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By Otto Mutzbauer

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Discontinuous Groups of Isometries in the Hyperbolic Plane

This booklet through Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex historical past. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World battle II - to write down the 1st chapters of the booklet (in German). whilst Fenchel,
who needed to get away 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 recognized
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 used to be deliberate to be released within the Princeton
Mathematical sequence. in spite of the fact that, end result of the swift improvement of the topic, they felt
that mammoth adjustments needed to be made prior to book.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed until eventually
1955), he was once a lot concerned with the foreign association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The data 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 a whole manuscript (manuscript I) in
English containing Chapters I-V (

1-27). The documents additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes precise 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 greatly concerned with a radical revision of the curriculum in al-
gebra and geometry, and centred his examine within the thought of convexity, heading
the foreign Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally positioned a lot attempt into his activity as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this job left him little time to complete the
Fenchel-Nielsen undertaking the way in which he desired to.
After his retirement from the collage, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the e-book straight forward Geometry in Hyperbolic area, which was once released by means of
Walter de Gruyter in 1989 almost immediately after his dying. concurrently, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal a few of the imprecise issues that have been within the unique
manuscript. Fenchel instructed me that he reflected removal elements of the introductory
Chapter I within the manuscript, due to the fact that this may be lined via the ebook pointed out above;
but to make the Fenchel-Nielsen e-book self-contained he finally selected to not do
so. He did choose to pass over
27, entitled Thefundamental crew.

As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide 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 activity 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 college) with the evidence studying of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a basic dialogue of the variation to the fashion
of TEX. In such a lot respects we determined to stick to Fenchel's intentions. even though, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of sure key-words, will be uniform in the course of the e-book. additionally,
we have indicated the start and finish of an explanation within the traditional sort of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and pride they agreed to put up the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and quickly 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 convey those in ultimate digital shape, yet by way of 1997 it grew to become transparent that he
would now not have the ability to locate the time to take action. despite the fact that, the writer provided an answer
whereby I should still convey certain drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might manage the construction 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 locations the Fenchel-Nielsen manuscript in its right viewpoint. In
this connection i need to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this e-book reproductions of pictures of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of targeted symbols, a listing of notation with brief
explanations and connection with the particular definition within the booklet has been integrated. additionally,
a finished index has been extra. In either instances, all references are to sections,
not pages.
We thought of including an entire record of references, yet made up our minds opposed to it as a result of
the overwhelming variety of examine papers during this region. in its place, a miles shorter
list of monographs and different accomplished bills appropriate to the topic has been
My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this e-book into lifestyles.

Statistics on Special Manifolds

This ebook is worried with statistical research at the precise manifolds, the Stiefel manifold and the Grassmann manifold, handled as statistical pattern areas together with matrices. the previous is represented by way of the set of m x ok matrices whose columns are together orthogonal k-variate vectors of unit size, and the latter via the set of m x m orthogonal projection matrices idempotent of rank okay.

Extra info for Analytische Geometrie

Example text

Abhandlungen. 11. 3 34 Anschauliche Geometrie. schneiden, weil sonst dieser Punkt ein gemeinsamer Asymptotenpunkt wäre und also gerade Linien, welche vorher imaginäre Asymptotenpunkte besaßen, nunmehr reelle hätten usw. Die Zahl der Ovale der parabolischen Kurve ist immer zehn; es kann sich höchstens ein oder das andere Oval in einen isolierten Punkt zusammenziehen, was nicht in Betracht kommt. Denn kein Oval kann verschwinden - es bleiben ja die Asymptotenpunkte, in denen es berührt, reell -, es können sich nie zwei Ovale vereinigen denn jedes Oval ist von einem Sechsseit von geraden Linien umgeben.

Sieben und dreizehn. In der Tat ist nun leicht zu sehen (Schläflis Betrachtungen in den Annali benutzt ganz ähnliche Momente), daß auch unsere fünfte Art sechs reelle Dreiecksebenen mehr besitzt als die vierte. Denn die drei Ebenen, welche man durch die drei einfach zählenden Geraden der Fläche und den Knotenpunkt legen kann, dessen Flächenteile im Falle IV verbunden, im Falle V getrennt werden, gehen eben deshalb im Falle V in sechs reelle, im Falle IV in imaginäre Tangentenebenen über, und diese Tangentenebenen sind Dreiecksebenen, da sie, durch eine Gerade der Fläche hindurchgelegt, in einem nicht der Geraden angehörigen ~unkte berühren.

Man findet nun nach Herrn Sturm, daß die Gerade völlig imaginär wird, wenn das letztere eintrat, daß 8ie dagegen im ersteren Falle punktiert imaginär wird. Doch gehe ich hier nicht näher auf die Betrachtung der einzelnen Fälle ein, als deren Summe eben die Sturmsche Regel resultiert 11». 15) [Indem man die reellen Tangentialebenen betrachtet, welche man im einzelnen FalIe durch die einfach zählenden Gel'aden an die Fläche legen kann, erfährt man zugleich, wo die reellen Punkte liegen, in denen sich gegebenenfalls zwei konjugiert imaginäre Gerade der Fläche schneiden.

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