By Konrad Schöbel
Konrad Schöbel goals to put the principles for a consequent algebraic geometric remedy of variable Separation, that is one of many oldest and strongest the right way to build particular recommendations for the elemental equations in classical and quantum physics. the current paintings finds a stunning algebraic geometric constitution at the back of the well-known record of separation coordinates, bringing jointly an excellent diversity of arithmetic and mathematical physics, from the overdue nineteenth century idea of separation of variables to fashionable moduli house thought, Stasheff polytopes and operads.
"I am quite inspired through his mastery of quite a few thoughts and his skill to teach sincerely how they have interaction to supply his results.” (Jim Stasheff)
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This publication by way of 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 struggle II - to jot 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 complete in 1948 and it used to be deliberate to be released within the Princeton
Mathematical sequence. in spite of the fact that, as a result of quick improvement of the topic, they felt
that immense adjustments needed to be made ahead of book.
When Nielsen moved to Copenhagen college in 1951 (where he stayed till
1955), he used to be a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The records 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 data additionally include a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes exact reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be greatly concerned with a radical revision of the curriculum in al-
gebra and geometry, and focused his learn within the conception of convexity, heading
the foreign 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 task left him little time to complete the
Fenchel-Nielsen undertaking the best way he desired to.
After his retirement from the collage, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - chanced on time to
finish the ebook easy Geometry in Hyperbolic area, which used to be released via
Walter de Gruyter in 1989 almost immediately after his demise. concurrently, and with an analogous
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, removal a few of the vague issues that have been within the unique
manuscript. Fenchel instructed me that he meditated elimination components of the introductory
Chapter I within the manuscript, because this may be lined by way of the booklet pointed out above;
but to make the Fenchel-Nielsen booklet self-contained he finally selected to not do
so. He did choose to miss
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 aid from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus college) with the facts 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 such a lot respects we determined to stick with Fenchel's intentions. in spite of the fact that, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of convinced key-words, will be uniform in the course of the ebook. additionally,
we have indicated the start and finish of an evidence within the ordinary kind of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice reduction and delight they agreed to post the manuscript of their sequence
Studies in arithmetic. i'm so much thankful for this confident and quickly response. One
particular challenge with the booklet became 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 through 1997 it grew to become transparent that he
would no longer have the capacity to locate the time to take action. notwithstanding, the writer provided an answer
whereby I may still convey designated drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they might arrange the creation of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his wonderful
collaboration in regards to the real construction of the figures.
My colleague Bent Fuglede, who has personaHy identified either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right standpoint. 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 few targeted symbols, a listing of notation with brief
explanations and connection with the particular definition within the ebook has been integrated. additionally,
a accomplished index has been additional. 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 because of
the overwhelming variety of examine papers during this region. as an alternative, a far shorter
list of monographs and different entire bills correct to the topic has been
collected.
My ultimate and so much honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into life.
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Extra info for An Algebraic Geometric Approach to Separation of Variables
Sample text
The space of algebraic curvature tensors on V and the space of symmetrised algebraic curvature tensors on V are isomorphic representations of GL(V ). 2a) S(w, y, x, z) − S(w, z, x, y) . 2b) Since the Nijenhuis torsion of K depends on K and its covariant derivative, ∇K, we need to express both in terms of the corresponding symmetrised algebraic curvature tensor S. 3. Up to a constant factor that can be neglected, we have Kx (v, w) = S(x, x, v, w) (∇u K)x (v, w) = 2S(x, u, v, w). 3b) Proof. 7). 3b) follows trivially.
4 Diagonal algebraic curvature tensors . . . . . 5 The residual action of the isometry group . . . . 2 Solution of the algebraic integrability conditions . 1 Reformulation of the first integrability condition . . 2 Integrability implies diagonalisability . . . . . 3 Solution of the second integrability condition . . . 4 Interpretation of the Killing-St¨ ackel variety . . 1 St¨ ackel systems and isokernel lines . . . . . . 2 Antisymmetric matrices and special conformal Killing tensors .
6b), we can interpret an algebraic curvature tensor R on V as a symmetric bilinear form on Λ2 V . We say that R is diagonal in an orthonormal basis {ei : 0 i n} of V , if it is diagonal as a bilinear form on Λ2 V in n}. In components, this the associated basis {ei ∧ ej : 0 i < j simply means that Rijkl = 0 unless {i, j} = {k, l}. We denote by K0 (Sn ) the vector space of Killing tensors on Sn that have a diagonal algebraic curvature tensor (with resprect to some fixed orthonormal basis in V ). We can now state the result that is central to Chapter 2.