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.