80 Submanifolds and Holonomy
Proposition 2.9.1 Let f :
¯
M →R be a transnormal function. Then f is isoparametric
if and only if the regular level hypersurfaces of f have constant mean curvature.
Each connected component of a regular level hypersurface of an isoparametric
function is called an isoparametric hypersurface. The transnormal system induced by
an isoparametric function is called an isoparametric system. By the previous proposi-
tion, any isoparametric hypersurface belongs to a family of equidistant hypersurfaces
with constant mean curvature.
A first thorough investigation of isoparametric functions on Riemannian mani-
folds was carried out by Wang in [342]. This was developed further more recently
by Ge and Tang in [135, 136] and Qian and Tang in [280], with some interesting
applications to exotic spheres. Survey articles about this topic were written by Thor-
bergsson [320], Cecil [68] and Miyaoka [210].
2.9.3 Homogeneous hypersurfaces
Let
¯
M be a connected complete Riemannian manifold and I(
¯
M) its isometry
group. Suppose that G is a connected closed subgroup of I(
¯
M) acting on
¯
M with co-
homogeneity one. We equip the orbit space
¯
M/G with the quotient topology relative
to the canonical projection
¯
M →
¯
M/G.Then
¯
M/G is a one-dimensional Hausdorff
space homeomorphic to the real line R,thecircleS
1
, the half-open interval [0,∞),or
the closed interval [0, 1]. This was proved by Mostert [220] for the compact case and
by B´erard Bergery [15] for the general case. The following basic examples illustrate
the four cases. Consider a one-parameter group of translations in R
2
. The orbits are
parallel lines in R
2
and the space of orbits is homeomorphic to R. Rotating a torus
around its vertical axis through the center leads to an orbit space homeomorphic to
S
1
, whilst rotating a sphere around some axis through its center yields an orbit space
homeomorphic to [0,1]. Eventually, rotating a plane around some fixed point leads
to an orbit space homeomorphic to [0,∞).
If
¯
M/G is homeomorphic to R or S
1
, each orbit of the action of G is principal
and the orbits form a codimension one Riemannian foliation of
¯
M. In the case
¯
M/G
is homeomorphic to [0,∞) or [0,1], there exist one or two singular orbits, respec-
tively. If a singular orbit h as codimension greater than one, then each regular orbit is
geometrically a tube around this singular one. And if the codimension of a singular
orbit is one, then each regular orbit is an equidistant hypersurface to it. Suppose that,
in addition,
¯
M is simply connected. If
¯
M is compact, then, for topological reasons,
¯
M/G is homeomorphic to [0,1] and each singular orbit has codimension greater than
one. Thus, every principal orbit is a tube around each of the two singular orbits, and
each singular orbit is a focal set of any principal orbit. If
¯
M is noncompact, then
¯
M/G
must be homeomorphic to R or [0,∞). In the latter case, the singular orbit must h ave
codimension greater than one and each principal orbit is a tube around the singular
one.
It is not difficult to deduce from the previous discussion that the orbits of G form
a transnormal system on
¯
M. According to Proposition 2.7.1, each principal orbit of