x m
Theorem Let M be an isoparametric hypersurface, and suppose that the isometry
group of M acts transitively on one of the focal manifolds. Then either the hyper-
surface itself is homogeneous (and explicitly known), or it is of Clifford type with
multiplicities (8, 7) or (3,4/c 4).
This theorem gives in particular a new, independent proof for the classification by
Hsiang and Lawson.
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Another application is in topological geometry. Every isotropic simple algebraic
group, in particular every non-compact real simple Lie group, gives rise to a spher-
ical Tits building. These buildings are characterized by the so-called Moufang
condition. In the case of a non-compact simple Lie group, the building inherits
a compact topology from the group action. These compact buildings are closely
related to symmetric spaces, Fiirstenberg boundaries and isoparametric submani-
folds. They play also a role in the theory of Hadamard spaces and rigidity results.
Tits classified all irreducible spherical buildings of rank at least 3 by showing
that they automatically satisfy the Moufang property. In contrast to this general-
ized polygons, i.e. spherical buildings of rank 2, need not be Moufang, and there is
no way to classify them without further assumptions.
In view of the examples above, it is natural to consider compact generalized
polygons and to try to classify them in terms of their automorphism groups. A
result of Knarr and the author states that such a building is of type Ai, C2, or G2.
Topologically, a compact generalized polygon looks very similar to an isoparametric
foliation with g = 3,4, 6 distinct principal curvatures, respectively; in particular, the
cohomology of these spaces can be determined. This is the analogue of Munzner's
theorem mentioned above.
An ^-building is the same as a projective plane; all compact homogeneous
projective planes have been classified by Lowen and Salzmann. The compact ho-
mogeneous GVbuildings have been classified by the author. The remaining cases
are the generalized quadrangles, i.e. the buildings of type Ci. As in the case of
isoparametric hypersurfaces, this is much more involved. The easiest case here are
the quadrangles with Euler characteristic 4 which have been classified by the au-
thor. The remaining case, namely compact quadrangles of Euler characteristic 0, is
very interesting, since the inhomogeneous isoparametric hypersurfaces discovered
by Ferus, Karcher and Miinzner are examples of such (non-Moufang) quadrangles.
By general arguments, a transitive automorphism group on a 1-connected com-
pact quadrangle contains a compact transitive Lie subgroup (the automorphism
group of a compact building is in general not compact). Thus, we can apply our
result to classify transitive actions of compact Lie groups on compact quadran-
gles. We obtain a list of all possible transitive actions. For three infinite series
we show that the group action determines the quadrangle up to isomorphism. In
fact, we can (to some extent) use the same geometric methods and arguments both
for isoparametric hypersurfaces with g 4 distinct principal curvatures and for
compact quadrangles.
Nevertheless, the classification of the compact quadrangles is much more diffi-
cult than the classification of isoparametric hypersurfaces. This is due to the fact
that an isoparametric hypersurface sits inside some Euclidean space on which the
group acts linearly. Even though this surrounding space also exists for general-
ized quadrangles, it has no natural Euclidean structure, and it is not a priori clear
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