By Werner Fenchel, Jakob Nielsen, Asmus L. Schmidt
This booklet via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex heritage. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World conflict II - to put in writing the 1st chapters of the e-book (in German). whilst Fenchel,
who needed to break out from Denmark to Sweden as a result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become identified
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) was once comprehensive in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. although, because of the quick improvement of the topic, they felt
that sizeable adjustments needed to be made prior to booklet.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed till
1955), he was once a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The information of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (
I -15), and an entire manuscript (manuscript I) in
English containing Chapters I-V (
1-27). The documents additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes unique reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen collage in 1956 (and stayed there
until 1974), was once a great deal concerned with an intensive revision of the curriculum in al-
gebra and geometry, and focused his study within the conception of convexity, heading
the foreign Colloquium on Convexity in Copenhagen 1965. for nearly two decades
he additionally placed a lot attempt into his activity as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this task left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
After his retirement from the college, Fenchel - assisted by means of Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - discovered time to
finish the e-book straight forward Geometry in Hyperbolic house, which used to be released by means of
Walter de Gruyter in 1989 almost immediately after his demise. at the same time, and with an analogous
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal the various vague issues that have been within the unique
manuscript. Fenchel informed me that he pondered removal components of the introductory
Chapter I within the manuscript, on the grounds that this could be coated via the booklet pointed out above;
but to make the Fenchel-Nielsen ebook self-contained he eventually selected to not do
so. He did choose to pass over
27, entitled Thefundamental team.
As editor, i began in 1990, with the consent of the felony heirs of Fenchel and
Nielsen, to supply a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb task of typing this manuscript in AMS- TEX. i've got additionally had
much support from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the evidence examining of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a basic dialogue of the difference to the fashion
of TEX. In so much respects we determined to stick to Fenchel's intentions. even if, turning
the typewritten variation of the manuscript into TEX helped us to make sure that the notation,
and the spelling of definite key-words, will be uniform through the booklet. additionally,
we have indicated the start and finish of an evidence within the traditional form of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and delight they agreed to post the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this confident and speedy response. One
particular challenge with the booklet grew to become out to be the replica of the numerous
figures that are a vital part of the presentation. Christian Siebeneicher had at
first agreed to bring those in ultimate digital shape, yet via 1997 it grew to become transparent that he
would no longer be ready to locate the time to take action. notwithstanding, the writer provided an answer
whereby I should still bring unique drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might set up the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his high quality
collaboration in regards to the real construction of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a brief biography of the 2 of them and their mathematical achievements,
and which additionally areas the Fenchel-Nielsen manuscript in its right point of view. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this ebook reproductions of images of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of specified symbols, an inventory of notation with brief
explanations and connection with the particular definition within the ebook has been incorporated. additionally,
a finished index has been further. In either instances, all references are to sections,
not pages.
We thought of including an entire checklist of references, yet determined opposed to it because of
the overwhelming variety of learn papers during this region. in its place, a miles shorter
list of monographs and different finished debts correct to the topic has been
collected.
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into life.