Homogeneous Structures on Submanifolds 201
Lemma 6.2.6 Let M = K ·X be an orbit of an s-representation. Then ∇
c
α
= 0.
Proof If Z ∈ m,then
γ
(t)=Exp(tZ)·X is a ∇
c
-geodesic in M and d
X
Exp(tZ) is the
∇
c
-parallel transport along
γ
. Then, using Lemma 3.1.5 we have
(∇
c
˙
γ
(0)
α
)(v,w)=
d
dt
t=0
α
(d
X
Exp(tZ)(v),d
X
Exp(tZ)(w))
⊥
=
d
dt
t=0
(Exp(tZ) ·
α
(v, w))
⊥
=[Z,
α
(v, w)]
⊥
= 0
for all v,w ∈ T
X
M.
C. 2-symmetric submanifolds
Definition 6.2.7 (Kowalski, Kulich [191]) Let M be a submanifold of a space form
¯
M.Aregular s-structure on M is a family of isometries {
σ
p
}
p∈M
of
¯
M such that
(1)
σ
p
(M) ⊂ M for all p ∈ M;
(2) p is an isolated fixed point of
σ
p
|
M
for all p ∈ M;
(3)
σ
p
◦
σ
q
=
σ
σ
p
(q)
◦
σ
p
for all p,q ∈M.
If there exists a positive integer k ≥ 2 such that
σ
k
p
= id holds for all p ∈ M,thenM
is called a k-symmetric submanifold of
¯
M and the s-structure is said to be of order k.
A k-symmetric submanifold is extrinsically homogeneous. Indeed, if M is a k-
symmetric submanifold, let Tr(M , {
σ
p
|
M
}) be the group of the transvections, that is,
the g roup generated by the isometries
σ
p
|
M
◦(
σ
q
|
M
)
−1
of M. Then we can define the
representation
F :Tr(M,{
σ
p
|
M
}) → I(
¯
M) ,
σ
p
|
M
◦(
σ
q
|
M
)
−1
→
σ
p
◦(
σ
q
)
−1
.
Then M is an orbit of Tr(M,{
σ
p
|
M
}) under the representation F.
Remark 6.2.8 Note th at the above definition is different from the one given by
S´anchez in [290]. S´anchez assumed in addition that d
p
σ
p
|
ν
p
M
= id
ν
p
M
for all p ∈M.
Using Exercise 6.4.6, one can show that a k-symmetric submanifold in the sense of
S´anchez is an orbit of an s-representation (see also [265]).
We will now consider the case k = 2 in more detail. It is clear that any symmet-
ric submanifold is 2-symmetric; the set of reflections
σ
p
in the affine normal spaces
defines a regular s-structure of order 2. However, the converse is not true; one can
prove that an s-structure of order 2 is generated by reflections with respect to (gener-
ally proper) subspaces of the affine normal spaces [56, 191].
Let M be a submanifold of
¯
M and p ∈ M.Thek-th osculating space O
k
p
Mof
Matpis the subspace of T
p
¯
M that is spanned by the first k derivativesin0of
202 Submanifolds and Holonomy
curves
γ
: (−
ε
,
ε
) → M with
γ
(0)=p. Note that O
1
p
M = T
p
M.Thek-th normal
space N
k
p
MofMatpis the orthogonal complement of O
k
p
M in O
k+1
p
M.So,for
instance, N
1
p
M = im(
α
p
) (see page 17). The submanifold M is called nicely curved
if for every k the dimension dim O
k
p
M is constant on M. I n this situation we can define
the k-th osculating bundle O
k
M and the k-th normal bundle N
k
M of M as the vector
bundle over M whose fiber at p is O
k
p
M and N
k
p
M, respectively. A metric connection
on N
k
M is defined by
∇
N
k
X
ξ
= proj
N
k
M
¯
∇
X
ξ
for all section X in TM and
ξ
in N
k
M, where proj is the orthogonal projection onto
N
k
M. We also can define higher order fundamental forms
α
k
by
α
k
(X,
ξ
)=proj
N
k
M
¯
∇
X
ξ
for all sections X in TM and
ξ
in N
k−1
M. Carfagna, Mazzocco and Romani proved
in [56 ] th e following character ization of 2-sym metric submanifolds of Euclidean
spaces and spheres, which was later proved for submanifolds in hyperbolic spaces
by Carfagna D’Andrea and Console in [55].
Theorem 6.2.9 A submanifold M of a space form
¯
Mis2-symmetric if and only if
∇
N
k
α
k
= 0 for all k ≥ 1.
Here, ∇
N
k
α
k
is defined in a natural way by using the Levi-Civita connection on
the tan gential part.
Let ∇
N
be the connection on the normal bundle
ν
M defined by ∇
N
X
ξ
= ∇
N
k
X
ξ
for
ξ
∈ N
k
, k ≥ 1 (and extended by linearity to any
ξ
∈
ν
M). We then define on
TM⊕
ν
M the connection
$
∇
N
= ∇ ⊕∇
N
,
and the tensor field
S
N
= ∇ ⊕∇
⊥
−
$
∇
N
.
Theorem 6.2.10 (Console [90]) A submanifold M of a space form
¯
Mis2-symmetric
if and only if S
N
is a homogeneous structure on M.
From Theorem 6.2.10 we see that a 2-symmetric submanifold M admits a homo-
geneous structure of type N . Actually, if M is a compact submanifold of R
n
,the
converse is also true.
Theorem 6.2.11 (Console [90]) Let M be a connected compact submanifold of R
n
.
Then M admits a homogeneous structure S ∈ N if and only if M is 2-symmetric.
Remark 6.2.12 (Historical note) S´anchez [291] was the first person to use the
canonical connection on the tangent bundle of a submanifold. This was in relation to
k-symmetric submanifolds (in the sense of S´anchez, see Remark 6.2.8). He proved
that a k-symmetric submanifold of R
n
can be characterized by the property that its
second fundamental form is parallel with respect to the canonical connection of a
k-symmetric space (generalizing Str¨ubing [301]).
Homogeneous Structures on Submanifolds 203
6.3 Isoparametric submanifolds of higher rank
In Section 4 .4 we introduced Thorbergsson’s Theorem 4.4.5, which states that
any irreducible full isoparametric submanifold of R
n
of rank at least three is an orbit
of an s-representation. The original proof by Thorbergsson in [319] uses Tits build-
ings and the Homogeneous Slice Theorem 4.3.6. There is an alternative proof of
Thorbergsson’s result by Olmos [256] using the theory of homogeneous structures
on submanifolds and normal holonomy. The idea is as follows. We know from The-
orem 6.2.5 that if there exists on a connected full compact submanifold M of R
n
a
canonical connection ∇
c
of type T ,thenM is an orbit of an s-representation. Given
an irreducible full isoparametric subm anifold of R
n
of codimension at least three,
we can focalize at the same time any two curvature distributions. The corresponding
fibers are, by the Homogeneous Slice Theorem, orbits of s-representations. A canon-
ical connection ∇
c
on M is constructed by gluing together the canonical connections
that occur naturally on these fibers. The proof of the compatibility between these
canonical connections is based on a relation between the normal holonomy g roups
of the different focal manifolds. The common eigenspaces of the shape operators
of M are parallel with respect to the canonical connection. This implies readily that
∇
c
α
= 0. To show that ∇
c
(∇−∇
c
)=∇
c
S
c
= 0 we have to use the geometric fact that
the ∇
c
-parallel transport alon g a horizontal cu rve with re spect to some focalization
equals the ∇
⊥
-parallel displacement in the focal manifold along the p rojection of the
curve.
6.3.1 The canonical connection on orbits of s-representations
To motivate the definition of canonical connection we discuss what properties we
would like it to have. With this in mind, we start with a homogeneous isoparametric
submanifold M.ThenM is a principal orbit K ·v of an s-representation, that is, of
the isotropy representation of a symmetric space G/K. Recall from Section 2.3 that
there is an orthogonal reductive decomposition k = k
v
⊕k
+
,wherek
+
=
∑
α
∈Ψ
+
k
α
and k
α
= {x ∈ k :ad(H)
2
x =
α
2
(H)x for all H ∈ a}. Recall also from Section 2.7
that the common eigenspaces of the shape operators of M are E
α
= p
α
+ p
2
α
,where
p
2
α
= {0} if 2
α
is not a restricted root. In particular, they correspond to positive
roots
α
of {ad(H)
2
: H ∈ a}.
Consider a focal orbit M
ξ
= K ·u with u = v +
ξ
(v) and define
¯
E = ker(A
ξ
−id).
Note that
¯
E corresponds to a sum f of restricted root spaces. Then k
u
= k
v
⊕f and k =
k
u
⊕n are r eductive decomposition of k
u
and k, respectively, where n is the orthogonal
complement of k
u
in k.
Proposition 6.3.1 Let M be an orbit of an s-representation, ∇
c
the canonical con-
nection on M associated with the reductive decomposition k = k
u
⊕n and
¯
∇
c
the
204 Submanifolds and Holonomy
canonical connection on M
ξ
associated with the reductive decomposition k
u
= k
v
⊕f.
Then ∇
c
and
¯
∇
c
have the following properties:
(1) The curvature distributions on M are ∇
c
-parallel. (This is due to the fact that
∇
c
α
= 0.)
(2) If X lies in an eigenspace or a sum of eigenspaces of {ad(H)
2
: H ∈ a} corre-
sponding to a curvature distribution E
i
=
¯
E, and if w ∈
¯
E, then
(a)
γ
(t)=Exp(tX)·visa∇
c
-geodesic in M and dExp(tX )(w) is ∇
c
-parallel
in M along
γ
;
(b)
¯
γ
(t)=Exp(tX) ·uisa∇
c
-geodesic in M
ξ
and dExp(tX )(w) is ∇
⊥
-
parallel in M
ξ
along
¯
γ
.
Next, we suppose that u = u
i
= v+
ξ
i
(v),where
ξ
i
is a parallel normal vector field
focalizing only E
i
(so that E
i
= ker(A
ξ
i
−id).Letn
i
be the curvature normal relative
to E
i
and V
i
(v) be the affine subspace through v parallel to the vector space spanned
by n
i
(v) and E
i
(v).Wedefine
˜
K
i
=
{k|
V
i
(v)
: k ∈ K and k ·S
i
(v)=S
i
(v)}
o
and
K
i
=
{k|
V
i
(v)
: k ∈ K
u
i
}
o
.
Clearly, we have
˜
K
i
⊂ K
i
. We claim th at K
i
⊂
˜
K
i
,andso
˜
K
i
= K
i
.Infact,ifM
u
i
is
full, then K
i
is the restricted normal holonomy group of M
u
i
at u
i
and so K
i
⊂
˜
K
i
follows from the Homogeneous Slice Theorem. If M
u
i
is not full, it is a factor of M
and obviously in this case K
i
=
˜
K
i
.
The representation of K
i
on V
i
(v) is an s-representation by the Normal Holonomy
Theorem. We can thus construct, as above, a canonical connection ∇
i
on S
i
(v).This
connection is associated with the reductive decomposition of the Lie algebra o f K
i
with reductive complement given by a sum of root spaces of {ad(H)
2
: H ∈a} corre-
sponding to E
i
. Thus, ∇
i
coincides with the connection induced by ∇
c
on the totally
geodesic submanifold S
i
(v).
6.3.2 The canonical connection on isoparametric submanifolds of rank
at least three
We first show how to focalize simultaneously two curvature distributions on an
irreducible full isoparametric submanifold M of R
n
with codim M ≥3. Let E
1
,...,E
g
be the curvature distributions on M (consisting of the common eigenspaces of the
shape operators on M)andn
1
,...,n
g
be the corresponding curvature normals. Let
ξ
i
be a parallel normal vector field on M that fo calizes only the curvature distribution
E
i
and let L
ij
be the span of n
i
and n
j
. Observe that L
ij
is a parallel subbundle of the
normal bundle
ν
M.
If we choose a parallel normal vector field
ξ
ij
on M with the property that
ξ
ij
,n
k
= 1 if and only if n
k
∈ L
ij
,
Homogeneous Structures on Submanifolds 205
then
ξ
ij
focalizes both E
i
and E
j
.Inotherwords,
S
ij
= ker(A
ξ
ij
−id)
is an autoparallel distribution on M that contains E
i
⊕E
j
if i = j .LetS
ij
(p) be the
maximal integral manifold of S
ij
through p, which, as we know, can be regarded
both as a totally geodesic submanifold of M an d as a compact full isoparametric
submanifold o f the affine subspace
V
ij
(p)=p + S
ij
(p) ⊕L
ij
(p),
of R
n
with curvature normals {n
k
|
S
ij
(p)
: n
k
∈ L
ij
}. The rank of L
ij
(p) is one if i = j
and two if i = j. By the Homogeneous Slice Theorem, S
ij
(p) is homogeneous under
the normal holonomy group of the focal manifold M
ξ
ij
.So,S
ij
(p) is an orbit of an
s-representation.
Consider now the submersions
π
ij
: M → M
ξ
ij
and
π
i
: M → M
ξ
i
. Observe that
ξ
ij
−
ξ
i
is constant on S
i
(p) since
A
i
ξ
ij
−
ξ
i
= A
ξ
ij
−
ξ
i
|
TS
i
(p)
= 0,
where A
i
is the shape operator of S
i
(p). Thus
ξ
ij
−
ξ
i
defines locally (on some open
neighborhood U ⊂ M
ξ
i
) a parallel normal vector field on M
ξ
i
, which we will denote
by
η
ij
.
Let us fix
η
ij
(
π
i
(p)) and consider the singular holonomy tube (M
ξ
i
)
η
ij
(
π
i
(p))
of
M
ξ
i
relative to the normal vector
η
ij
(
π
i
(p)). Observe that this singular holonomy tube
coincides (locally) with the parallel focal manifold (M
ξ
i
)
η
ij
to M
ξ
i
. Because a f ull
focal manifold of an ir reducible isoparametric submanifold determines the foliation
(see Corollary 4.3.7 of Homogeneous Slice Theorem 4.3.6 and Exercise 6.4.1) we
have that M
ξ
ij
=(M
ξ
i
)
η
ij
(
π
i
(p))
(and also M
ξ
i
=(M
ξ
ij
)
−
η
ij
(
π
i
(p))
).
Locally (on U ) we also have a submersion p
ij
: M
ξ
i
⊃U → p
ij
(U) ⊂M
ξ
ij
defined
by s → s +
η
ij
(s).Sowehave
p
ij
◦
π
i
=
π
ij
.
Remark 6.3.2 If
γ
is a horizontal curve in M with respect to
π
ij
,then
γ
is also
horizontal with respect to
π
i
and
π
i
◦
γ
is horizontal with respect to p
ij
.
We can now define a connection on M, which will turn out to be canonical. Let
∇
ij
q
be the canonical connection on S
ij
(q) naturally induced by the restricted normal
holonomy group of M
ξ
ij
(which acts as an s-representation). Denote by D
ij
= ∇−∇
ij
q
the corresponding homogeneous structure of type T . For the sake of simplicity, we
will still denote by D
ij
the value of the tensor field D
ij
at q. Recall that D
ij
X
is a
skewsymmetric endomorphism for all X ∈ T
q
S
ij
(q). We now decompose X ,Y ∈ T
q
M
as
X =
g
∑
i=1
X
i
and Y =
g
∑
i=1
Y
i
with X
i
,Y
i
∈ E
i
(q)
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