X l l
It follows also that G has a semisimple normal transitive subgroup KG which
has at most two almost simple factors. By a general result about transitive per-
mutation groups it suffices to determine the pair (K, K H H) C (G, H) in order
to determine all possibilities for the larger group G. Replacing G by K we may
therefore assume that G is almost simple or that G is semisimple with two almost
simple factors. The condition rt\ 3 guarantees then that H is also semisim-
ple. We determine all such pairs (G,H) with the right rational cohomology and
all possible embeddings H c G using representation theory. Thus, we obtain
an explicit classification of these homogeneous spaces together with the transitive
group actions.
In the special case that G/H has the same integral cohomology as S n i x § n 2
we obtain the list of homogeneous spaces given in the theorem above. In the
course of the proof we determine also all compact connected Lie groups which act
transitively on 1-connected rational homology spheres; in particular, we reprove the
classification of transitive actions on spheres and on spaces which have the same
homology as the Stiefel manifolds V2(R 2 n + 1 ).
: * * *
We apply our result to a problem in submanifold geometry. A closed hypersurface in
a sphere is called isoparametric if its principal curvatures are constant. Hsiang and
Lawson classified all isoparametric hypersurfaces which admit a transitive group of
isometries; these homogeneous isoparametric hypersurfaces arise as principal orbits
of isotropy representations of non-compact symmetric spaces of rank 2.
By a result of Miinzner, the number of distinct principal curvatures of an
isoparametric hypersurface is g = 1,2,3,4,6; the hypersurfaces with g = 1,2,3
have been classified in the 30s by Segre and Cart an. Some hypersurfaces with g = 6
were classified by Dorfmeister and Neher; the full classification for g = 6 still seems
to be an open problem. The case of hypersurfaces with g = 4 is much more diffi-
cult. Takagi proved uniqueness for isoparametric hypersurfaces with g = 4 distinct
principal curvatures and multiplicities (1,&). On the other hand, Ferus, Karcher
and Miinzner showed that there are many non-homogeneous isoparametric hyper-
surfaces with g = 4 distinct principal curvatures, and Stolz recently obtained sharp
number theoretic restrictions on the possible dimensions of such hypersurfaces.
Some of the known inhomogeneous examples have homogeneous focal manifolds, so
the question arises if the classification by Hsiang and Lawson can be generalized to
transitive actions on focal manifolds. In view of the classification by Segre, Cart an,
Dorfmeister and Neher, the cases g 1,2,3 are less interesting. If g = 4 and if
the multiplicities (rai,?7i2) of the isoparametric hypersurface are large enough, then
the focal manifolds have the same integral cohomology as a product of spheres. We
apply our result about homogeneous spaces to this problem and obtain a complete
T h e o r e m Let M be an isoparametric hypersurface with 4 distinct principal curva-
tures. Suppose that the isometry group of M acts transitively on one of the focal
manifolds, and that this focal manifold is 2-connected. Then either the hypersurface
itself is homogeneous (and explicitly known), or it is of Clifford type with multiplic-
ities (8,7) or (3,4fc-4) .
Recently, Wolfrom showed in his Ph.D. Thesis that the theorem above holds also
if one drops the assumption that the focal manifold is 2-connected, and proved a
similar result for g = 6. The final result is as follows.
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