246 Submanifolds and Holonomy
Then g
is an ideal of g.LetG
be the connected Lie subgroup of G with Lie algebra
g
. Decompose
V = V
0
⊕...⊕V
k
,
where V
0
is the set of fixed points of G
and G
acts irreducibly on V
i
, i ∈{1,...,k}.
Let g
i
= {Θ
x
i
: x
i
∈V
i
, Θ ∈F } be the subalgebra of g with associated Lie subgroup
G
i
⊂G. Observe that G
0
= {0},sinceΘ
x
0
y = −Θ
y
x
0
= 0forallx
0
∈V
0
and y ∈ V
⊥
0
.
For i ≥ 1wedefine
C
i
(g
i
)={x ∈ so(V
i
) : [x, g
i
]=0}.
Proposition 9.3.1 (see, e.g., [7], Section 4) With the above notation we have
(i) G
= G
1
×...×G
k
,andG
i
acts irreducibly on V
i
and trivially on V
j
for 0 =
i = j.
(ii) The decomposition V = V
0
⊕...⊕V
k
is unique, up to order.
(iii) C
i
(g
i
)={0} for all i ≥ 1. In particular, g
i
is semisimple.
(iv) G = G
0
×G
= G
0
×G
1
×...×G
k
,whereG
0
acts on V
0
and trivially on V
⊥
0
(G
0
can be arbitrary).
Proof Let Θ ∈F , x
i
∈V
i
, x
j
∈V
j
and x ∈V (i = j). Then Θ
x
x
i
,x
j
= 0 = Θ
x
i
x
j
,x.
Thus Θ
x
i
x
j
= 0 and therefore, if y = y
1
+ ...+ y
k
with y
i
∈V
i
, Θ
y
= Θ
y
1
+ ...+ Θ
y
k
,
where Θ
y
i
∈ SO(V
i
). This implies parts (i) and (ii).
Let B ∈ C
i
(g
i
).Thenker(B) is G
i
-invariant and so, since G
i
acts irreducibly on
V
i
,ker(B)={0} or ker(B)=V
i
. Let us assume that B = 0. Then B is invertible. For
Θ ∈ F and x,y,z ∈ V
i
we have
Θ
x
By,z = BΘ
x
y,z = −BΘ
y
x,z = −Θ
y
Bx,z= Θ
Bx
y,z.
Interchanging x and z we also get
Θ
x
By,z = −Θ
z
By,x = −Θ
Bz
y,x = Θ
x
y,Bz.
Since B is skewsymmetric, Θ
x
y,Bz = −BΘ
x
y,z = −Θ
x
By,z,andso
Θ
x
By,z = Θ
x
y,Bz = −BΘ
x
y,z = −Θ
x
By,z.
Thus we conclude that Θ = 0 when restricted to V
i
for all Θ ∈ F , which is a contra-
diction. I t fo llows that B = 0, which proves (iii).
Since g
is an ideal of g and the decomposition in part (ii) is unique, G leaves
V
i
invariant for all i ∈{0,...,k}.Sog
|
V
i
is an ideal of g|
V
i
. Hence g
|
V
i
= g|
V
i
by
part (iii). In fact, if B belongs to the complementary ideal of g
in g,thenB|
V
i
∈
C (g
|
V
i
)={x ∈ so(V
i
) : [x,g
|
V
i
]=0} and so B|
V
⊥
0
= 0. From this and part (i) we
getpart(iv).
Let [V,Θ,G], Θ = 0, be an irreducible skew-torsion holonomy system and let
ν
v
(G ·v) be the normal space of the orbit G ·v at the point v ∈ V, namely
ν
v
(G ·v)={
ξ
∈ V : g ·v,
ξ
= 0}.
The Skew-Torsion Holonomy Theorem 247
As for holonomy systems (see Lemma 8.2.5), the normal space
ν
v
(G ·v ) is a Θ-
invariant subspace, that is, Θ
ν
v
(G·v)
ν
v
(G ·v) ⊂
ν
v
(G ·v).Infact,if
ξ
∈
ν
v
(G ·v) and
x ∈ V,then
0 = Θ
x
v,
ξ
= −Θ
ξ
v,x
and so Θ
ξ
v = 0. Thus Θ
ξ
∈ g
v
,whereg
v
is the Lie algebra of the isotropy group
G
v
.SinceG
v
leaves the normal space
ν
v
(G ·v ) invariant, we see that
ν
v
(G ·v ) is
Θ-invariant.
With an analogous proof as the one given in Chapter 8 for the Simons Holonomy
Theorem we obtain:
Theorem 9.3.2 (Weak Skew-Torsion Holonomy Theorem) Let [V, Θ, G], Θ = 0,
be an irreducible nontransitive skew-torsion holonomy system. Then [V, Θ, G] is sym-
metric.
In fact, the proof is even simpler since Θ has less variables than an algebraic
Riemannian curvature tensor.
Proposition 9.3.3 Let [V,Θ,G], Θ = 0, be an irreducible symmetric skew-torsion
holonomy system. Then, with the above notations, we have
(i) G = G
, and hence the linear span of {g(Θ)
x
: g ∈ G, x ∈ V} = {Θ
x
: x ∈ V}
coincides with the Lie algebra g of G;
(ii) (V,[·,·]) is an (orthogonal) simple Lie algebra with respect to the bracket
[x,y]=Θ
x
y;
(iii) G = Ad(H), where H is the (connected) Lie group associated with the Lie
algebra (V,[·,·]);
(iv) Θ is unique, up to a scalar multiple.
Proof Part (i) follows from Proposition 9.3.1. If B ∈ g then, since [V,Θ,G] is sym-
metric, B.Θ = 0andso
0 =(B.Θ)
x
y = BΘ
x
y −Θ
x
By −Θ
Bx
y.
By putting B = Θ
z
we get the Jacobi identity for [·,·], which implies that (V, [·,·]) is
a Lie algebra. From this we get part (iii). Since G acts irreducibly, the Lie algebra
(V,[·,·]) is simple, which implies part (ii).
Part (iv) follows from the fact that (V,[·, ·]) is simple. In fact, if [V, Θ
,G] is an-
other symmetric skew-torsion holonomy system, then Θ
x
is a derivation of (V,[·,·])
and so Θ
x
=[(x),·],where : V → V is linear. Since Θ
and [·, ·] are both G-
invariant, is G-invariant, that is, commutes with G.SinceG acts by isometries,
both the skew-symmetric part
1
and the symmetric part
2
of commute with G.
Using Proposition 9 .3.1 we obtain
1
= 0. Moreover, since G acts irreducibly, we get
2
=
λ
id, which proves part (iv).
The following remark is well known (see, e.g., Remark 2.6 in [261]). This topo-
logical result is used for the classification of connected compact simple Lie groups.
248 Submanifolds and Holonomy
Remark 9.3.4 We recall here, for the sake of completeness, that the universal cov-
ering group Spin
3
of SO
3
is, up to isomorphism, the only simply connected compact
Lie group of rank 1. The main step of the proof is topological and uses exact homo-
topy sequences.
Let H be a compact simply connected Lie group of rank 1 and dimension n.
Then, as a symmetric pair, we can write H =(H ×H)/diag(H ×H).SinceH is a
Riemannian symmetric space of rank one, diag(H ×H) H acts transitively on the
unit sphere S
n−1
in T
e
H. Thus S
n−1
= H/S
1
,whereS
1
is a compact one-dimensional
Lie subgroup of H (and so, S
1
is homeomorphic to the circle). Recall that the ho-
motopy groups of S
1
are all trivial except for the first one. If n −1 = 2 this yields a
contradiction to the exact homotopy sequence induced by 0 →S
1
→ H →S
n−1
→0.
Thus we must have n = 3. In this case the bracket is unique, since there is a unique
3-form, up to multiples, in dimension 3. This bracket gives rise to the Lie a lgebra so
3
of SO
3
.
9.4 The derived 2-form with values in a Lie algebra
Let Θ be a totally skew 1-form on a Euclidean vector space V with values in a
Lie algebra g ⊂ so(V) so
n
. We will construct a totally skew 2-form Ω on V with
values in g, called th e derived 2-form, which measures how far Θ deviates from being
a L ie bracket on g.
Let [V, Θ, G], Θ = 0, be a skew-torsion holonomy system. Let us define
Ω
x,y
=(Θ
x
.Θ)
y
=[Θ
x
,Θ
y
] −Θ
Θ
x
y
.
It is clear that Ω
x,y
∈ g for all x,y ∈ V. From the definition we obtain that Ω
x,y
is
skewsymmetric in x and y. Moreover, for any fixed x ∈ V, Ω
x,y
z,w is a 3-form in
the other three variables, since Θ is totally skew (and hence B.Θ is totally skew for
all B ∈ so(V)). Thus Ω
x,y
z,w is an algebraic 4-form on V.
Remark 9.4.1 If v ∈ V,thenΩ
v,·
is a (totally skew) 1-form on V with values in the
isotropy algebra g
v
= {B ∈ g : Bv = 0}. In fact, since Ω is totally skew, Ω
v,·
v = 0.
Lemma 9.4.2 Let [V,Θ,G] be a skew-torsion holonomy system and let Σ be the set
of fixed points of H, where H is a subgroup of the normalizer N
O(V)
(G) of G in O(V).
Assume that Σ contains a principal point for the G-action o n V.LetG
Σ
be the identity
component of the subgroup of G leaving Σ invariant.
(i) The cohomogeneity of the action of G
Σ
on Σ is less or equal than the cohomo-
geneity of the action of G on V.
(ii) There exists a totally skew 1-form Θ
Σ
on Σ with values in the Lie algebra
¯
g
Σ
of
¯
G
Σ
= {g|
Σ
: g ∈ G
Σ
} such that Θ
Σ
·
·,· coincides with the restriction of Θ
·
·,·
to Σ.
The Skew-Torsion Holonomy Theorem 249
Proof Part (i) follows immediately from Lemma 9.1.1.
We may assume, by possibly taking the closure, that H is compact. Let us define
˜
Θ
Σ
=
H
h
∗
Θdh,
where h
∗
Θ is the pullback of Θ by h. Observe that
˜
Θ
Σ
is a totally skew 1-form with
values in g.Ifw
1
,w
2
,w
3
∈ Σ,then
˜
Θ
Σ
w
1
w
2
,w
3
=
H
Θ
h(w
1
)
h(w
2
),h(w
3
)dh =
H
Θ
w
1
w
2
,w
3
dh = Θ
w
1
w
2
,w
3
.
Let now v ∈ Σ
⊥
. Observe that Σ
⊥
is H-invariant and so
H
h(v)dh = 0, since it
belongs to Σ
⊥
and is fixed by H.Then
˜
Θ
Σ
w
1
w
2
,v=
H
Θ
h(w
1
)
h(w
2
),h(v)dh =
H
Θ
w
1
w
2
,h(v)dh
=
+
Θ
w
1
w
2
,
H
h(v)dh
,
= 0.
Thus, since v ∈ Σ
⊥
is arbitrary,
˜
Θ
Σ
w
1
w
2
∈ Σ.IfΘ
Σ
denotes the restriction of
˜
Θ
Σ
to Σ,
wegetpart(ii).
Remark 9.4.3 Let H be a compact Lie group with Lie algebra h.Let0= v ∈ h.The
normal space
ν
v
(H ·v) of the orbit H ·v = Ad(H)v at v is given by
ν
v
(H ·v)=C (v)={
ξ
∈h :ad
v
(
ξ
)=0}= {
ξ
∈h : [v,
ξ
]=0}.
In fact,
ξ
∈
ν
v
(H ·v) if and only if {0}= T
v
(H ·v),
ξ
= [g,v],
ξ
= g,[v,
ξ
],where
·,· is an Ad(H)-invariant inner product on g.
This implies that the set of fixed points in g of the group {Ad(Exp(tv)) : t ∈ R}
is just the normal space
ν
v
(H ·v). In fact, this follows since
d
dt
t=0
Exp(tv)=ad
v
.
Lemma 9.4.4 Let [V,Θ,G], Θ = 0, be an irreducible skew-torsion holonomy system.
Then G acts on V as the isotropy representation of a simple Riemannian symmetric
space.
Proof We d efine the following algebraic (Riemannian) curvature tensor R on V with
values in the Lie algebra g of G:
R
v,w
=[Θ
v
,Θ
w
] −
2
3
Ω
v,w
with Ω
v,w
=(Θ
v
.Θ)
w
.
Then R
v,w
∈ g for all v, w ∈ V. Let us verify the Bianchi identity. Let B denote the
cyclic sum over the first three variables. Since Ω
v,w
z is skewsymmetric in v,w,z,we
have B(
2
3
Ω
v,w
z)=2Ω
v,w
z. We then compute
B([Θ
v
,Θ
w
]z)=[Θ
v
,Θ
w
]z +[Θ
w
,Θ
z
]v +[Θ
z
,Θ
v
]w
= Θ
v
Θ
w
z −Θ
w
Θ
v
z + Θ
w
Θ
z
v −Θ
z
Θ
w
v + Θ
z
Θ
v
w −Θ
v
Θ
z
w.
250 Submanifolds and Holonomy
Now observe that, in the above sum, the first term is equal to the last one, the second
term is equal to the third one, and the remaining two terms are also equal. Thus
B([Θ
v
,Θ
w
]z)=2(Θ
v
Θ
w
z −Θ
w
Θ
v
z + Θ
z
Θ
v
w)
= 2[Θ
v
,Θ
w
]z −2Θ
Θ
v
w
z = 2Ω
v,w
z,
which implies B(R
v,w
z)=0.
Let us compute the scalar curvature s(R) of R.Lete
1
,...,e
n
be an orthonormal
basis of V.SinceΩ
v,w
z,u is a 4-form we have
s(R)=
∑
i< j
R
e
i
,e
j
e
j
,e
i
=
∑
i< j
[Θ
e
i
,Θ
e
j
]e
j
,e
i
=
∑
i< j
Θ
e
i
Θ
e
j
e
j
,e
i
−Θ
e
j
Θ
e
i
e
j
,e
i
= −
∑
i< j
Θ
e
j
Θ
e
i
e
j
,e
i
=
∑
i< j
Θ
e
i
e
j
,Θ
e
j
e
i
= −
∑
i< j
Θ
e
i
e
j
,Θ
e
i
e
j
= Θ= 0.
In particular, R = 0. Then [V,R,G] is an irreducible holonomy system in the sense of
Simons [295] with s(R) = 0. Hence G acts as the isotropy representation of a simple
Riemannian symmetric space by Theorem 5 in [295]. In fact, let
¯
R =
G
g(R)dg.
Then s(
¯
R)=s(R) = 0and[V,
¯
R,G] is an irreducible symmetric holonomy system.
Using the Cartan construction (see pages 107 and 237) we obtain that G actsasan
irreducible s-representation.
9.5 The Skew-Torsion Holonomy Theorem
In this part we state and prove the main result about skew-torsion holonomy sys-
tems. Let us first prove that the transitive ones are generic. This result was essentially
proved by Agricola and Friedrich [5, 7] using the classification of transitive actions
on spheres.
Theorem 9.5.1 Let [V,Θ,G], Θ = 0, be a transitive skew-torsion holonomy system.
Then G = SO(V).
Proof Let F be the family of totally skew 1-forms on V with values in the
Lie algebra g of G. From Proposition 9.3.3 (i) we know that the linear span of
{
˜
Θ
v
:
˜
Θ ∈ F ,v ∈ V} coincides with g.
We shall prove the theorem by induction on the dimension n of V. Observe that
n ≥ 3, because otherwise Θ = 0. For n = 3 the theorem is true since there is only
one such a form Θ, up to a scalar multiple, since Θ
·
·,· is a 3-form. In this case
{Θ
v
: v ∈ V} = so
3
and the theorem holds.
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