By Thomas E. Cecil
This exposition presents the state-of-the paintings at the differential geometry of hypersurfaces in genuine, advanced, and quaternionic area kinds. exact emphasis is put on isoparametric and Dupin hypersurfaces in actual area types in addition to Hopf hypersurfaces in advanced house types. The ebook is available to a reader who has accomplished a one-year graduate path in differential geometry. The textual content, together with open difficulties and an intensive checklist of references, is a wonderful source for researchers during this area.
Geometry of Hypersurfaces starts with the elemental thought of submanifolds in genuine area varieties. subject matters comprise form operators, central curvatures and foliations, tubes and parallel hypersurfaces, curvature spheres and focal submanifolds. the focal point then turns to the idea of isoparametric hypersurfaces in spheres. vital examples and category effects are given, together with the development of isoparametric hypersurfaces in response to representations of Clifford algebras. An in-depth remedy of Dupin hypersurfaces follows with effects which are proved within the context of Lie sphere geometry in addition to those who are acquired utilizing commonplace tools of submanifold thought. subsequent comes a radical therapy of the speculation of actual hypersurfaces in complicated house types. A vital concentration is a whole facts of the class of Hopf hypersurfaces with consistent valuable curvatures as a result of Kimura and Berndt. The e-book concludes with the fundamental concept of genuine hypersurfaces in quaternionic house types, together with statements of the key class effects and instructions for extra research.
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This ebook via Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate heritage. In 1938-39, Nielsen gave a chain of lectures on
discontinuous teams of motions within the non-euclidean aircraft, and this led him - in the course of
World conflict II - to jot down the 1st chapters of the ebook (in German). while Fenchel,
who needed to get away from Denmark to Sweden due to the German career,
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. besides the fact that, end result of the fast improvement of the topic, they felt
that gigantic alterations needed to be made sooner than booklet.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed till
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 comprise 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 records additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes distinctive 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), used to be a great deal concerned with an intensive revision of the curriculum in al-
gebra and geometry, and centred his learn within the conception of convexity, heading
the overseas 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 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 - stumbled on time to
finish the booklet uncomplicated Geometry in Hyperbolic house, which used to be released by means of
Walter de Gruyter in 1989 presently after his demise. concurrently, and with a similar
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination some 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, on account that this is able to be coated by way of the ebook pointed out above;
but to make the Fenchel-Nielsen publication self-contained he eventually selected to not do
so. He did choose to pass over
27, entitled Thefundamental workforce.
As editor, i began in 1990, with the consent of the criminal 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 an excellent activity 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 evidence interpreting 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 made up our minds to keep on 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 sure key-words, will be uniform through the booklet. additionally,
we have indicated the start and finish of an evidence within the ordinary type 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 post the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this confident and fast response. One
particular challenge with the book became out to be the replica of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to bring those in ultimate digital shape, yet by means of 1997 it grew to become transparent that he
would now not manage to locate the time to take action. despite the fact that, the writer provided an answer
whereby I should still convey distinct 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 nice
collaboration about the genuine creation of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally areas 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 ebook reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a couple of unique symbols, a listing 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 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 because of
the overwhelming variety of examine papers during this region. as an alternative, a far shorter
list of monographs and different entire bills suitable 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 book into life.
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Extra info for Geometry of Hypersurfaces
Example text
This is a local calculation, so we will consider M as an embedded hypersurface in SnC1 and suppress the mention of the embedding f . We use stereographic projection W SnC1 fPg ! x/ D1 hx; Pi. P Using the fact that hx; i D 0 for x 2 M and we get D e h. 63) where D is the Euclidean covariant differentiation on RnC2 . Since hX; i D 0, it follows that DX D rQ X , where rQ is the Levi-Civita connection on SnC1 . Then, we know that rQ X D AX D X, so DX D X. X /: From this it is clear that X D 0 if and only if X D 0.
1) is constant along each leaf of its principal foliation T . (2) The leaves of T are plane curves of constant curvature. n 1/-dimensional space of leaves U=T Q nC1 . into M We first give the proof in the Euclidean case and then handle the other cases via stereographic projection. Proof (Euclidean case). 8 concerning the rank of the focal map f and from the connectedness of the leaves of the foliation T . 1/ This follows easily from the Frenet equations for plane curves. t; v/ given by the coordinate chart W .
T/ D 0 if t Ä 0. t//; t 2 . 1; 1/: Then the curve itself is the sheet of the focal set of K corresponding to the principal curvature D 1 with appropriate choice of unit normal field. Let N be the intersection of K with the closed upper half-space given by z 0, with the points satisfying z D 0; x 0 removed. Let M be the union of N with its mirror image in the plane z D 0. Then D 1 is still a constant principal curvature on all of M. However, the leaf space M=T is not Hausdorff, since the two open semi-circular leaves L1 and L2 in the plane x D 0 cannot be separated by disjoint neighborhoods in the quotient topology.