Vol.3, No 13, 2003 pp. 775-777
PROFESSOR AKITSUGU KAWAGUCHI
Talk on occasion of the Anniversary of Akisugu
KAWAGUCHI's 100 years birth,
who is the Founder of Tensor Society August
5-9, 2002
We have gathered here to commemorate and celebrate the centenary of the
birth of the prominent mathematician Professor Akitsugu Kawaguchi.
He was born on April 8, 1902. He was only four years old when his father
sud-denly died. Difficult times started for the family. His mother worked
hard as a teacher, so the young boy was accustomed at an early age to work
without any-one's help and work harder than other boys, but later he received
much reward. He was admitted into Tohoku Imperial University, Faculty of
Science, Department of Mathematics at Sendai in 1922. In 1925 he obtained
the Bachelor degree, was admitted to the postgraduate school of the same
University, and at the same time he gained a research fellowship from the
Japanese Ministry of Education. In 1930 he was engaged as an arranging
officer to establish the foundation of the new Fac-ulty of Science, Hokkaido
Imperial University, where he first obtained an associate professorship,
and in 1932 he was appointed as a full professor of the Department of Mathematics.
He had served this University very well until 1966, when he reached the
retirement age of 63. During this period he became an internationally renown
mathematician. He played a decisive role also in the development of his
Depart-ment at Hokkaido University. On his retirement he was granted the
title of Professor Emeritus of Hokkaido University. At the same time, at
the height of his creative power, he was inaugurated as a distinguished
guest professor of Nihon University College of Science and Technology,
and in the next year he became simultaneously the Director of Department
of Mathematical Engineering in Sagami Institute of Technology.
He was only 26 years old when was appointed a research fellow to Europe
and the United States of America for two years. Later this experience was
followed by a number of fruitful scientific visits to Germany, France,
Poland, USA, Greece, Austria, Romania, Italy, Hungary, etc. He visited
almost all countries of Europe, and later India and South Asia. He accepted
many invited lectureships and was visiting professor many times at leading
universities. At that time abroad he was already one of the most famous
Japanese mathematicians.
He worked much and efficiently. He had several brilliant ideas, but
some of them are not yet completely developed and exhausted. Among these,
the first concerned the projective differential geometry. In this area
he wrote about 20 papers between 1927 and 1931. Also since then this area
was developed and investigated by many mathematicians all over the world.
Like Hilbert, he worked on a theme through several years, and then
he changed to another one. - After the projective diffierential geometry,
from 193I to 1937, he investigated different concepts of parallelisms,
displacements and general connec-tions. Among these there are interesting
papers on Finsler geometry. We cannot say that Finsler geometry would have
been the main field of his most important investigations, however he made
essential contributions to this field, and he was the initiator of the
investigations in this field in Japan. He can be considered as the founder
of the worldwide famous Finsler geometric school with numerous collaborators
led later by Professor M. Matsumoto. The flourishing of this school started
in the 1970's and lasts even now.
The problem which attracted him for the longest time (mainly from 1937
to 1944, but also after that) is the foundation and developing of the higher
order geometry. Spaces dealt with in this geometry are commonly called
Kawaguchi spaces. It is well known that the Riemannian arc length of a
curve x(t) is given by the integral of the square root of a quadratic form
in with coefficients dependent in x. This integrand is of course
homogeneous of the first order in . If we drop the quadratic property,
and retain the homogeneity only, then we obtain Finsler geometry. It is
most natural to suppose that the integrand depends not only on x and
, but also on the higher derivatives , . . . up to the k-th derivative
x(k). This is an important, natural, clear and simple idea for the metric
in general.
Big ideas usually are simple. The father of the differential geometry
of the eu-clidean space, Gauss, derived the metric tensor gij of a hypersurface
? from the coordinates of ?, and he could express every metrical relation
on ? by this gij. Riemann's idea was basically that gij should not be derived
from ?, but given arbitrarily; and he obtained the Riemann geometry. This
was just as important, natural, clear and simple an idea as Kawaguchi's.
However, detailed development, acceptance and dissemination of a new idea
often need reasonable time, sometimes a long time. Riemann's idea lacked
the notion of parallelism and connection. The rapid development of Riemannian
geometry started only 40-50 years after Rie-mann's discovery. Bolyai's
and Lobachewski's ideas were accepted also only 50 years after their discovery.
A similar case happened with G. Cantor and many others. ? The idea of the
higher order geometry may be very clear and natural, its development was
not yet quite simple. More recently Prof. Miron and his col-laborators
have made considerable and successful progress to give the higher order
geometry a more simple form and attach to it a well treatable apparatus.
It is my deep personal conviction that higher order geometry has still
a great future before it.
After world war II, he started to develop the geometry of areal spaces.
It is well known that in the euclidean geometry from the length (from the
distance) one can derive the area of different dimension. It is the same
in the Riemannian and also in the Finsler geometry, however in the latter
the notion of the area found little application until now. Neverthless
one can begin directly with the notion of the area, and try now to build
a geometry on it. One can see that the apriori given area is more general
than that which can be derived from a V n or F n. Development of a geometry
built on the notion of the k-dimensional area seems to have the same importance
and to be the counterpart of the geometries built on the distance (arc
length i.e. 1-dimensional area). But this seems to be a definitely difficult
task. We have not yet found the necessary tools, apparatus, perhaps not
even the appropriate notions. I guess it is not only a problem of the differential
geometry. It certainly needs also direct geometric, and maybe also other
considerations. In this direction still less has been done than in the
higher order geometry.
He published nearly one hundred and fifty papers, most of them in leading
Japan-ese journals, such as Tohoku Mathematical Journal, Proceedings of
the Imperial Academy of Tokyo, also in Tensor, Old and New Series; and
a number of papers in foreign journals, such as Monatshefte für Mathematik
und Physik, Transaction of the Amer. Math. Soc., Comptes Rendus Paris,
Rendiconti di Palermo, and in many other journals over the world, and in
proceedings of conferences in different fields. He published also a number
of books (exactly 36) mainly textbooks. From 1930 to 1934, 16 university
textbooks appeared by him covering all fields of geometry. His enormous
activity is admirable.
We cannot finish the appreciation of his life-work without mentioning
Tensor Society and the journal Tensor. In the period between the two wars
(1920-1939) considerable development took place in the Japanese mathematics.
The number of geometers materially increased and their cooperation needed
an organization. This was recognized by Akitsugu Kawaguchi, who at that
time was already an established, experienced and energetic, yet relatively
young mathematician with international fame. In 1938 at Sapporo he founded
the Tensor Society in order to promote research, further development and
facilitate cooperation among differ-ential geometers. At that time the
Society was a national organization and its journal, the Tensor, published
articles in Japanese. After the world war II the So-ciety became international,
and the new series of the Tensor published papers in the usual languages
of international mathematical journals (now almost always in English).
It became a well known international journal of high prestige, with an
international Editorial Board. Although Professor Kawaguchi obtained essential
aid from his pupils with the editorial work, he invested very much time
and energy into the journal which sometimes needed also his private financial
support. He also established and maintained Kawaguchi Research Institute
in Chigasaki in order to foster research and cooperation with his guests,
pupils and students, and to house the editorial office of the Tensor. -
The journal and the Society are essential es-tablishments of his life which
are successfully continued and sustained now by his sons.
Professor Akitsugu Kawaguchi died in 1984. He was one of the most famous
Japanese differential geometers acknowledged all over the world. His achievements
often were acknowledged by high state decorations. He had a number of excellent
pupils. For this time they became successful mathematicians, and most of
them are professors at different universities or colleges. He has 7 children,
among them 4 boys. There of them succeeded him in his profession and they
are professors of mathematics. The journal Tensor and Tensor Society have
survived and flourished. His ideas act even now. He is honored and esteemed
in these days in his country and over the world not less than in his life.
We received from him a rich scientific heritage. If he looks back upon
us and upon this generation he can be satisfied. He did not live in vain.
His ideas will live among us still for a long time.
He was born 100 years ago. On this occasion we pay our tribute and
bow our head to him. I was fortunate enough to meet him several times.
I remember him well on all of these occasions. I would like to pay also
may personal tribute to his memory.
Lajos Tamássy
University of Debrecin, Hungary
