INTRODUCTION xv

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

fields.

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.