Our classification of homogeneous spaces is carried out in Chapters 5 and 6. The
classification is stated at the end of Chapter 3. Readers who are willing to accept
this result without entering into the details can skip Chapters 5 and 6. However,
the representation theory of Chapter 4 is used again in the last two chapters.
In Chapter 7 we classify transitive actions of compact Lie groups on compact
quadrangles. The homogeneous focal manifolds of isoparametric hypersurfaces with
g = 4 distinct principal curvatures are classified in Chapter 8, together with all
possible transitive group actions. Here, the reader is assumed to be familiar either
with topological geometry or with isoparametric hypersurfaces.
The logical dependencies of the chapters are as follows:
In particular, Chapter 7 (topological geometry) and Chapter 8 (submanifold ge-
ometry) are in principle independent of each other. Nevertheless, the subjects of
these two chapters, isoparametric hypersurfaces and compact polygons, share many
geometric properties which I tried to emphasize. Often, the proofs in Chapters 7
and 8 are quite similar and geometric. Besides the global properties of isoparamet-
ric hypersurfaces, very little differential geometry is needed in the classification. I
hope that the present book will be useful both for differential geometers and for
topological geometers, and that it helps to broaden the bridge between the two
The book is a revised, corrected and expanded version of my Habilitationsschrift.
I would like to thank my teachers Theo Grundhofer and Reiner Salzmann for their
constant interest and support. Harald Biller, Oliver Bletz, Norbert Knarr, Gerhard
Rohrle, Stephan Stolz, Markus Stroppel, Hendrik Van Maldeghem, and Martin
Wolfrom made helpful suggestions or spotted errors. I used a computer program
by Richard Bodi and Michael Joswig to check the tables for the representations of
simple Lie groups. Robert Bryant and Friedrich Knop helped me with a question
about a certain representation. The commutative diagrams in the original manu-
script were drawn with Paul Taylors diagrams T^X-package. Finally, I would like
to thank my wife, Katrin Tent, not only for reading the manuscript. Without her
support, this book would not have been possible.
Gerbrunn, February 2001
Linus Kramer
Man vergilt seinem Lehrer schlecht,
wenn man imrner nur der Schiller bleibt.
F. N.
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