Submanifold Geometry of Orbits 77
Proof Let p ∈ M. Recall that an isometry F of R
n+1
is uniquely determined by F(p)
and the differential d
p
F at p. We can thus define an isometry F of R
n+1
by F(p)=
η
and requiring that d
p
F|
T
p
M
coincides with the above identification of V and T
η
(K ·
η
),
and d
p
F|
ν
p
M
coincides with the above id entification of W and
ν
η
(K ·
η
). We will
show that F maps the connected component of M containing p to K ·
η
. The method
we use is similar to the one we used for the proof of Theorem 2.8.2. The set
C = {q ∈ M : F(q) ∈K ·
η
,d
q
F(T
q
M)=T
F(q)
K ·
η
,d
q
F
ϕ
v
=
ϕ
d
q
F(v)
for all v ∈ T
q
M}
is clearly a closed subset of M and, since p ∈C, C is nonempty. To see that C is the
connected component of M containing p , it is enough to show that C is an open subset
of M. With this in mind we will prove that, for each q ∈ C, there is an open neigh-
borhood of q in M that is contained in C. As a consequence of the Gauss equation,
the linear isometry
d
q
F|
T
q
M
: T
q
M → T
F(q)
K ·
η
preserves the curvature tensors of M and K ·
η
at q and F(q), respectively. Hence, it
can be extended to a local isometry f from U ⊂M into K ·
η
.Let
γ
be a geodesic in M
with
γ
(0)=q and let v
1
=
˙
γ
(0),v
2
,...,v
n
be a Darboux frame at q. Parallel translation
of this frame along
γ
(with respect to the Levi-Civita and normal connections) yields
γ
,V
1
,...,V
n
, which satisfies the system (2.8) of linear differential equations.
Let
¯
γ
= f ◦
γ
and ¯v
i
= d
q
F(v
i
), and parallel tr anslate this fram e along
¯
γ
. Moreover,
set
˜
γ
= F ◦
γ
and in an analoguous way consider
˜
γ
,
˜
V
1
,...,
˜
V
n
. Just as in the proof
of Theorem 2.8.2, bearing in mind d
q
F
ϕ
v
=
ϕ
d
q
F(v)
,wehavethat
¯
γ
,
¯
V
1
,...,
¯
V
n
and
˜
γ
,
˜
V
1
,...,
˜
V
n
both satisfy the system (2.8) of linear differential equations with the
same initial conditions. Thus
¯
γ
=
˜
γ
,thatis,F(
γ
(t)) = f (
γ
(t)).IfW is a normal
neighborhood of q,thenF|(U ∩W )= f |(U ∩W ) and U ∩W ⊂C. This shows that C
is open in M and completes the proof.
Case 2: Spheres.
The classification of symmetric submanifolds of S
n
(r) is a simple consequence
of the one of symmetric submanifolds of R
n+1
. An important observation is that a
symmetric submanifold of S
n
cannot have have a Euclidean factor, because otherwise
it could not be contained in a sphere.
Theorem 2.8.14 Let M be a locally symmetric submanifold of S
n
(r). Then:
(a) If M is not contained in any extrinsic sphere of S
n
(r), then M is lo cally a
submanifold product
M
1
×...×M
s
→ S
m
1
−1
(r
1
) ×...×S
m
s
−1
(r
s
) → S
n
(r),
where M
i
→ S
m
i
−1
(r
i
) is the standard embedding of an irreducible symmetric
R-space M
i
, the second map is the submanifold product of extrinsic spheres as
in Theorem 2 .8.6, n =
∑
m
i
+ s −1 and
∑
r
2
i
= r
2
.
78 Submanifolds and Holonomy
(b) If M is contained in an extrinsic sphere S
m
(r
) of S
n
(r),thenM→S
n
(r) factors
as M → S
m
(r
) → S
n
(r) with r
−2
= r
−2
+ H
2
and M → S
m
(r
) as in (a).
Proof It is easy to see that M has parallel second fundamental form in R
n+1
if and
only if M has parallel second fundamental form in S
n
(r). Moreover, M is not con-
tained in any extrinsic sphere of S
n
(r) if and only if M is full in R
n+1
.IfM is con-
tained in an extrinsic sphere of S
n
(r),itsuffices to apply Theorem 1.6.2. The result
is then a direct consequence of Theorem 2.8.8.
Case 3: Hyperbolic spaces.
The classification of symmetric submanifolds of H
n
(r) was carried out indepen-
dently by Takeuchi [311] and Backes and Reckziegel [13]. In the latter paper, Jordan
triple systems were used for the proof (as in [130]). Here, we adopt the more geomet-
ric approach of [311]. What comes out from the classification in hyperbolic spaces is
that there is no “interesting” new example. This is a consequence of Lemma 2.8.10
and actually follows from the more general results in [107] (see Lemma 2.6.4).
Theorem 2.8.15 Let M be a locally symmetric submanifold of H
n
(r) ⊂ R
n,1
.
(a) If M is full in R
n,1
, then M is locally a submanifold product
M
m
0
(r
0
) ×M
1
×...×M
s
→ M
m
0
(r
0
) ×S
n−m−1
(r
) → H
n
(r),
where r
0
< 0,r
> 0,r
−2
0
−r
−2
= r
−2
, and M
1
×...×M
s
is a symmetric sub-
manifold of S
n−m−1
(r
) as in Theorem 2.8.14 (a).
(b) If M is contained in an extrinsic sphere
¯
M
m
(
κ
) of H
n
(r),thenM→ H
n
(r)
factorsasM→
¯
M
m
(
κ
) → H
n
(r) with
κ
= r
−2
+ H
2
, and M →
¯
M
m
(
κ
) is
as in (a) above, as described by Theorem 2.8.14 (a), or as in Theorem 2.8.8
(according to the sign of
κ
).
Proof We regard M as a submanifold of R
n,1
. Recall that M is not contained in any
extrinsic sphere of H
n
(r) if and only if M is full in R
n,1
.
(a) In this case we can proceed as in the proof of Theorem 2.8.8 (a) b y applying
the version of Moore’s Lemma for submanifolds of Lorentzian space (Lemma 1.7.4).
Then M is locally a submanifold product M
0
×M
1
×...×M
s
→ R
m
0
,1
×R
m
1
+1
×
...×R
m
s
+1
,whereM
0
is a minimal submanifold of a necessarily negatively curved
hypersphere of R
m
0
,1
. According to Lemma 2.8.10, M
0
is totally geodesic in this
hypersphere, so it must coincide with it, for M is full in R
n,1
. The other factors M
i
are minimal in a hypersphere of R
m
i
+1
, so they can be treated as above.
(b) It suffices to apply Theorem 1.6.2 to reduce this case to case (a).
Submanifold Geometry of Orbits 79
2.9 Isoparametric hypersurfaces in space forms
Isoparametric submanifolds are one of the main topics in this book. In this section
we will focus on isoparametric hypersurfaces. These hypersurfaces were introduced
at the beginning of the 20th century, motivated by questions in geometrical optics,
and studied by Segre, Levi-Civita and Cartan, among others. The generalization to
higher codimension came much later, starting from the 1980s. For more details about
the historical development we refer to [320].
Isoparametric hypersurfaces of space forms can be characterized by the prop-
erty of their principal curvatures being constant. They are defined as regular level
sets of isoparametric functions, so that they determine an orbit-like foliation of the
space form. Isoparametric hypersurfaces share many properties with homogeneous
hypersurfaces.
2.9.1 Transnormal functions
Let
¯
M be a connected Riemannian manifold. A transnormal function on
¯
M is
a nonconstant smooth function f :
¯
M → R such that grad f
2
= a ◦ f for some
smooth function a : I → R,whereI = f (
¯
M) is an interval in R. Basic properties of
transnormal functions can be found in [135, 214, 342]. First of all, f has n o critical
values in the set I
o
of interior points of I. Therefore the level set M
c
= f
−1
({c})=
{p ∈
¯
M : f (p)=c} is a smooth hypersurface of
¯
M for each c ∈ I
o
.Ifc
1
,c
2
∈ I
o
,
then M
c
1
and M
c
2
are equidistant to each other, that is,
¯
d(p
1
,M
c
1
)=
¯
d(M
c
2
, p
2
) for
each p
1
∈ M
c
1
and each p
2
∈ M
c
2
. So regular level sets o f a transnormal function
f :
¯
M → R form a foliation on
¯
M by equidistant hypersurfaces, except possibly for
one or two singular level sets. One often calls this a transnormal system.
2.9.2 Isoparametric functions and isoparametric hypersurfaces
An isoparametric function is a transnormal function f :
¯
M → R such that Δ f =
b◦ f for some continuous function b : I →R,whereΔ f = div(grad f ) is the Laplacian
of f . Suppose that f is a transnormal function on
¯
M and M
c
is a regular level set. Then
ξ
=
grad f
a(c)
is a unit normal vector field on M
c
.LetE
1
,...,E
n−1
be a local orthonormal frame
field of M
c
. Using the Weingarten formula, the mean curvature h
c
of M
c
is given by
(n −1)h
c
=
n−1
∑
i=1
A
ξ
E
i
,E
i
= −
1
a(c)
n−1
∑
i=1
¯
∇
E
i
grad f ,E
i
= −
1
a(c)
Δ f .
Taking into account that a transnormal function has at most two nonregular values,
we can now conclude:
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
Submanifold Geometry of Orbits 81
the action of G has constant principal curvatures, hence in particular constant mean
curvature. Thus, using Proposition 2.9.1, we get the following.
Proposition 2.9.2 Let
¯
M be a connected complete Riemannian manifold and G be a
connected closed subgroup of the isometry group of
¯
M. If G acts on
¯
M with cohomo-
geneity one, then the orbits of the action of G on
¯
M form an isoparametric system on
¯
M whose p rincipal orbits have constant principal curvatures.
2.9.4 Hypersurfaces with constant principal curvatures
Clearly, the condition of constant principal curvatures is stronger than just having
constant mean curvature. A natural question is whether any isoparametric hypersur-
face has constant principal curvatures.
´
Elie Cartan [57] gave an affirmative answer
for the case that
¯
M is a space form.
Theorem 2.9.3 (Cartan) Any isoparametric hypersurface in a space of constant
curvature has constant principal curvatures.
This is a consequence of a more general result in the next chapter. In particular,
it is a special case of Exercise 3.6.5.
Theorem 2.9.3 does not extend to more general Riemannian manifolds. In fact,
Wang [339] gave an example of an isoparametric hypersurface in complex projective
space CP
n
with nonconstant principal curvatures. Further examples are provided by
distance spheres in Damek-Ricci spaces. The story brieflygoesasfollows(forde-
tails see [41]): Using the Iwasawa decomposition of semisimple real Lie groups, the
complex hyperbolic space CH
n
can be realized as a solvable Lie group S equipped
with a left-invariant Riemannian metric. As a group, S is the semidirect product o f
R and the (2n −1)-dimensional Heisenberg group. In this construction one can re-
place the Heisenberg group by a so-called generalized Heisenberg group. For certain
generalized Heisenberg groups this yields the quaternionic hyperbolic spaces and
the Cayley hyperbolic plane, but, in all other cases, one gets a nonsymmetric homo-
geneous Hadamard manifold, a so-called Damek-Ricci space. These m anifolds are
named after Damek and Ricci, who proved that these spaces provide counterexam-
ples to the Lichnerowicz Conjecture, stating that any harmonic manifold is locally
isometric to a two-point homogeneous space.
There are various ways to define or characterize harmonic manifolds. One char-
acterization is that a Riemannian manifold is harmonic if and only if its geodesic
hyperspheres have constant mean curvature. Hadamard’s Theorem implies that in a
Hadamard manifold the square of the distance function to a po int is a well-defined
transnormal function. The result of Damek and Ricci says that in a Damek-Ricci
space this function is even isoparametric. It was then p roved by Tricerri and Van-
hecke that the corresponding isoparametric hypersurfaces, which are geodesic hy-
perspheres, have non-constant principal curvatures. This lead Tricerri and Vanhecke
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