The Normal Holonomy Theorem 103
As a linear operator, or more precisely as an element of so(
ν
p
M),wehave
R
⊥
(u,v)=−
1
2
d
2
dt
2
t=0
τ
⊥
γ
t
.
Note that
τ
⊥
γ
t
is a curve in the normal holonomy group Φ
p
that is the identity at t = 0
and has zero velocity. So the second derivative
d
2
dt
2
t=0
τ
⊥
γ
t
= −2R
⊥
(u,v) makes sense
and belongs to the Lie algebra L (Φ
p
) of the normal holonomy group Φ
p
. Moreover,
if
γ
is a curve from p to q and x,y ∈ T
q
M,then(
τ
⊥
γ
)
−1
R
⊥
(x,y)
τ
⊥
γ
∈ L (Φ
p
).
In a suitable sense, the holonomy algebra is spanned by the curvature tensors
produced in this way. This is due to a remarkable result by Ambrose and Singer
(which holds for the holonomy of any linear connection on a vector bundle; see, for
instance, [178] or [42], [279]).
Theorem 3.1.9 (Ambrose-Singer Holonomy Theorem) Let M be a connected
submanifold of a standard space form
¯
M
n
(
κ
). The normal holonomy algebra L (Φ
p
)
is the subalgebra of so(
ν
p
M) generated by the endomorphisms
τ
⊥−1
γ
R
⊥
(x,y)
τ
⊥
γ
,
where
τ
⊥
γ
is the ∇
⊥
-parallel transport on M along a piecewise differentiable curve
γ
: [0,1] → M starting from p and x,y ∈ T
γ
(1)
M.
For a proof of this theorem see Exercise 10.6.4.
Remark 3.1.10 The Ambrose-Singer Holonomy Theorem implies that the Lie alge-
bra of Φ
p
can also be constructed as linear span of
{
τ
⊥−1
γ
R
⊥
(
τ
γ
x,
τ
γ
y)
τ
⊥
γ
}
where
τ
γ
is the tangential parallel transpo rt on M along a piecewise differentiable
curve
γ
: [0,1] → M starting from p and x, y ∈ T
p
M (actually, the tangential connec-
tion is irrelevant).
For the local study of the geometry of submanifolds we will use the local normal
holonomy group. The local normal holonomy group Φ
loc
p
at p is defined as intersec-
tion of all normal holonomy groups Φ
∗
p
(U),whereU runs through all open neigh-
borhoods of p. Observe that there always exists an open neighborhood V of p such
that the normal holonomy group of V at p coincides with Φ
loc
p
,andthesameistrue
for smaller neighborhoods of p, so we can assume that V is diffeomorphic to an open
ball. If the dimension of Φ
loc
p
is constant, then Φ
loc
p
= Φ
∗
p
.
A property that will be useful later is the following proposition (see [126, Ap-
pendix] for a proof).
Proposition 3.1.11 If
ε
> 0 is sufficiently small, then parallel transport along loops
of length at most
ε
contains a neighborhood of the identity of Φ
loc
p
and belongs to
Φ
loc
p
.
104 Submanifolds and Holonomy
3.2 The Normal Holonomy Theorem
We begin with some motivating facts about holonomy of Riemannian manifolds.
The two fundamental results regarding the restricted holonomy group of a Riemann-
ian manifold are the de Rham Decomposition Theorem and the Berger Holonomy
Theorem. Both have local and global versions, and we will refer only to the first one.
De Rham’s Theorem asserts that a Riemannian manifold M is locally irreducible
around p if and only if its local holonomy group acts irreducibly on T
p
M.Berger’s
Theorem says that if M is irreducible around p and not locally symmetric, then the
restricted holonomy group Hol
o
p
(M) acts transitively on the unit sphere in T
p
M.In
particular, one has the following property: For each p ∈ M there exist a unique (up to
order) orthogonal decomposition T
p
M = V
0
⊕...⊕V
k
of T
p
M into Hol
o
p
(M)-invariant
subspaces V
1
,...,V
k
and normal subgroups G
0
,...,G
k
of Hol
o
p
(M) such that:
(i) Hol
o
p
(M)=G
0
×...×G
k
(direct product);
(ii) G
i
acts trivially on V
j
if i = j;
(iii) G
0
= {1} and, if i ≥ 1, G
i
acts transitively on the unit sphere in T
p
M or it
acts irreducibly on V
i
as the isotropy representation of a simple Riemannian
symmetric space.
We call this result the algebraic de Rham-Berger Theorem.
A key property for dealing with holonomy groups of Riemannian manifolds is
that the holonomy algebra is generated by algebraic curvature tensors, that is, linear
tensors satisfyin g all th e algebraic iden tities of a curvature tensor. More precisely,
the holonomy algebra of a Riemannian manifold is generated by endomorphisms of
the form
γ
∗
R(x,y)=(
τ
γ
)
−1
R(
τ
γ
x,
τ
γ
y)
τ
γ
where x,y ∈ T
p
M,
γ
: [0, 1] → M is a piecewise differentiable curve starting at p,and
τ
γ
denotes parallel transport along
γ
.
It is surprising that, for the normal connection of a submanifold of a standard
space form, the algebraic de Rham-Berger Theorem holds in a simpler version.
Theorem 3.2.1 (Normal Holonomy Theorem [255]) Let M be a connected subma-
nifold of a standard space form
¯
M
n
(
κ
).Letp∈ M and let Φ
∗
be the restricted nor-
mal holonomy group at p. Then Φ
∗
is compact and there exist a unique (up to order)
orthogonal decomposition
ν
p
M = V
0
⊕...⊕V
k
of the normal space
ν
p
MintoΦ
∗
-
invariant subspaces and normal subgroups Φ
0
,...,Φ
k
of Φ
∗
such that
(i) Φ
∗
= Φ
0
×...×Φ
k
(direct product);
(ii) Φ
i
acts trivially on V
j
if i = j;
(iii) Φ
0
= {1} and, if i ≥ 1, Φ
i
acts irreducibly on V
i
as the isotropy representation
of an irreducible Riemannian symmetric space.
The Normal Holonomy Theorem 105
This result is an important tool for investigating the geometry of submanifolds
of space forms. Indeed, in Section 3.4, we will construct so-called holonomy tubes,
many properties of which rely on the Normal Holonomy Theorem. Holonomy tubes
have many applications, as we will see later.
We first sketch the proof by listing the ingredients of the Normal Holonomy
Theorem recipe before providing the arguments in full detail. We will define a tensor
field, the so-called adapted normal curvature tensor
R
⊥
: ⊗
3
ν
M →
ν
M,
that contains the same geometric information as the normal curvature tensor R
⊥
,yet
bears the algebraic properties of a Riemannian curvature tensor, that is, it is an alge-
braic curvature tensor (see, for instance, Section A.1). A very important property of
R
⊥
is that its scalar curvature is nonpositive and vanishes if and only if R
⊥
vanishes.
This implies that normal holonomy groups look like nonexceptional Riemannian hol-
onomy groups.
The argument follows some ideas of Cartan and the methods used by Simons
[295] in his proof of Berger’s Theorem. We define a holonomy system to be a triple
[V, R, G],whereV is a Euclidean vector space, R is an algebraic curvature tensor on
V and G is a compact Lie group acting effectively on V by isometries, such that
R
xy
belongs to the Lie algebra of G for all x,y ∈ V . Some reduction resu lts allow
us to concentrate on irreducible holonomy actions. Roughly speaking, we will take
G = Φ
∗
p
, V =
ν
p
M and R = R
⊥
p
. Note that, since a connected Lie subgroup of the
orthogonal group acting irreducibly on a vector space is compact, one gets that Φ
∗
is
compact.
A prominent role among holonomy systems is played by the so-called sym-
metric holonomy systems, which are closely related to symmetric spaces and s-
representations.
3.3 Proof of the Normal Holonomy Theorem
3.3.1 Holonomy systems
Let us consider an n-dimensional Euclidean vector space (V,·,·) and the real
vector space P of all tensors of type (1,3) on V . We identify such a tensor with a
bilinear map P : V ×V → End(V ), (x,y) → P
x,y
. Then the group O
n
of isometries of
V acts on P by
(g ·P)
x,y
= gP
g
−1
(x),g
−1
(y)
g
−1
, g ∈ O
n
, x,y ∈V.
By differentiation we get an action o f so
n
on P by
(A ·P)
x,y
= −P
Ax,y
−P
x,Ay
−[P
x,y
,A] , A ∈ so
n
, x,y ∈V.
Next, we recall the following definition (cf. Section A.1).
106 Submanifolds and Holonomy
Definition 3.3.1 AtensorR ∈ P is called an algebraic curvature tensor if
(1) R
xy
= −R
yx
;
(2) R
xy
z,w = −R
xy
w, z;
(3) R
xy
z,w = R
zw
x,y;
(4) R
xy
z + R
yz
x + R
zx
y = 0 (algebraic or first Bianchi identity).
Note that every linear com bination of algebraic curvature tensors is again an
algebraic curvature tensor. If R is an algebraic curvature tensor, g ∈SO
n
and A ∈ so
n
,
then g ·R and A ·R are algebraic curvature tensors as well. Thus the real vector space
of all algebraic curvature tensors on V is an SO
n
-module.
Associated with every alg ebraic curvature tensor R is its scalar curvature
k(R)=2
∑
i< j
R
e
i
e
j
e
j
,e
i
,
where e
1
,...,e
n
is an orthonormal basis of V . Note that k(g ·R)=k(R) holds for all
g ∈ O
n
.
Definition 3.3.2 Let R be an algebraic curvature tensor on V . A compact subgroup
G of O
n
is called a holonomy group of R if R
xy
∈ g for all x, y ∈V ,whereg denotes
the Lie algebra of G.
If R
xy
∈ g holds for all x,y ∈V ,then(g ·R)
xy
∈ g and (A ·R)
xy
∈ g holds for all
g ∈G, A ∈g and x, y ∈V .Thus,ifG is a holonomy group for R, it is also a holonomy
group for g ·R and A ·R for all g ∈ G and A ∈ g.
Definition 3.3.3 AtripleS =[V,R,G],whereV is a Euclidean vector space, R an
algebraic curvature tensor on V and G a connected holonomy group of R is called a
holonomy system.
The definition of holonomy systems is motivated by the fact that, on a Rie-
mannian manifold M with Riemannian curvature tensor R, the elements R(x,y),
x,y ∈ T
p
M, lie in the holonomy algebra at p.Thus,ifM is a Riemannian manifold,
then [T
p
M,R
p
,Hol
o
p
(M)] is a holonomy system for each p ∈ M.
3.3.2 Symmetric holonomy systems and holonomy of symmetric spaces
We now turn to a fundamental class of holonomy systems that are closely related
to symmetric spaces.
Definition 3.3.4 A holonomy system S =[V, R, G] is called symmetric if g ·R = R
for all g ∈ G, or equivalently, if A ·R = 0forallA ∈ g.
The Normal Holonomy Theorem 107
In other words, a holonomy system is symmetric if its algebraic curvature ten-
sor is G-invariant. The definition of symmetric holonomy systems is motivated by
Cartan’s theory of symmetric spaces. Indeed, given a symmetric holonomy system
S =[V,R,G], one can carry out the following construction due to Cartan:
Consider the real vector space
L = g ⊕V
and define on it a bilinear skewsymmetric map [·,·] : L ×L → L by
[A,B]=[A, B]
g
, [x,y]=−R
xy
, [A,x]=Ax , A,B ∈ g , x,y ∈V,
where [·,·]
g
denotes the Lie algebra structure on g . It turns out that [·,·] defines a Lie
algebra structure on L, that is, it satisfies the Jacobi identity. To verify this, note first
that the only nontrivial case occurs when two elements are in V and one element is
in g. Indeed, if all three elements are in V , the Jaco bi identity is just the algebraic
Bianchi identity for R,andifA,B ∈g and x ∈V we have
[[A,B],x]+[[B, x], A]+[[x,A],B]=(AB −BA)x −A(Bx)+B(Ax)=0.
For A ∈ g and x,y ∈V , the definition of symmetric holonomy systems implies
[A,[x,y]] + [y,[A,x]] + [x,[y,A]] = −[A, R
x,y
]+R
y,Ax
+ R
x,Ay
= −(A ·R )
xy
= 0.
From the very definition it follows that the Cartan relations
[g,g] ⊂g , [g,V ] ⊂V , [V,V ] ⊂ g
hold. Then L corresponds to a Riemannian symmetric space M, whose tangent space
at a point p can be identified with V , whose curvature tensor at p is R and whose
holonomy algebra at p is g. To construct M explicitly, consider the involutive auto-
morphism
σ
: L = g ⊕V → L = g ⊕V , A + x → A −x.
The pair (L,
σ
) is an orthogonal symmetric Lie algebra, since G is compact and
hence g is a compact Lie algebra. If L is a simply connected Lie group with Lie
algebra L and G is the connected closed subgroup of L with Lie algebra g,then(L,G)
is a Riemannian symmetric pair and M = L/G is a simply connected Riemannian
symmetric space (cf. [151, page 213]).
Let
π
: L → M be the canonical projection and p =
π
(G)=eG ∈ M.LetR
p
be
the Riemannian curvature tensor of M at p.Then
R
p
(x,y)z = −[[x, y],z] , x,y, z ∈T
p
M.
Using the definition of [·,·] on L,wehave
R
p
(x,y)z = −[[x, y],z]=−[−R
xy
,z]=R
xy
z,
and hence R
p
= R. Note also that, by the Ambrose-Singer Holonomy Theorem, the
holonomy algebra of M at p is generated by the endomorphisms
τ
−1
γ
R
τ
γ
x,
τ
γ
y
τ
γ
,where
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