homogeneous spaces, Riemann manifolds whose geodesics are
orbits of one-parameter subgroups, etc.
The central place in D'Atri's research is occupied by homogeneous
en,
bounded domains in which are not symmetric. Such domains were
discovered by Piatetskii-Shapiro in 1959, and given Joe's strong interest
in the generalization of symmetric spaces, it was very natural for him to
direct his research along this path. For many years he moved towards
the solution to the problem, which without doubt was principal for his
work in this area: finding the geometrical characterization of symmetrical
domains among homogeneous ones. The final result is surprisingly elegant:
symmetric domains are characterized by the condition that its sectional
curvature in its Bergmann metric is nonpositive.
An area of interest to Joe during the last years of his life were homoge
neous pseudo-Kahlerian manifolds. I was happy to have had an opportunity
to collaborate with him on this project. We investigated the problem of
the realization of pseudo-Hermitian symmetric spaces as nonconvex tube
domains. We had a lot of other plans. Joe continued to work even though
he was terminally ill. We reproduce here one of his last notes, which he
sent to me from the hospital.
A significant part of this collection is composed of papers in geomet
rical areas, where D'Atri made outstanding contributions: Lie groups and
homogeneous manifolds, first of all, D' Atri spaces, nonsymmetric homoge
neous bounded domains, Siegel domains, homogeneous cones and different
problems of symmetric spaces. The other part of this collection (which in
tersects with the first) is comprised of contributions of D'Atri's colleagues
at Rutgers University. Joe D'Atri did a lot for the development of the
Department of Mathematics. From 1985 to 1990 he was the Chair of the
Department.
x Preface
The topics of these papers reflects a broad view of the subject of ge
ometry, which was inherent to him. We have here papers on differential
equations of geometrical nature, automorphic functions, topology, Lorentz
surfaces, etc. We also included biographical information and a summary
of research prepared by D'Atri in 1992.
Joe D' Atri was an incredibly talented person. He had very broad
interests not only in mathematics, but in other areas as well. He was dis
tinguished by high professionalism in everything he did: research, teaching,
administrative service, photography, cactus and orchid cultivation, model
railroading, woodworking. Also he had a talent to be a devoted and gen
erous husband, father, grandfather and friend. We felt that during his life
and we feel it even more so now after his death.
Simon Gindikin
November 1995