Normal Holonomy of Complex Submanifolds 221
Proof From Proposition 7 .3.6 we obtain that
π
(H
ξ
(x)) is a submanifold with con-
stant principal curvatures. If
π
(H
ξ
(x)) is reducible or non-full, then N would be
reducible or non-full since
N =
!
y∈
π
(Σ
ξ
(x))
(
π
(H
ξ
(x)))
y−
π
(x)
(locally)
and
π
(Σ
ξ
(x)) ∼
locally
(
π
(
˜
ν
ξ
))
π
(x)
=
ν
0
(
π
(H
ξ
(x))) (see Lemma 7.3.5). But the normal
holonomy group of
π
(H
ξ
(x)) is irreducible and non-transitive (on the unit sphere in
the orthogonal complement of its set o f fixed vectors). Then, using Thorbergsson’s
Theorem 4.4.5, we see that
π
(H
ξ
(x)) is a focal manifold of a homogeneous isopara-
metric submanifold (we have used that an isoparametric submanifold is always con-
tained in a complete one [275]). Then there exists a compact group of isometries K
of R
n
, acting as the isotropy representation of a simple Riemannian symmetric space,
such that (locally) K ·
π
(x)=
π
(H
ξ
(x)). This holds for any fixed x. But, for x
= x,
π
(H
ξ
(x
)) is a parallel manifold to
π
(H
ξ
(x)) in the ambient space. Since the group
K gives the parallel transport in
ν
0
(K ·x) (see Proposition 2.3.5), we conclude that
K ·
π
(x)=
π
(H
ξ
(x)) for all x ∈M. It is clear that K is not transitive on the unit sphere
and so the corresponding symmetric space is of rank ≥ 2.
We summarize in the following theorem the main result of this section, which
will be of a great importance for our subsequent investigations.
Theorem 7.3.8 Let N be a submanifold of R
n
and assume that its normal holonomy
group acts irreducibly on the normal space and non-transitively on the unit sphere
in the normal space. Let
η
q
∈
ν
q
N be a principal vector for the normal holonomy
action of Φ
q
on
ν
q
N. Let us consider the normal holonomy tube M =(N)
η
q
,which
has flat normal bundle (
η
q
short, in a neighborhood of a generic point q).
Assume that there exist two nontrivial parallel normal vector fields
ξ
,
ξ
on (N)
η
q
such that
H ⊂ ker(A
M
ξ
)+ker(A
M
ξ
),
where H is the horizontal distribution of the holonomy tube M (we assume, since we
are working locally, that the sum on the right hand side and its two terms are smooth
distributions).
Then there exists a compact subgroup K of I(R
n
), acting as the isotropy repre-
sentation of an irreducible Riemannian symmetric space with rank ≥ 2, such that
N =
!
v∈(
ν
0
(K·q))
q
(K ·q)
v
(near q),
that is, N is locally the union of the parallel orbits to K ·q. Moreover, (
ν
0
(K ·q))
q
is
contained in the nullity space N
N
q
of the second fundamental form
α
N
at q.
Remark 7.3.9 We keep the assumptions in Theor e m 7.3.8.
222 Submanifolds and Holonomy
(i) The orbit K ·q is not isoparametric, because otherwise,
R
n
=
!
v∈
ν
q
(K·q)
(K ·q)
v
=
!
v∈(
ν
0
(K·q))
q
(K ·q)
v
= N.
(ii) Observe that dim
ν
0
(K ·x) ≥ 1 since the position vector field, from the fixed
point of K, gives a parallel normal vector field. So the nullity space is non-
trivial. Note that
!
v∈
ν
0
(K·p)
(K ·q)
v
is globally never a subm anifold since there are always focal parallel orbits. We
will come back to the discussion about completeness of N in the case when N
is a complex submanifold of C
n
.
7.4 Applications to complex submanifolds of C
n
with
nontransitive normal holonomy
Let N be a complex submanifold of C
n
, not necessar ily complete, which is locally
irreducible and full. The standard complex structur e on C
n
is denoted by J. We denote
by A
N
the shape operator of N.
It is a well-known result from complex geometry that A
N
J
ξ
= −JA
N
ξ
holds for all
normal vector fields
ξ
on N. Moreover, J anti-commutes with all shape operators A
N
ξ
.
Then, if N is full, there are no parallel normal vector fields on N. In fact, if
ξ
is a
parallel normal vector field on N,then
0 = R
⊥
x,y
J
ξ
,
ξ
=[A
N
J
ξ
,A
N
ξ
]=A
N
J
ξ
A
N
ξ
−A
N
ξ
A
N
J
ξ
= −JA
N
ξ
A
N
ξ
+ A
N
ξ
JA
N
ξ
= −2J(A
N
ξ
)
2
.
(7.3)
Then A
N
ξ
= 0 and hence
ξ
= 0, since N is full (see Remark 1.5.2). This implies that
the restricted normal holonomy group of N has no nonzero fixed vectors if N is full.
It is remarkable that for the normal holonomy group of complex submanifolds
of C
n
there is a de Rham type decomposition result due to Di Scala [104, Corollary
4.2].
Theorem 7.4.1 (Di Scala) The normal holonomy group of a locally irreducible and
full submanifold of C
n
acts irreducibly on the normal space.
Let Φ
∗
q
be the normal holonomy group at q ∈ N, which acts by complex transfor-
mations on
ν
q
N. Choose
ξ
1
q
∈
ν
q
N such that the orbit Φ
∗
q
·
ξ
1
q
projects down to the
(unique) complex orbit in the (complex) projectivization P(
ν
q
N) of the normal space
Normal Holonomy of Complex Submanifolds 223
ν
q
N (see [293]). This implies that the orthogonal complement (R
ξ
1
q
)
⊥
∩
ν
ξ
1
q
(Φ
∗
q
·
ξ
1
q
)
of R
ξ
1
q
in the normal space of the holonomy orbit is a complex subspace of
ν
q
N.
Since Φ
∗
q
is not transitive on the sphere, (R
ξ
1
q
)
⊥
∩
ν
ξ
1
q
(Φ
∗
q
·
ξ
1
q
) is a nontrivial sub-
space of
ν
q
N.
Now choose 0 =
ξ
2
q
∈ (R
ξ
1
q
)
⊥
∩
ν
ξ
1
q
(Φ
∗
q
·
ξ
1
q
).SinceR
⊥
X,Y
always lies in the hol-
onomy algebra, we g et 0 = R
⊥
X,Y
ξ
1
q
,
ξ
2
q
. So, by the Ricci identity, [A
N
ξ
1
q
,A
N
ξ
2
q
]=0.
The same is true if we replace
ξ
2
q
by J
ξ
2
q
.SoA
N
ξ
1
q
also commutes with A
N
J
ξ
2
q
. Then, by
the same argument as in (7.3), we obtain
A
N
ξ
1
q
A
N
ξ
2
q
= 0 = A
N
ξ
2
q
A
N
ξ
1
q
. (7.4)
We may assume that the slice representation orbit
(Φ
∗
q
)
ξ
1
q
·
ξ
2
q
is a principal orbit in the normal space of the holonomy orbit Φ
∗
q
·
ξ
1
q
,where(Φ
∗
q
)
ξ
1
q
is the isotropy group of Φ
∗
q
at
ξ
1
q
. Note that we can find such a
ξ
2
q
perpendicular to
ξ
1
q
,since
ξ
1
q
is a fixed point for the slice representation of (Φ
∗
q
)
ξ
1
q
.
Note that, by construction, we also have that
A
N
τ
⊥
c
(
ξ
1
q
)
A
N
τ
⊥
c
(
ξ
2
q
)
= 0,
where
τ
⊥
c
is the normal parallel transport along an arbitrary curve c in N which starts
at q.
Consider the iterated holonomy tube
((N)
ξ
1
q
)
ξ
2
q
,
which coincides with the full holonomy tube (N)
ζ
q
,where
ζ
q
=
ξ
1
q
+
ξ
2
q
(see Theo-
rem 4.4.12 or appendix of [256]). For this purpose we h ave to choose
ξ
1
q
sufficiently
small and after that also
ξ
2
q
sufficiently small. The vector
ξ
1
q
gives rise to a paral-
lel normal vector field
˜
ξ
on the partial holonomy tube (N)
ξ
1
q
so that ((N)
ξ
1
q
)
˜
ξ
= N.
This parallel normal vector field can be lifted to a parallel normal vector field
ξ
on
((N)
ξ
1
q
)
ξ
2
q
=(N)
ζ
q
. We can do this, since
˜
ξ
(x) is fixed by the normal holonomy group
of (N)
ξ
1
q
at x and hence it is perpendicular to any holonomy orbit. Similarly,
ξ
2
q
gives
rise to a parallel normal vector field
ξ
on ((N)
ξ
1
q
)
ξ
2
q
=(N)
ζ
q
.
From (7.4) and the tube formula relating shape operators of parallel focal mani-
folds we obtain
A
M
ξ
A
M
ξ
|
H
= 0 = A
M
ξ
A
M
ξ
|
H
where M =(N)
ζ
q
and H is the horizontal distribution on M.SoA
M
ξ
|
H
and A
M
ξ
|
H
are simultaneously diagonalizable and
H ⊂ ker(A
M
ξ
)+ker(A
M
ξ
).
From Theorem 7.3.8 we can then deduce the following result.
224 Submanifolds and Holonomy
Theorem 7.4.2 Let N be a complex, locally irreducible, full (proper) submanifold
of C
n
such that the normal holonomy group, which must act irreducibly by Theorem
7.4.1, is not transitive on the unit sphere of the normal space. Then there exists a
compact Lie group K, acting as the isotropy representation of an irreducible Hermi-
tian symmetric space with rank ≥ 3, such that N is locally given, around a generic
point q, as
N =
!
v∈(
ν
0
(K·q))
q
(K ·q)
v
.
Moreover, (
ν
0
(K ·q))
q
is contained in the nullity space N
N
q
of the second fundamen-
tal form
α
N
of N at q.
Proof It remains only to show that K is of Hermitian type of rank at least 3. We
may assume that the origin 0 ∈ C
n
is the fixed point of K.Ifp ∈ N, then the position
vector p, by the description given above, belongs to T
p
N.So,Jp ∈ T
p
N. Then the
orbits of the S
1
-action (t,x) → e
it
x on C
n
are tangent to N at the points in N. In fact,
d
dt
t=0
e
it
p = Jp ∈ T
p
N. This implies that N is (locally) S
1
-invariant. Let now
¯
K be
the subgroup of linear isometries of C
n
generated by K and S
1
.Then
¯
K · p ⊂ N and
so
¯
K is not transitive on the sphere, because otherwise N = C
n
. By Simons’ Theorem
3.3.7 (see Remark 8.3.5) and since K acts irreducibly, we get
¯
K = K and so K is of
Hermitian type.
The normal holonomy group of K ·q at q, restricted to the orthogonal complement
of (
ν
0
(K ·q))
q
, coincides with the normal holonomy group of N.Infact,thisisa
consequence of the fact that R
⊥
X,Y
= 0ifX is tangent to the K-orbits in N and Y is
perpendicular to these orbits (see the arguments in the proof of Lemma 7.3.5). If the
rank of the symmetric space is 2, then the normal holonomy group of the singular
orbit K ·q would be transitive (on the orthogonal complement of the position vector).
So the rank is at least 3.
From the above local theorem we obtain the following corollary in [93] for com-
plete submanifolds.
Theorem 7.4.3 (Console, Di Scala, Olmos) The restricted normal holonomy group
Φ
∗
of a complete, irreducible, full immersed complex submanifold of C
n
acts tran-
sitively on the unit sphere of the normal space. Indeed, Φ
∗
= U
k
, where k is the
codimension of the submanifold.
Proof If the normal holonomy group of N is not transitive on the unit sphere, then,
locally,
N =
!
v∈(
ν
0
(K.q))
q
(K.q)
v
,
where K acts irreducibly as in Theorem 7.4.2 (we may assume that 0 is the fixed
point of K). Recall that we assume N to be complete (not necessarily injectively
immersed). So, if p ∈ N and since N is real analytic, then the line Rp is contained in
N (that is, this line is the image, via the immersion, of a geodesic in N). In order to
simplify the notation we omit the immersion map. By construction we have T
tp
N =
Normal Holonomy of Complex Submanifolds 225
T
p
N for all t = 0, as subspaces of C
n
. But the same must be true for the limit point,
that is, T
0
N = T
p
N. So the isotropy group K
0
= K must leave this subspace invariant.
This is a contradiction, since K acts irreducibly. Thus, the normal holonomy group
must be transitive.
7.5 Applications to complex submanifolds of CP
n
with
nontransitive normal holonomy
7.5.1 Complex submanifolds of CP
n
The complex projective space CP
n
is the quotient of C
n+1
{0} by the equiv-
alence relation that identifies points in the same complex line through the origin
0 ∈ C
n+1
. We denote the projection by
π
: C
n+1
{0}→CP
n
. We can also regard
CP
n
as a quotient of the unit (2n + 1)-dimensional unit sphere in C
n+1
by the stan-
dard action of U
1
,thatis,CP
n
= S
2n+1
/U
1
(since every line in C
n+1
intersects the
unit sphere in a circle; for n = 1 this construction gives the classical Hopf bundle
S
3
→ S
2
). So we also have a submersion
˜
π
: S
2n+1
→ CP
n
. We equip CP
n
with the
usual Fubini-Study Riemannian metric ·, ·
FS
that turns
˜
π
into a Riemannian sub-
mersion. This metric has constant holomorphic sectional curvature 4.
Let M be a full complex submanifold of CP
n
. Here, full means that M is not
contained in a proper totally geodesic submanifold of CP
n
.Let
˜
M =
π
−1
(M) and V
be the vertical distribution on
˜
M that is induced from the submersion
π
|
˜
M
:
˜
M → M.
It is standard to show that V ⊂ N
˜
M
,whereN
˜
M
is the relative nullity distribution,
that is, the union of the nullity spaces of the second fundamental form
α
˜
M
of
˜
M in
C
n+1
,
N
˜
M
q
=
(
v ∈ T
q
˜
M :
α
˜
M
(·,v)=0
)
=
ξ
∈
ν
q
˜
M
ker A
˜
M
ξ
.
If X is a tangent vector field on M we will write
˜
X for its horizontal lift to C
n+1
{0}.
The submersion
π
: C
n+1
{0}→CP
n
is not Riemannian, but the following
O’Neill type formula holds:
Proposition 7.5.1 (O ’Neill type formula) Let X,Y b e vector fields on CP
n
and
˜
X,
˜
Y
be their horizontal lifts. Then, if ∇ and
ˆ
∇ denote the Levi-Civita connections on C
n+1
and CP
n
respectively, we have
∇
˜
X
˜
Y =
*
ˆ
∇
X
Y + O(
˜
X,
˜
Y ) (7.5)
where O(
˜
X,
˜
Y ) ∈ V is vertical.
The proof is the same as for the Riemannian submersion case, see [267, 268].
Indeed, the restriction d
π
: V
⊥
→ T CP
n
is a dilation, that is,
π
∗
·,·
FS
=
λ
2
·,·,
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