Tiểu Luận On the classi cation of isoparametric hypersurfaces with four distinct principal curvatures in spher

On the classication of isoparametric hypersurfaces with four distinct principal curvatures in spheres
By Stefan Immervoll
Abstract
In this paper we give a new proof for the classication result in [3]. We
show that isoparametric hypersurfaces with four distinct principal curvatures
in spheres are of Cliord type provided that the multiplicities m1;m
2
of the
principal curvatures satisfy m2  2m1 1. This inequality is satised for all
but ve possible pairs (m1;m
2
) with m1  m2
. Our proof implies that for
(m1;m
2
) 6 = (1; 1) the Cliord system may be chosen in such a way that the
associated quadratic forms vanish on the higher-dimensional of the two focal
manifolds. For the remaining ve possible pairs (m1;m
2
) with m1  m2
(see
[13], [1], and [15]) this stronger form of our result is incorrect: for the three
pairs (3; 4), (6; 9), and (7; 8) there are examples of Cliord type such that
the associated quadratic forms necessarily vanish on the lower-dimensional of
the two focal manifolds, and for the two pairs (2; 2) and (4; 5) there exist
homogeneous examples that are not of Cliord type; cf. [5, 4.3, 4.4].
1. Introduction
In this paper we present a new proof for the following classication result
in [3].
Theorem 1.1. An isoparametric hypersurface with four distinct prin-cipal curvatures in a sphere is of Cliord type provided that the multiplicities
m1;m
2
of the principal curvatures satisfy the inequality m2  2m1 1.
An isoparametric hypersurface M in a sphere is a (compact, connected)
smooth hypersurface in the unit sphere of the Euclidean vector space
V = R
dim V
such that the principal curvatures are the same at every point.
By [12, Satz 1], the distinct principal curvatures have at most two dierent
multiplicities m1
, m2
. In the following we assume that M has four distinct
principal curvatures. Then the only possible pairs (m1;m
2
) with m1 = m2
are
(1; 1) and (2; 2); see [13], [1]. For the possible pairs (m1;m
2
) with m1 < m2
we have (m1;m
2
) = (4; 5) or 2
(m1 1)
divides m1 + m2
+ 1, where  : N ! N
1012 STEFAN IMMERVOLL
is given by
(m) = fi j 1  i  m and i  0; 1; 2; 4 (mod 8)g ;
see [15]. These results imply that the inequality m2  2m1 1 in Theorem 1.1
is satised for all possible pairs (m1;m
2
) with m1  m2
except for the ve
pairs (2; 2), (3; 4), (4; 5), (6; 9), and (7; 8).
In [5], Ferus, Karcher, and Munzner introduced (and classied) a class of
isoparametric hypersurfaces with four distinct principal curvatures in spheres
dened by means of real representations of Cliord algebras or, equivalently,
Cliord systems. A Cliord system consists of m + 1 symmetric matrices
P0; : : : ; Pm with m  1 such that P
2
i
= E and Pi Pj + Pj Pi
= 0 for i; j =
0; : : : ;m with i 6 = j , where E denotes the identity matrix. Isoparametric
hypersurfaces of Cliord type in the unit sphere S
2l 1
of the Euclidean vector
space R
2l
have the property that there exists a Cliord system P0; : : : ; Pm of
symmetric (2l  2l)-matrices with l m 1 > 0 such that one of their two
focal manifolds is given as
fx 2 S
2l 1
j hPi
x; xi = 0 for i = 0; : : : ;mg;
where h
;
i denotes the standard scalar product; see [5, Section 4, Satz (ii)].
Families of isoparametric hypersurfaces in spheres are completely determined
by one of their focal manifolds; see [12, Section 6], or [11, Proposition 3.2].
Hence the above description of one of the focal manifolds by means of a Cliord
system characterizes precisely the isoparametric hypersurfaces of Cliord type.
For notions like focal manifolds or families of isoparametric hypersurfaces, see
Section 2.
The proof of Theorem 1.1 in Sections 3 and 4 shows that for an isopara-metric hypersurface (with four distinct principal curvatures in a sphere) with
m2  2m1 1 and (m1;m
2
) 6 = (1; 1) the Cliord system may be chosen in such
a way that the higher-dimensional of the two focal manifolds is described as
above by the quadratic forms associated with the Cliord system. This state-ment is in general incorrect for the isoparametric hypersurfaces of Cliord type
with (m1;m
2
) = (3; 4), (6; 9), or (7; 8); see the remarks at the end of Section 4.
Moreover, for the two pairs (2; 2) and (4; 5) there are homogeneous examples
that are not of Cliord type. Hence the inequality m2  2m1 1 is also a
necessary condition for this stronger version of Theorem 1.1.
Our proof of Theorem 1.1 makes use of the theory of isoparametric triple
systems developed by Dorfmeister and Neher in [4] and later papers. We need,
however, only the most elementary parts of this theory. Since our notion
of isoparametric triple systems is slightly dierent from that in [4], we will
present a short introduction to this theory in the next section. Based on
the triple system structure derived from the isoparametric hypersurface M in
the unit sphere of the Euclidean vector space V = R
2l
, we will introduce in