206 Submanifolds and Holonomy
and define a tensor field D on M by
D
X
Y =
g
∑
i, j=1
D
ij
X
i
Y
j
.
Also here we write D for the value of the tensor field D at q for the sake of simplicity.
Let u s discuss this definition in more detail. If i = j,sayi = j = 1, then we
operate on the curvature sphere S
1
(q),whichforanyk ≥ 2 is a totally geodesic
submanifold of the orbit S
1k
(q) of the s-representation. Take, for instance, k = 2.
From what we wrote in Section 6.3.1 we know that the connection on S
1
(q) induced
by the canonical connection ∇
12
on S
12
(q) coincides with the connection relative
to the isotropy representation of the isotropy group Φ
12
z
, z =
ξ
1
(q) −
ξ
12
(q),onthe
normal space
ν
z
(Φ
12
·z),whereΦ
12
is the restricted normal holonomy group of M
ξ
12
.
On the other hand, M
ξ
1
can be regarded as holonomy tube of M
ξ
12
with respect
to z =
ξ
1
(q) −
ξ
12
(q) and, by Theorem 4.4.12, the normal holonomy of M
ξ
1
at z
is the image under the slice representation of Φ
12
z
in
ν
z
(Φ
12
·z). Thus, the canonical
connection ∇
11
= ∇
1
on S
1
(q), regarded as an orbit of the normal holonomy group of
M
ξ
1
, coincides with ∇
12
. This is the key point for the proof of the following lemma:
Lemma 6.3.3 If X,Y ∈T
q
S
ij
(q),thenD
X
Y = D
ij
X
Y.
Proof If, as above, we decompose X,Y as sums of vectors in curvature distributions,
then D
X
Y decomposes into a sum of terms of type D
kl
X
k
Y
l
(k = l) and of type D
kk
X
k
Y
k
.
Observe that the indices k,l are not arbitrary but correspond to curvature distributions
contained in S
ij
(q). In particular, if k = l,thenS
ij
(q)=S
kl
(q).Forthefirst kind of
term, there is clearly no problem. For the second kind of term, by what we remarked
above, D
kk
X
k
Y
k
= D
ij
X
k
Y
k
. This proves the lemma.
As a consequence we see that D
X
is a skewsymmetric endomorphism of T
q
M and
hence it determines a metric connection on M by
∇
c
= ∇ −D.
Consider the connection on TM⊕
ν
M givenbythesumof∇
c
and ∇
⊥
. We denote
this connection also by ∇
c
.
Remark 6.3.4 From Lemma 6.3.3 we see that S
ij
(q) is a totally geodesic submani-
fold of M with respect to the connection ∇
c
.
Thorbergsson’s Theorem 4.4.5 follows now from Theorem 6.2.5, the results on
homogeneous isoparametric submanifolds in Section 4.4 and the following result:
Theorem 6.3.5 ∇
c
is a canonical connection of type T on M.
Proof We must prove that ∇
c
α
= 0and∇
c
D = 0.
For ∇
c
α
= 0, it suffices to show by Exercise 6.4.2 that every curvature distri-
bution E
i
on M is ∇
c
-parallel. By the previous remark, S
ij
(q) is a totally geodesic
Homogeneous Structures on Submanifolds 207
submanifold of M with respect to ∇
c
. Moreover, E
i
is ∇
c
-parallel in S
ij
(q) because
the second fundamental form of S
ij
(q) is ∇
c
-parallel. Thus ∇
c
E
j
E
i
⊂ E
i
and, since j
is ar bitrary, ∇
c
E
i
⊂ E
i
,thatis,E
i
is ∇
c
-parallel.
It remains to prove that ∇
c
D = 0. If X ,Y,Z ∈ T
q
S
ij
(q) then (∇
c
X
D)
Y
Z =
(∇
ij
X
D
ij
)
Y
Z = 0, because S
ij
(q) is an orbit of an s-representation and ∇
c
is its canoni-
cal connection. Let u s assume that i = j and k is chosen so that E
k
(q) is not contained
in T
q
S
ij
(q).
We first remark that (∇
c
X
D)
Y
Z = 0, with X ,Y ∈T
q
S
ij
(q) and Z ∈E
k
(q), is equiva-
lent to the property that for any ∇
c
-geodesic
γ
k
in S
k
(q) with
γ
k
(0)=q and
˙
γ
k
(0)=Z
we have
D
τ
c
X
τ
c
Y =
τ
c
D
X
Y,
where
τ
c
denotes the ∇
c
-parallel transport along
γ
k
from q to p =
γ
k
(1).Now,to
prove ∇
c
D = 0 using the above reformulation, it is crucial to establish the relation
between pa rallel transport in a norma l direction on a focal manifold and parallel
transport (with respect to the canonical connection) on an orbit of an s-representation
(see Proposition 6.3.1), and also comparing parallel transport in normal directions
(Lemma 3.4.6).
Let us consider the focal maps
π
i
: M → M
ξ
i
and
π
ij
: M → M
ξ
ij
and the local
submersion p
ij
of M
ξ
i
on M
ξ
ij
(regarded as holonomy tube of M
ξ
i
). Let X
i
∈ E
i
(q)
and observe that its ∇
c
-parallel transport
$
X
i
(t) along
γ
k
can be done in S
ki
(q) (which
contains S
k
(q)). S
ki
(q) focalizes in M
ξ
i
into (S
ki
(q))
ξ
i
and, by Proposition 6.3.1,
$
X
i
(t)
is ∇
⊥
-parallel in (S
ki
(q))
ξ
i
along
π
i
◦
γ
k
(t). Moreover, by Lemma 3.4.6,
$
X
i
(t) is still
∇
⊥
-parallel along p
ij
◦(
π
i
◦
γ
k
)(t)=
π
ij
◦
γ
k
(t) in M
ξ
ij
. We can repeat this argument
for any E
h
(q) ⊂T
q
S
ij
(q) to obtain that for any X ∈T
q
S
ij
(q),
˜
X(t) is ∇
c
-parallel along
γ
k
(t) if and only if it is ∇
⊥
-parallel alo ng
π
ij
◦
γ
k
(t).Inotherwords,onS
ij
(q),the
∇
c
-parallel transport
τ
c
along
γ
k
from q to p agrees with the ∇
⊥
-parallel transport
τ
⊥
along
π
ij
◦
γ
k
(t). By the Homogeneous Slice Theorem and the fact that d
τ
⊥
sends
the normal holonomy of M
ξ
ij
at q to the normal holonomy of M
ξ
ij
at p,wehavethat
d
τ
⊥
(D
q
)=D
p
. Thus, if X,Y ∈ T
q
S
ij
(q),wehave
D
τ
c
X
τ
c
Y = D
τ
⊥
X
τ
⊥
Y =
τ
⊥
D
X
Y =
τ
c
D
X
Y,
and therefore ∇
c
D = 0.
Remark 6.3.6 A proof of Thorbergsson’s Theorem that relies on normal holonomy,
but avoids the use of homogeneous structures, follows from the remarkable result
of Heintze and Liu [144] about homogeneity of infinite-dimensional isoparametric
submanifolds.
208 Submanifolds and Holonomy
6.4 Exercises
Exercise 6.4.1 Let F be the parallel singular foliation on R
n
{0} induced by a
compact irreducible isoparametric submanifold . Let M and N be two leaves of F ,
p ∈ M and choose q ∈ N such that
ξ
= q − p is normal to M at p. Prove that N
coincides with the holonomy tube (M)
ξ
. Deduce that −
ξ
= q − p is normal to N at
q and M is the holonomy tube (N)
−
ξ
.
Exercise 6.4.2 Let M be an isoparametric submanifold of R
n
. Prove that a connec-
tion ∇
c
on M with (∇
c
)
⊥
= ∇
⊥
is canonical if and only if any curvature distribution
of the shape operator of M is parallel with respect to ∇
c
.
Exercise 6.4.3 (see [89, 90] ) Let S be a homogeneous structure o n a submanifold M
of R
n
.Letp ∈M and consider the pair (S
p
,
α
p
),where
α
p
is the second fundamental
form of M at p. The purpose of this exercise is to describe how one can associate with
(S
p
,
α
p
) a subalgebra g of the Lie algebra of I(R
n
) and a homogeneous submanifold
of R
n
that contains M as an open subset.
To simplify the notation, we put S = S
p
and
α
=
α
p
.Let
γ
(t) be a curve in M
with
γ
(0)=p and put x =
˙
γ
(0). Denote by
τ
γ
(t )
the linear isometry from T
p
¯
M =
T
p
M ⊕
ν
p
M to T
γ
(t )
¯
M = T
γ
(t )
M ⊕
ν
γ
(t )
M determined by the parallel transport with
respect to
$
∇ along
γ
.Foreveryt there exists a unique isometry F
t
∈I(
¯
M) such that
F
t
(p)=
γ
(t) and d
p
F
t
=
τ
γ
(t )
.
(a) Prove that the tangent vector at id to the curve t → F
t
∈I(R
n
) gives an element
Ψ
x
in the Lie algebra of I(
¯
M) that has the following expression in terms of S
and
α
:LetY ∈R
n
= T
p
M ⊕
ν
p
M and set Y (t)=
τ
γ
(t )
Y . The linear part LΨ
x
of
the transformation Ψ
x
is given by
LΨ
x
Y = S
x
Y +
α
(x,Y
) −A
Y
⊥
x = Γ
x
Y
where Y
(resp. Y
⊥
) is the tangential (resp. normal) component of Y .The
translational part of Ψ
x
is p → x.
(b) Let k be the Lie algebra spanned by the operators
$
R
xy
and m be the span of the
transformations Ψ
x
.Set
g = k ⊕m.
Prove that g is a subalgebra of the Lie algebra of I(R
n
) with Lie bracket
[Ψ
x
,Ψ
y
]=Ψ
S
x
y−S
y
x
+
$
R
xy
,
[
$
R
xy
,Ψ
z
]=Ψ
$
R
xy
z
,
[
$
R
xy
,
$
R
zw
]=
$
R
$
R
xy
zw
+
$
R
z
$
R
xy
w
.
(Note that [
$
R
xy
,
$
R
zw
]=
$
R
$
R
xy
zw
+
$
R
z
$
R
xy
w
follows from
$
R
xy
·
$
R = 0, which is a
Homogeneous Structures on Submanifolds 209
consequence, or more precisely, an integrability condition, for the equations
$
∇
α
= 0and
$
∇S = 0.)
(c) Let G be the connected Lie subgroup of I(
¯
M) with Lie algebra g. Prove that
$
M = G ·p is a homogeneous submanifold R
n
that contains M as an open subset.
[Hint: Modify the proof of Lemma 6.1.10.]
(d) Gener alize all this to a submanifold of a space form.
Exercise 6.4.4 (cf. [193]) Let M = K ·v be a p rincipal orbit of an s-representation
(that is, a homogeneous isoparametric submanifold o f R
n
) corresponding to a sym-
metric space G/K with a reduced root system (that is, if
α
is a root, then 2
α
is not
a root). Prove that the canonical connection ∇
c
associated with the reductive decom-
position k = k
v
⊕k
+
(see Section 2.3 and Section 6.3.1) agrees with the projection
connection
ˆ
∇ defined by projecting the Levi-Civita connection onto the various cur-
vature distributions E
i
,thatis,
ˆ
∇
X
Y =
g
∑
i=1
(∇
X
Y
i
)
i
,
where (·)
i
denotes the projection onto E
i
.
Exercise 6.4.5 Let M = K ·v be a principal orbit of an s-representation (that is, a
homogeneous isoparametric submanifold) corresponding to a symmetric space G/K
with a reduced root system. Let
ˆ
S be the homogeneous structure relative to the pro-
jection connection
ˆ
∇ in Exercise 6.4.4. Show that
ˆ
S
x
v = −(id −A
ξ
)
−1
(∇
⊥
x
A)
ξ
v for all x ∈ (E
i
)
p
, v ∈(E
j
)
p
, i = j
and
ˆ
S
x
v = 0forallx, v ∈ (E
i
)
p
.
Using Exercise 6.4.3, deduce that M is uniquely determined by the values at p of the
second fundamental form and its covariant derivative.
Exercise 6.4.6 Let M be a connected embedded submanifold of R
n
and let G =
{g ∈ I(R
n
) : g(M)=M} be its family of extrinsic isometries. Assume that G acts
transitively on M.Letp ∈ M and assume that the subgroup
H = {g ∈ G
p
: d
p
g|
ν
p
M
= id
ν
p
M
}
of the isotropy group G
p
has no fixed points in T
p
M.Let
g = k ⊕m
be a r eductive decomposition of the Lie algebra g of G,wherek is the Lie algebra
of G
p
and m is an Ad(G
p
)-invariant subspace of g (G is not assumed to be con-
nected). Let S be the homogeneous structure on M associated with this reductive
210 Submanifolds and Holonomy
decomposition (observe that g is also the Lie algebra of the identity component G
o
of G, but it is important for the applications to extrinsic k-symmetric submanifolds
in the sense of S´anchez not to assume that G is connected). Prove that S is of type
T and so M is an orbit of an s-representation. [Hint: If h ∈ H,thenS
X
ξ
,
η
=
S
d
p
h(X)
d
p
h(
ξ
),d
p
h(
η
) = S
d
p
h(X)
ξ
,
η
for all X ∈ T
p
M and
ξ
,
η
∈
ν
p
M. Hence,
S
d
p
(h−id)(X)
ξ
,
η
= 0. So, S
X
ξ
,
η
= 0ifX is perpendicular to the fixed vectors o f
h in T
p
M.]
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