Table Of ContentSeries in Pure Mathematics - Volume 14
Selected Papers of
Wilhelm P. A. Klingenberg
SERIES IN PURE MATHEMATICS
Editor: C C Hsiung
Associate Editors: S S Chern, S Kobayashi, I Satake, Y-T Siu, W-T Wu
and M Yamaguti
Part I. Monographs and Textbooks
Volume 1: Total Mean Curvature and Submanifolds on Finite Type
BY Chen
Volume 3: Structures on Manifolds
KYano&MKon
Volume 4: Goldbach Conjecture
Wang Yuan (editor)
Volume 6: Metric Rigidity Theorems on Hermitian Locally
Symmetric Manifolds
Ngaiming Mok
Volume 7: The Geometry of Spherical Space Form Groups
Peter B Gilkey
Volume 9: Complex Analysis
T O Moore &EH Hadlock
Volume 10: Compact Riemann Surfaces and Algebraic Curves
Kichoon Yang
Volume 13: Introduction to Compact Lie Groups
Howard D Fegan
Part II. Lecture Notes
Volume 2: A Survey of Trace Forms of Algebraic Number Fields
PE Conner & R Perils
Volume 5: Measures on Infinite Dimensional Spaces
Y Yamasaki
Volume 8: Class Number Parity
P E Conner & J Hurrelbrink
Volume 11: Topics in Mathematical Analysis
Th M Rassias (editor)
Volume 12: A Concise Introduction to the Theory of Integration
Daniel W Stroock
Series in Pure Mathematics - Volume 14
Selected Papers of
Wilhelm P. A. Klingenberg
World Scientific
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SELECTED PAPERS OF WILHELM P A KLINGENBERG
Copyright © 1991 by World Scientific Publishing Co. Pte. Lid,
All rights reserved. This book, or parts thereof, may not be reproduced in any form
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PREFACE
It was a happy day for me when I received a letter at the beginning of the year
from the editors of the Series in Pure Mathematics of the World Scientific Publishing
Co. in Singapore, asking me whether I would like to have my papers published.
Of couse, I liked it very much and I opted for Selecta. I have included some papers
here which may not seem to be too typical for somebody who has established his
name in the first place in Global Riemannian Geometry. But, let me remind you that
in the literature one occasionally encounters the name Klingenberg plane, denoting
a plane permitting homomorphisms. As a matter of fact, a common thread runs
through all my work, which may be described as geometry in its most intuitive
meaning. It was Goethe who in the epigram no. 40 in the appendix to his novel
"Wilhelm Meisters Wanderjahre1' most clearly gave expression to the essence of
geometry. Geometry possesses a character which is quite different from the character
of abstract mathematics — a subject which was quite alien to Goethe and on which
he had few good words to spare.
I can only hope that these Selecta will keep alive the memory of a mathematician
and a man who tried hard and with enthusiasm to make his contribution for a better
understanding of the cosmos and the chaos. Finally, I wish to thank the editors -
all of them from the Far East, to which I feel a special attachment - as well as the
publisher for giving me the chance to present myself in their Series.
Bonn, July 1991 Wilhelm Klingenberg
v
PERSONAL AND PROFESSIONAL HISTORY
I was born on January 28, 1924 in Rostock, the oldest child of a Protestant
minister, Paul Klingenberg, and his wife Henny Bunker. I grew up in a small village
on the Baltic Sea. Five more children were to follow.
When I was ten years old, my father moved to Berlin where good schools could
be found for his children and where, at the same time, it was easier to escape
the pressure to which he was subjected as an active member of the "Bekennende
Kirche", a group of his church opposed to the excesses of the Hitler government,
which had formed around M. Niemoller in Berlin - Dahlem. In 1937 I entered the
prestigious boarding school "Joachimsthalsces Gymnasium" where I obtained my
high school diploma in 1941.
I enrolled to study mathematics at Berlin University, but due to the war, I was
not allowed to begin my studies until I had completed the military service. In total,
I had to stay four and a half years with the army. During the last two years I was
a second lieutenant in an artillery unit at the eastern front. When the end of the
war finally gave me my freedom, I changed my handwriting and started looking for
a place to study. The devastated and Soviet-occupied city of Berlin was out of the
question, Gottingen and Hamburg were filled up, so I went to Kiel University. Here,
K. H. Weise became my teacher. From him I learnt the tricks of tensor analysis
and in 1950 under his supervision I wrote my thesis on Affine Differential Geometry
(see [1] and [2]). In a later paper [4] I solved the problem of rigging a generic
submanifold in affine space, generalizing the work of L. Berwald, W. Blaschke, B.
Laptev, J. A. Schouten and others. I did not continue with this type of work, but
it must have had some impact since some thirty years later it helped me to get my
first invitation to China - to Fudan University in Shanghai where Su Buchin had
taken notice of my papers.
In Kiel there was also F. Bachmann. Through him I became interested in
the foundations of geometry. I solved in [5] an old problem on the equivalences of
configurations in a general affine plane, something Hilbert's student, R. Moufang,
had worked on for many years. Subsequently I extended the work of J. Hjelmslev
and introduced the concept of homomorphism in Projective Geometry (see [18] and
[35]). This led me to investigate the structure of the classical groups over local
rings (see [30], [34], [39]). For my work, I found Artin's book "Geometric Algebra"
invaluable.
But to return to the early fifties, my fiancee Christine was eager to travel to
Sicily in the spring of 1951. K. H. Weise knew that W. Blaschke went to Taormina
every year - he had many friends in Italy. When I was introduced to him and
told him my plan, he suggested that I should lecture in Catania and Palermo, in
Italian of course. I have loved to speak Italian ever since. I had many chances to
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Vlll
practice it during later visits to Italy, and I had (and still have) good friends there.
Among other posts I spent six months as "Borsista di Istituto di Alta Matematica"
in Rome in 1952/53. F. Severi was the director, and E. Bompiani and B. Segre were
the professors with whom I had professional contact
After my return from Rome I had a two-year position as an Assistant in Applied
Mathematics under L. Collatz, but this was not something that gave me much
satisfaction. So Blaschke introduced me to K. Reidemeister in Gottingen, and
there I found a permanent position after my "Habilitation" in Hamburg. In the
meantime I had married; our oldest son Christian was born in Hamburg, Wilhelm
and Karin were born in Gottingen where we had a small house of our own in the
village of Geismar, which is today a part of Gottingen. I have fond memories of
our years there - Reidemeister had a brilliant mind and a wide range of interests,
his wife Elisabeth was a renown photographer. I was also treated very kindly by C.
L. Siegel, who had a reputation of being somewhat eccentric. I never had to suffer
from this; on the contrary he treated me as a respected younger colleague.
My years in Gottingen were interrupted by visits to the USA, which were
decisive for my scientific career. In 1954/5 I spent four months in Bloomington,
Indiana, by invitation from V. Hlavaty. By that time I had already lost interest
in affine and projective differential geometry. On my way back from Bloomington
I stopped over for a month at the Institute for Advanced Study in Princeton; M.
Morse, an old friend of Blaschke, had invited me.
Then in 1956,1 got the coveted invitation to Princeton as a temporary member,
which in 1957 was extended for another year. While the Sputnik raced over the
evening sky in the fall of 1957, I managed to make some important contributions
to Riemannian geometry in the Large (see [25]). My starting point was a paper by
H. E. Rauch in the Annals of Mathematics (1951), which was considered important
but at the same time somewhat mysterious and very difficult to understand. With
geometric rather than analytic arguments I started a new approach which with
hindsight can be viewed as the inauguration of a new area in global Riemannian
geometry. It was M. Gromov who later had the power and the vision to push wide
open the door which I had discovered.
But even my limited contribution was so unusual that both Rauch and Morse
openly disputed the validity of my arguments. I am grateful to D. Montgomery
that he did not hesitate for a moment to accept my announcement [22] for the
Proceedings of the National Academy of Sciences.
The principal application of my result on the injectivity radius was to improve
Rauch's pinching hypothesis for the so-called sphere theorem. Shortly afterwards,
M. Berger arrived at the final result using Toponogov's triangle comparison theorem.
All this did apply to manifolds of even dimension only. During the International
Colloquium on Differential Geometry and Topology in Zurich in 1960, I got the
idea on how I could extend my good lower estimate for the injectivity radius to
manifolds of odd dimension as well. I mentioned this to I. Singer and he understood
my arguments at once, while H. Hopf remained most sceptical - an experience which
was not new to me. (See [29] for the final version of the sphere theorem.)
IX
In the USA news about my results circulated more quickly than in Europe.
I received offers for a permanent position, but I only accepted an invitation as
Visiting Full Professor to Berkeley, which S. S. Chern must have been instrumental
in arranging. I had met him in Hamburg in 1953 when he visited his former teacher
Blaschke, and since that time he had actively helped my career whenever he had a
chance to do so.
In February 1962 we arrived in Berkeley. By then we had three children. We
first moved into a wonderful house in upper Shattuck Avenue and later to another
one in the Oakland Hills. All my life I was fortunate in that my wife took care of
the children by herself, so that I could concentrate on my work as a mathematician.
We have the fondest memories of our year in California which included a Summer
School in Santa Barbara.
During my stay I got offers to Full Professorships from Wiirzburg and from
Mainz. I accepted Mainz since it seemed to be the more liberal and open-minded
place, for which I give credit to H. Rohrbach. In 1963 I moved to Mainz; in 1964-65
we again spent the academic year in Princeton.
In 1966 I got a very attractive offer from Zurich University. F. Hirzebruch at
Bonn must have heard that I was seriously considering going to Zurich. Shortly
before that, he had convinced J. Tits to move to Bonn, and now he used his ex-
cellent connections with the bureaucracy to match the Zurich offer. During the
International Congress in Moscow, where I gave an invited lecture, I accepted the
position in Bonn and I have stayed there ever since.
In the first years in Bonn, Hirzebruch, Tits and I regularly had lunch together
and planned invitations and other matters related to the Mathematical Institute.
As time went by, G. Harder, S. Hildebrandt and E. Brieskorn came to Bonn as
professors. The number of students, guests and staff grew, and some of the intimate
charm of a close-knit group thereby went down the drain. Not without some pain
and struggle, I finally accepted the change and concentrated my activities on my
own differential geometry group.
In doing so, I always found the full support of my colleagues. I was fortunate to
attract H. Karcher and E. Ruh to Bonn as professors. Together, we always manage
to draw some of the best students to our fields of research and I am very proud of
the many excellent PhD theses written under my supervision. The last two were by
Ursula Hamenstadt - my only female PhD student - and Xiao-wei Peng - my only
student from China.
But first I have to mention my work on closed geodesies. It started on the Boa
Vista beach in Recife, Brazil, where I tried to understand the work of A. S. Svarc
and S. L. Al'ber on the existence of periodic solutions in the calculus of variations
(see [41]). My first main contribution was to introduce the Hilbert manifold of
closed curves of class Hl on a Riemannian manifold. This Hilbert manifold carries
a canonical Riemannian metric and possesses a natural 0(2)-action (cf. [49]). In
developing this I took up the previous work of R. Palais and S. Smale on the
loop space. Over the following years I worked on the problem of finding infinitely
many different closed geodesies. Today this has been proved to be true for generic
x
metrics; the best results are due to my students D. Gromoll, W. Meyer and II.
B. Rademacher. In connection with my work on closed geodesies I also became
interested in the properties of the geodesic flow - after all, closed geodesies may
be viewed as the periodic orbits of this flow. Many classical results from people
working in Dynamical Systems yield useful tools for the study of closed geodesies.
I credit myself for having drawn Riemannian Geometry and Hamiltonian Systems
closer together after they had grown apart from the times of H. Poincare and G.
Birkhoff.
Besides publishing research papers, I wrote a dozen books and lecture note vol-
umes - some of them jointly with younger colleagues. Among these, "Riemannsche
Geometrie im GroBen" by Gromoll, Klingenberg and Meyer became an impor-
tant reference work and was translated into Russian. I would like to also men-
tion my short "Course in Differential Geometry", originally published in German
and translated into English, Spanish and Japanese. My "Closed Geodesies" in the
Grundlehren Series seems to be useful even today for people working in String The-
ory. And my "Riemannian Geometry" in 1982 was the first book on this subject
since the monograph of L. P. Eisenhart in 1926.
I wish to conclude this article with some remarks on my present life. The
mandatory retirement age at Bonn is 65. I still have my office at the University
and students still show up for examinations. At present I am Visiting Professor in
Leipzig, the famous university where S. Lie, 0. Holder, G. Herglotz, B. L. van der
Waerden and many others were once active. Long before retirement I have pursued
other interests which are related to my last two PhD students mentioned above:
Ursula is an accomplished pianist and an avid opera fan. I found it a challenge
to take up again the piano playing on my baby grand and I at least enjoy it if
nobody else does. And going to the opera really is a rewarding activity whether
in Bayreuth or at the Met. And as for Xiao-Wei - I met him during my first visit
to Hefei in China in 1980. By that time, I had already developed a deep interest
in the civilizations of Asia. It had started with a stay at the Tata Institute in
Bombay in 1963. During several visits to Japan I got some taste of the old Chinese
culture which developed into my collecting Chinese art. In 1986 I began a series of
backpacking tours in China. Thereby I could profit from two trekking expeditions
with Christine to the newly opened regions of Ladakh in Kashmir in 1975 and 1977.
My first tour took the ancient Silk Road from Pakistan to Xian. This was
followed by trips to Lhasa and the base camp of the Qomolangma (Mt. Everest) to
East Tibet and finally, last year, to the holy mount Kailas in West Tibet. I enjoyed
these trips thoroughly - they have taken me back my boyhood days when I read
of the exploits of famous travellers in Central Asia. All these places were closed to
foreigners until recently, and even now it is a true adventure to go there. On each
of my trips I wrote a story. They are about to be published in a book, cf. [10] in
the section C of the Bibliography.
Also I should not forget to mention the sessions of the Academy of Sciences and
Literature in Mainz. There I met interesting people, among them many writers.
Of course, I do not know what is still in store for me. But at this particular point
