108 Submanifolds and Holonomy
γ
is any piecewise differentiable curve in M with
γ
(0)=p and
τ
γ
denotes parallel
transport along
γ
.SinceS is a symmetric holonomy system, the holonomy algebra
coincides with
span{R
xy
: x, y ∈ T
p
M} = im(R)=[V,V ].
Hence a symmetric holonomy system S =[V,R,G] with R = 0 corresponds to a sim-
ply connected Riemannian symmetric space M = L/G in such a way that R iden tifies
with the Riemannian curvature tensor of M at p = eG.
Note that the restriction of the Killing form B
L
of L to g is given by
B
L
(A,B)=tr
g
(ad(A) ◦ad(B))+ tr
V
(ad(A) ◦ad(B))
= B
g
(A,B)+
1
n −2
B
so
n
(A,B)
for all A,B ∈ g.
Now suppose that the action of G on V is irreducible. In this case we say that the
holonomy system is irreducible. This obviously implies that M is irreducible, since
by the de Rham Decomposition Theorem (Appendix, page 397) this is equivalent to
the holonomy acting irreducibly on T
p
M = V .
Without loss of generality we can also assume that G acts faithfully on V (so L is
effective, that is, g does not contain a nontrivial ideal of L). From Schur’s Lemma it
follows that L, and thus also L,aresemisimple,i.e.B
L
is nondegenerate. Indeed, any
effective orthogonal symmetric algebra L = g ⊕V with g acting irred ucibly on V and
[V,V ] = 0 is the direct sum of at most two simple ideals [178, Chapter 11, Proposition
7.5]. Therefore the restriction of B
L
to V , which is also nondegenerate, must equal a
nonzero constant multiple c
−1
of the inner product on V (via the identification of V
and T
p
M), since both bilinear forms are G-invariant. Hence we have
R
xy
z,w = cB
L
([[x,y],z],w)=cB
L
([x,y],[z,w]). (3.1)
Thus we obtain the following result, due to Kostant, which allows us to read off the
curvature tensor of the irreducible symmetric space corresponding to the orthogonal
symmetric algebra L = g ⊕V from the Lie algebras of g and V alone (see also [289]).
Theorem 3.3.5 (Kostant) The Riemannian curvature tensor R of the irreducible
Riemannian symmetric space M corresponding to the orthogonal symmetric alge-
bra L = g ⊕V has the form
R
xy
z,w = cB
L
([x,y],[z,w]) = c
B
g
+
1
n −2
B
so
n
([x,y],[z,w]), (3.2)
and M is Einstein with nonzero scalar curvature.
Remark 3.3.6 This result implies that, if S =[V,R,G] and S
=[V,R
,G] are two
irreducible symmetric holonomy systems with dimV ≥ 2, then R
=
λ
R for some
constant
λ
.
The Normal Holonomy Theorem 109
Observe now that [V,V ]=hol = g,wherehol is the holonomy algebra. Indeed,
since we are assuming that L is effective, so that L is semisimple, the orthogonal
complement of [V,V]⊕V with respect to B
L
is an ideal of L contained in g and hence
vanishes. So we conclude: The holonomy algebra of M coincides with g and an irre-
ducible symmetric holonomy system S =[V, R, G] corresponds to an s-representation
with
G = holonomy group = isotropy group of the symmetric space.
The following remarkable result due to Simons [295] is the essential tool in Si-
mons’ proof of Berger’s Holonomy Theorem. It enables one to prove that an irre-
ducible Riemannian manifold with nontransitive holonomy group is locally symmet-
ric. Simon’s Holonomy Theorem is not needed for proving the Normal Holonomy
Theorem. We will prove Theorem 3.3.7 in Chapter 8 by using the normal holonomy
of the holonomy orbits.
Theorem 3.3.7 (Simons) Let [V,R,G] be an irreducible holonomy system. If G is
not transitive on the unit sphere in V , then [V,R,G] is symmetric.
Remark 3.3.8 If M is a locally irreducible Riemannian manifold and p ∈M satisfies
R
p
= 0, then [T
p
M,R,Hol
o
p
(M)] is an irreducible holonomy system. Using the above
theorem and standard facts, Simons provided a conceptual proof of the following
result (which was obtained seven years earlier, using classification results, by Berger
[17] and is therefore known as Berger’s Theorem): Let M be a Riemannian manifold
that is locally irreducible at some point p. If the holonomy group is not transitive on
the unit sphere in T
p
M,thenM is locally symmetric.
The (restricted) holonomy group of a locally irreducible Riemannian manifold is
called nonexceptional if there exists a symmetric space with an isomorphic holonomy
group. By Berger’s Theorem, exceptional holonomies are always transitive on the
unit sphere of the tangent space. If the scalar curvature of M is not identically zero,
then the holonomy is nonexceptional (this is a consequence of Lemma 3.3.14).
From Theorem 3.3.7 we deduce the following important property of s-
representations.
Proposition 3.3.9 Let K and K
be two irreducible s-representations on V with
dimV ≥ 2 that are not transitive on the unit sphere in V . If K and K
have the same
orbits, then K = K
and the s-representations coincide.
Proof The group
˜
K generated by K and K
is not transitive on the unit sphere in
V , since it has the same orbits as K and K
.LetR and R
be the curvature tensors
corresponding to the s-representations of K and K
.Then[V, R,
˜
K] is an irreducible
nontransitive holonomy system. By Theorem 3.3.7, [V, R,
˜
K] is symmetric and hence
the Lie algebra of
˜
K is generated by R.ThenK =
˜
K and, by Remark 3.3.6, R
=
λ
R.
110 Submanifolds and Holonomy
3.3.3 Normal curvature tensor and proof of the Normal Holonomy
Theorem
Let M be a submanifold of a standard space form
¯
M
n
(
κ
). We star t with the normal
curvature tensor R
⊥
(at a point p ∈ M) in order to construct a holonomy system. For
each x, y ∈ T
p
M the endomorphism R
⊥
(x,y) lies in the normal holonomy algebra.
But it follows from the very definition that R
⊥
is not an algebraic curvature tensor.
To construct a tensor field of type (1,3) on the normal bundle of M we consider R
⊥
as a homomorphism
R
⊥
: Λ
2
T
p
M → Λ
2
ν
p
M.
The homomorphism R
⊥
composed with its adjoint homomorphism R
⊥∗
gives rise to
an endomorphism
R
⊥
= R
⊥
◦R
⊥∗
: Λ
2
ν
p
M → Λ
2
ν
p
M,
which can be identified with a (1 , 3)-tensor on
ν
p
M.
The Ricci equation R
⊥
(x,y)
ξ
,
η
= [A
ξ
,A
η
]x,y implies
R
⊥∗
(
ξ
∧
η
)=[A
ξ
,A
η
]
and hence
R
⊥
(
ξ
1
,
ξ
2
)
ξ
3
,
ξ
4
= R
⊥∗
(
ξ
1
∧
ξ
2
),R
⊥∗
(
ξ
3
∧
ξ
4
)
= −tr([A
ξ
1
,A
ξ
2
] ◦[A
ξ
3
,A
ξ
4
]), (3.3)
since the inner product on Λ
2
T
p
M is given by A,B = −tr(AB).Using(3.3)weget
Lemma 3.3.10 ( [255]) R
⊥
is an algebraic curvature tensor on
ν
p
M.
Proof We must verify that conditions (1)–(4) in Definition 3.3.1 hold. Now (1)–(3)
are clear from (3.3). To verify the algeb raic Bianch i identity (4), easy computations
transform (3.3) into the form
R
⊥
(
ξ
1
,
ξ
2
)
ξ
3
,
ξ
4
= 2tr(A
ξ
1
A
ξ
2
A
ξ
3
A
ξ
4
) −2tr(A
ξ
3
A
ξ
1
A
ξ
2
A
ξ
4
). (3.4)
Cyclic sum over the indices 1, 2, 3 yields the algebraic Bianchi identity.
Note that (3.3) implies th a t R
⊥
has nonpositive scalar curvature and its scalar
curvature vanishes if and only if R
⊥
is identically zero. This tensor will be called the
adapted normal curvature tensor.
Remark 3.3.11 The existence of this algebraic curvature tensor with nonpositive
scalar curvature is of great importance for the structure theory of normal holonomy. It
implies that normal holonomy groups behave like nonexceptional holonomy groups
of Riemannian manifolds (see Remark 3.3.8) and simplifies the proof of the Normal
Holonomy Theorem, compared with the one of Berger’s Theorem.
The Normal Holonomy Theorem 111
Using again the Ricci equation [255], an alternative expression for R
⊥
is given
by
R
⊥
(
ξ
1
,
ξ
2
)
ξ
3
= 2
m
∑
i=1
(R
⊥
)(A
ξ
1
e
i
,A
ξ
2
e
i
)
ξ
3
, (3.5)
where e
1
,...,e
m
is an orthonormal basis of T
p
M. Moreover, since ker(R
⊥
)=
(im(R
⊥∗
))
⊥
,theimageofR
⊥
is the same as that of R
⊥
(cf. [255]). The Ambrose-
Singer Holonomy Theorem and the previous lemma therefore imply:
Lemma 3.3.12 Let M be a submanifold of a standard space form
¯
M
n
(
κ
) and p ∈M.
Then the Lie algebra of the normal holonomy group is spanned by the tensors of the
form
γ
∗
R
⊥
(
ξ
,
η
)=
τ
⊥−1
γ
R
⊥
(
τ
⊥
γ
ξ
,
τ
⊥
γ
η
)
τ
⊥
γ
,where
γ
is any piecewise differentiable
curve in M with p as endpoint.
We denote by S the real vector space of tensors on
ν
p
M that is spanned by
all
γ
∗
R
⊥
,where
γ
runs through all piecewise differentiable curves in M with p as
endpoint. Since R
⊥
is an algebraic curvature tensor, any R ∈ S has the algebraic
properties of a curvature tensor.
Next, we define a holonomy system. At this stage it is still not clear that Φ
∗
p
is
compact, but this obstacle is avoided by splitting algebraically both
ν
p
M and the Lie
algebra of Φ
∗
p
using the algebraic properties of the tensors in S . What we achieve
is an “algebraic” de Rham decomposition. Explicitly, we decompo se
ν
p
M into Φ
∗
p
-
invariant orthogonal subspaces,
ν
p
M = V
0
⊕...⊕V
k
,
where Φ
∗
p
acts trivially on V
0
and irreducibly on V
i
for all i ≥ 1. If
ξ
i
denotes the
orthogonal projection of
ξ
∈
ν
p
M onto the factor V
i
, the following holds for any
R ∈ S :
(a) R(
ξ
i
,
ξ
j
)=0ifi = j;
(b) R(
ξ
,
η
)=
∑
i
R(
ξ
i
,
η
i
);
(c) R(
ξ
i
,
η
i
)V
j
= {0},ifi = j;
(d) R(
ξ
i
,
η
i
)V
i
⊆V
i
.
These statements are all obtained from the properties of algebraic curvature ten-
sors. As for (a), if i = j and
η
,
η
∈
ν
p
M,thenR(
ξ
i
,
ξ
j
)
η
,
η
= R(
η
,
η
)
ξ
i
,
ξ
j
=
0, because R(
η
,
η
) ∈ g and g leaves V
i
invariant. This readily implies (b). For
(c), if
ζ
j
∈ V
j
, the algebraic Bianchi identity together with (a) give R(
ξ
i
,
η
i
)
ζ
j
=
−R(
η
i
,
ζ
j
)
ξ
i
−R(
ζ
j
,
ξ
i
)
η
i
= 0. Part (d) is clear.
We denote b y g
i
the real vector space spanned b y all
R(
ξ
i
,
η
i
) ,
ξ
i
,
η
i
∈V
i
, R ∈ S .
The following lemma can be easily verified.
112 Submanifolds and Holonomy
Lemma 3.3.13 The following statements hold:
(i) g
0
= {0} and each g
i
,i≥ 1, is an ideal of g;
(ii) g = g
1
⊕...⊕g
k
and [g
i
,g
j
]={0} if i = j;
(iii) g
i
·V
i
= V
i
for all i ≥ 1;
(iv) g
i
·V
j
= {0} if i = j;
(v) g
i
acts irreducibly on V
i
for all i ≥ 1.
Let now Φ
i
be the connected Lie subgroup of Φ
∗
p
with Lie algebra g
i
.Then
Φ
∗
p
= Φ
0
×...×Φ
k
,
with Φ
i
acting trivially on V
j
for all i = j and irreducibly on V
i
for all i ≥ 1. Since
a connected Lie group of orthogonal transformations on a vector space acting irre-
ducibly is compact, each Φ
i
is compact. Thus Φ
∗
p
is compact.
We can say m ore. For each i ≥1 we can choose R
i
∈S so that R
i
is not identically
zero on V
i
.Then[V
i
,R
i
,Φ
i
] is an irreducible holonomy system, R
i
has nonvanishing
scalar curvature (cf. Remark 3.3.11) and we can apply the following
Lemma 3.3.14 Let G be a connected Lie subgroup of SO(V ) acting irreducibly on a
Euclidean vector space V , and let R be an algebraic curvature tensor on V such that
R
xy
∈g for all x,y ∈V. If k(R) = 0, then G is compact, S =[V,R,G] is an irreducible
holonomy system, and G acts on V as an s-representation.
Proof The compactness of G follows from a general result stating that a connected
Lie subgroup of SO
n
acting irreducibly on a vector space is compact (see, for in-
stance, [178], vol 1, Appendix 5). Since G is compact, there exists a Haar measure
on G and we can define
ˆ
R =
G
g ·R.
The scalar curvature of
ˆ
R coincides w ith the one of R, since averaging does not
change the scalar curvature (recall that k(
ˆ
R)=k (g ·
ˆ
R) for all g ∈ G ). Moreover,
ˆ
R = 0 because k (
ˆ
R) = 0. Clearly, g ·
ˆ
R =
ˆ
R for all g ∈ G,so[V,
ˆ
R,G] is a symmetric
holonomy system. According to previous remarks, G acts o n V as an s-representation.
It follows from Lemma 3.3.14 that Φ
i
acts on V
i
as an s-representation for all
i ≥ 1. This concludes the proof of the Normal Holonomy Theorem 3.2.1.
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