By Thomas E. Cecil, Shiing-Shen Chern (auth.), Boju Jiang, Chia-Kuei Peng, Zixin Hou (eds.)
From the contents: T.E. Cecil, S.S. Chern: Dupin Submanifolds in Lie Sphere Geometry.- R.L. Cohen, U. Tillmann: Lectures on Immersion Theory.- Li An-Min: Affine Maximal floor and Harmonic Functions.- S. Murakami: unheard of uncomplicated Lie teams and similar subject matters in contemporary Differential Geometry.- U. Simon: Dirichlet difficulties and the Laplacian in Affine Hypersurface Theory.- Wang Shicheng: crucial Invariant Circles of floor Automorphism of Finite Order.
Read or Download Differential Geometry and Topology: Proceedings of the Special Year at Nankai Institute of Mathematics, Tianjin, PR China, 1986–87 PDF
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Discontinuous Groups of Isometries in the Hyperbolic Plane
This publication through Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and complex heritage. 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 struggle II - to jot down the 1st chapters of the booklet (in German). whilst Fenchel,
who needed to break out from Denmark to Sweden end result of the German profession,
returned in 1945, Nielsen initiated a collaboration with him on what turned 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 accomplished in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. notwithstanding, end result of the quick improvement of the topic, they felt
that tremendous alterations needed to be made ahead of ebook.
When Nielsen moved to Copenhagen college 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 used to be left to Fenchel. The documents 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 files additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes certain 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 centred his study within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally positioned a lot attempt into his task 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 venture the best way he desired to.
After his retirement from the college, Fenchel - assisted by way of Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the e-book easy Geometry in Hyperbolic area, which was once released by way of
Walter de Gruyter in 1989 almost immediately after his loss of life. concurrently, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal some of the imprecise issues that have been within the unique
manuscript. Fenchel advised me that he reflected removal elements of the introductory
Chapter I within the manuscript, on account that this is able to be lined by means of the e-book 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 crew.
As editor, i began in 1990, with the consent of the felony 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 an excellent task of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a scholar of Kate Fenchel
at Aarhus collage) with the facts analyzing 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 made up our minds to stick with Fenchel's intentions. notwithstanding, 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 during the ebook. additionally,
we have indicated the start and finish of an evidence within the traditional kind of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and pride they agreed to submit the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and speedy response. One
particular challenge with the e-book grew to become out to be the replica of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to carry those in ultimate digital shape, yet by way of 1997 it turned transparent that he
would no longer have the ability to locate the time to take action. even if, the writer provided an answer
whereby I should still carry special drawings of the figures (Fenchel didn't depart such
for Chapters IV and V), after which they might set up the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his advantageous
collaboration about the genuine creation of the figures.
My colleague Bent Fuglede, who has personaHy identified 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 booklet reproductions of images of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few particular 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 circumstances, all references are to sections,
not pages.
We thought of including a whole checklist of references, yet made up our minds opposed to it as a result of
the overwhelming variety of learn papers during this region. as an alternative, a miles shorter
list of monographs and different entire debts suitable 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 book into lifestyles.
Statistics on Special Manifolds
This e-book is anxious with statistical research at the specific manifolds, the Stiefel manifold and the Grassmann manifold, taken care of as statistical pattern areas together with matrices. the previous is represented through the set of m x ok matrices whose columns are together orthogonal k-variate vectors of unit size, and the latter by way of the set of m x m orthogonal projection matrices idempotent of rank ok.
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Extra info for Differential Geometry and Topology: Proceedings of the Special Year at Nankai Institute of Mathematics, Tianjin, PR China, 1986–87
Example text
Way from s23 - s 3 2 , [Y1,YT] is still right-hand not -IY2 + laY7 ' Y6 a-lY6 determined, completely - ]aYl " we h a v e u n d e r t h i s The etc. ,4 + P~I ' ' 1" -1 1 3 ~)3 = a ~ 3 - '/~)7 " which implies t h a t a * = c~- 2 a + ~{- 1 ~/ , p * =a -2 p -cx -1 p We c h o o s e p to make a = p . After . 37) -((P31+P13)/2)o2 position to - u. 34) and get that 4PP3°3 ' t o w a r d the Ultimately, we show that the frame can be chosen so t h a t main results. the function p is constant, and the c l a s s i f i c a t i o n naturally s p l i t s into the two cases p = 0 and p40.
18) to a s, so the correspond- field is Ya = u a~ YB " In particular Ym(S) = (exp~),~(Xm) = Em(S ). lae , ~(%~)2 = I. 1) From the following proposition we know that Ym(g)=Em(S) is the unit normal vector field to N . ,~ Y~ along XV defined as in §i satisfy Y1 ( ~)A'''AYn(~)iYn+ I(~)A'-'AYm_ I(E:) = O. 2) Let J denote the Jacobian of map exp~ at ~. AYm(e)I = C" IYI(g)A---AYn(e)AYn+I(g)A'o'AYm_I(g) where C is a non-zero constant. 2) holds. QED. 4) c=0 sh( -/~c s), c<0, f-c f c (s) = gc(S). We put gc(S) f (s).
54) c a n be w r i t t e n for r and p, . 61) Suppose that P 3 neighborhood # 0 at U of b. 62) on U. 60) Using 2 -2 = -P3(s+r-p) b e B. 58} + 4Ptp2p2 for -3 = 0 . 63). 63) reduces to = 0 . Since p ~ O, t h i s implies t h a t P2 = 0 on U. 46) equation PP21 must vanish to our o n U. Since assumption. + PlP2 = p2p3 p ~ O, we c o n c l u d e Hence, P3 must that vanish P 3 = 0 o n U, identically a contradiction on B and the 1emma i s proven. 54) and = 0 , s+r-p our to formulas Lemma simplify 7 we h a v e w2 = O.