Chapter 9
The Skew-Torsion Holonomy Theorem
This chapter is based on [261, 262] and related to the work of Agricola and Friedrich
in [5, 7]. We develop the theory of the so-called skew-torsion holonomy systems,
which extend Simons’ holonomy systems from Chapter 8. Our approach is geomet-
ric and based on Euclidean submanifold geometry and normal holonomy. The main
result in this chapter is the Skew-Torsion Holonomy Theorem, which extends the
Simons Holonomy Theorem. In contrast to the latter case, the only transitive group
that can occur is the full special orthogonal group.
The basic object of a skew-torsion holonomy system is a totally skew one-form
taking values in a subalgebra of the orthogonal Lie algebra so
n
. Such a one-form
naturally arises as the torsion of a metric connection with the same geodesics as the
Levi-Civita connection. The study of such connections was introduced by Cartan
and is nowadays a very active field o f research in both mathematics and theoretical
physics. We do not explore deeply this broad subject, but will only be concerned
with naturally reductive spaces, for which there are interesting applications of the
Skew-Torsion Holonomy Theorem.
9.1 Fixed point sets of isometries and homogeneous submanifolds
We are concerned here with submanifolds of homogeneous spaces arising as
fixed point sets of families of isometries. We have the following result, based on
Lemma 5.2 in [261]. We add a condition here on the existence of principal points in
the totally geodesic submanifold. This condition was missing in [261], but it has no
consequences at all for the main results there.
Lemma 9.1.1 Let M be a complete Riemannian manifold, G be a connected closed
subgroup of the isometry group I(M) of M, and H be a subset of the normalizer
N
I(M)
(G) of G in I(M).Let
Σ = {q ∈ M : h(q)=q for all h ∈ H}
be the fixed point set of H (note that Σ is a closed and totally geodesic submanifold
of M). Assume that there exists a point q ∈ Σ such that G ·q is a principal orbit of
the G-action on M. Let G
Σ
be the identity component of the subgroup of G leaving
241
242 Submanifolds and Holonomy
Σ invariant. Then the cohomogeneity of the action of G
Σ
on Σ is less or equal than
the cohomogeneity of the action o f G on M. In particular, if G acts transitively on M,
then G
Σ
acts transitively on Σ.
Proof We may assume that H is a closed subgroup of N
I(M)
(G). The group H is
compact since each point in Σ is a fixed point of H. We equip H with an H-invariant
volume form dh such that
H
dh = 1. Let X ∈ K
G
(M) g,whereK
G
(M) is the Lie
algebra of Killing vector fields on M induced by G.Wedefine
¯
X ∈ K
G
(M) by
¯
X =
H
h
∗
Xdh,
where the vector field h
∗
X on M is defined by h
∗
X
h(p)
= d
p
h(X
p
) for all p ∈ M (for
details about invariant integration on compact Lie groups see, e.g., [52]). Then
¯
X
q
is
the orthogonal projection of X
q
to T
q
Σ for all q ∈ Σ. In fact,
¯
X
q
=
H
d
h
−1
(q)
h(X
h
−1
(q)
)dh =
H
d
q
h(X
q
)dh
=
H
d
q
h(v)dh+
H
d
q
h(w)dh,
where X
q
= v + w with v ∈ T
q
Σ and w ∈
ν
q
Σ. Observe that
H
d
q
h(v)dh =
H
vdh = v.
On the other hand, the vector z =
H
d
q
h(w)dh is perpendicular to T
q
Σ and fixed by
H, which implies z = 0. It follows that
¯
X|
Σ
is always tangent to Σ. Moreover,
¯
X|
Σ
coincides with the orthogonal projection of X|
Σ
to T Σ.
Let K
G
(Σ) be the Killing vector fields in K
G
(M) whose restrictions to Σ are
tangent to Σ everywhere. Then K
G
(Σ)|
Σ
coincides with the orthogonal projection to
Σ of the Killing vector fields in K
G
(M) restricted to Σ. It is now clear that a vector
in T
q
Σ which is perpendicular to the orbit G
Σ
·q must be perpendicular to the orbit
G ·q ⊂ M. This imp lies the lemma.
We recall Lemma 5.1 in [261].
Lemma 9.1.2 Let M = G/G
p
be a homogeneous Riemannian manifold, H be a nor-
mal subgroup of G
p
and V be the subspace of fixed vectors of H in T
p
M. Then V is
G
p
-invariant. Moreover, if D is the G-invariant distribution on M defined by D
p
= V,
then D is integrable with totally geodesic integral manifolds (or, equivalently, D is
autoparallel).
Proof We may assume that G is a closed subgroup of I(M). Otherwise, we con-
sider its closure. We will construct explicitly the integral manifold Σ(q) of D con-
taining q. We choose g ∈ G so that q = gp and let Σ(q) be the connected compo-
nent containing q of the set of fixed points of gHg
−1
in M.ThenΣ(q) is a totally
geodesic submanifold of M and T
q
Σ(q)=D
q
.Letr ∈ Σ(q).SinceG is transitive
on M, G
Σ(q)
is transitive on Σ(q) by Lemma 9.1.1. So Σ(q) is a homogeneous sub-
manifold of M and therefore there exists g
∈ G
Σ(q)
with g
q = r.Thenwehave
T
r
Σ(q)=d
q
g
(T
q
Σ(q)) = d
q
g
(D
q
)=D
r
,sinceD is G-invariant. Thus Σ(q) is an
integral manifold of D.
The Skew-Torsion Holonomy Theorem 243
9.2 Naturally reductive spaces
Let M = G/H be a homogeneous Riemannian manifold with a G -invariant Rie-
mannian metric ·,· and p ∈ M so that H = G
p
.Leth and g be the Lie algebras of
H and G, respectively. The homogeneous space M is said to be naturally reductive
if there exists a reductive decomposition g = h ⊕m (that is, m is an Ad(H)-invariant
complementary subspace of h in g) such that the geodesics in M through p are given
by
t → Exp(tX)p
for all X ∈ m. In other words, the geodesics in M coincide with the ∇
c
-geodesics,
where ∇
c
is the canonical connection (which is a metric connection on M) associated
with the reductive deco mposition g = h⊕m (see Ch apter 6). This is in fact equivalent
to the property that [X, ·]
m
: m → m is a skewsymmetric transformation for every
X ∈ m T
p
M (see Appendix A.3).
The Levi-Civita connection ∇ and the canonical connection ∇
c
are given by
∇
v
¯w =
1
2
[ ¯v, ¯w]
p
and ∇
c
v
¯w =[¯v, ¯w]
p
,
where, for u ∈ T
p
M,¯u = X
∗
is the Killing vector field on M induced by the unique
vector X ∈ m with
d
dt
t=0
Exp(tX)p = u. The difference tensor field D between both
connections is
D
v
w = ∇
v
¯w −∇
c
v
¯w = −
1
2
[ ¯v, ¯w]
p
= −∇
v
¯w.
The tensor field D is totally skew, that is, D
v
w, z is a 3-form on M.
Remark 9.2.1 Let M = G/H be a naturally reductive space and p ∈ M with H =
G
p
.Letv ∈ T
p
M beavectorfixed by H and let ˜v be the G-invariant vector field
on M with ˜v
p
= v.Then˜v is a Killing vector field on M. In fact, ∇
c
˜v = 0, since
any invariant tensor is parallel with respect to the canonical connection ∇
c
.Then
∇
x
˜v, y = D
x
v,y is skewsymmetric in x and y. Hence ˜v is a Killing vector field.
Remark 9.2.2 Let M = G/H be a homogeneous Riemannian manifold with G com-
pact and let p ∈M with H = G
p
.SinceG is compact, there exists an Ad(G)-invariant
inner product b on g.Letm = h
⊥
and identify m T
p
M. The restriction of b to m
gives an Ad(H)-invariant inner product on T
p
M, which extends to a G-invariant Rie-
mannian metric ·,· on M. Such a metric is naturally reductive and is called a normal
homogeneous metric on M.
Theorem 9.2.3 Let M = G/H be a simply connected, irreducible, naturally reduc-
tive, homogeneous Riemannian manifold. Assume that T
p
M = V
0
⊕...⊕V
k
(orthog-
onally) and that H = H
0
×...×H
k
,wherep∈M with H = G
p
.Furthermore,assume
that
244 Submanifolds and Holonomy
(1) H
0
acts only on V
0
and, if k ≥ 1,H
i
= {e} acts irreducibly on V
i
and trivially
on V
j
if i = j, i ≥ 1;
(2) If i ≥ 1,thenC
i
(h
i
)={0},whereC
i
(h
i
)={x ∈ so(V
i
) : [x,h
i
]=0} and h
i
is
the Lie algebra of H
i
.
Then k = 0 or k = 1. Moreover, if k = 1,thenV
0
= {0}.
Proof Let W
0
⊂ T
p
M be the subspace of vectors which are fixed by H.Fromas-
sumption (1) we get W
0
⊂ V
0
. Assume that k ≥ 1andleti ≥ 1befixed, say i = 1.
Let H
1
= H
0
×H
2
×...×H
k
, which is a normal subgroup of H. The set of vectors
fixed by H
1
in T
p
M is W
0
⊕V
1
.LetD
1
be the G-invariant distribution on M with
D
1
(p)=W
0
⊕V
1
.ThenD
1
is an autoparallel distribution on M by Lemma 9.1.2.
Let us consider the (totally ge odesic) integral manifold Σ of D
1
containing p.
Let v ∈ W
0
be arbitrary and let ˜v be the G-invariant vector field on M with ˜v
p
= v.
Note that ˜v is a Killing vector field on M by Remark 9.2.1. So its restriction ˜v
1
= ˜v|
Σ
,
which is always tangent to Σ, is a Killing vector field on Σ. Then, by assumption (2),
(∇ ˜v
1
)
p
∈ so(W
0
),
since it commutes with H
1
, which leaves Σ invariant and fixes p.So∇
W
0
⊕V
1
˜v
1
⊂W
0
.
Since ˜v
1
is arb itrary, this implies that the G
Σ
-invariant distribution
¯
D
0
on Σ, which de-
fines W
0
, is a parallel distribution on Σ. Then its complementary distribution, which
has initial condition V
1
at p, is a parallel distribution on Σ.
Let Σ
1
be the integral manifold containing p of the G
Σ
-invariant distribution on Σ
with initial condition V
1
at p.ThenΣ
1
is totally geodesic in Σ and hence in M.Note
that Σ
1
must be an integral manifold containing p of the G-invariant distribution D
1
on M with D
1
(p)=V
1
.ThenD
1
is an autoparallel distribution on M.However,
the complementary distribution D
⊥
1
to D
1
is just the G-invariant distribution that
coincides at p with the set of fixed vectors of the normal subgroup H
1
of H.So
D
⊥
1
is also an autoparallel distribution by L emma 9.1.2. Then, by Exercise 1.8.9, the
autoparallel distributions D
1
and D
⊥
1
must be parallel. Since M is irreducible, the
de Rham Decompo sition Theorem implies V
1
= T
p
M. Then, if k > 0, we must have
k = 1.
If M is compact, we can drop the assumption that the metric is naturally reductive.
Corollary 9.2.4 Let M = G/H be a compact, simply connected, irreducible, homo-
geneous Riemannian manifold. Assume that T
p
M = V
0
⊕...⊕V
k
(orthogonally) and
that H = H
0
×...×H
k
,wherep∈ M with H = G
p
. Furthermore, assume that
(1) H
0
acts only on V
0
and, if k ≥ 1,H
i
= {e} acts irreducibly on V
i
and trivially
on V
j
if i = j, i ≥ 1;
(2) If i ≥ 1,thenC
i
(h
i
)={0},whereC
i
(h
i
)={x ∈ so(V
i
) : [x,h
i
]=0} and h
i
is
the Lie algebra of H
i
.
Then k = 0 or k = 1. Moreover, if k = 1,thenV
0
= {0}.
The Skew-Torsion Holonomy Theorem 245
Proof Let ·, · denote the Riemannian metric on M. We may assume that G acts
effectively on M.SinceM is compact, the isometry group I(M) of M is a compact
Lie group and so its Lie algebra admits an Ad(I(M))-invariant inner product. Thus g
admits an Ad(G)-invariant inner product. Then we can define a normal homogeneous
Riemannian metric ·,·
on M = G/H.Apriori,(M,·,·
) may be reducible. For
i ∈{0,...,k} let D
i
be the G-invariant distribution on M with (D
i
)
p
= V
i
. It is not
hard to see, using assumptions (1) and (2), that the G-invariant distribution D
i
is
orthogonal to D
j
for i = j with respect to any G-invariant Riemannian metric on M.
Moreover, if i ≥ 1, then ·,·|
D
i
×D
i
=
λ
i
·,·
|
D
i
×D
i
for some
λ
i
> 0. By following
the arguments in the proof of Theorem 9.2.3 we obtain that (M,·,·
) splits as M =
M
0
×...×M
k
,whereM
i
is the integral manifold of D
i
containing p, i ∈{0,...,k}.
Then, by the previous observations, (M, ·,·) also splits in this way. Since (M,·,·)
is irreducible, k = 0andM = M
0
or k = 1andM = M
1
. This proves the corollary.
Remark 9.2.5 Corollary 9.2.4 does not hold in general if M is not compact. In fact,
let H
n
be the real hyperbolic space of dimension n ≥4andletF be a foliation of H
n
by parallel horospheres that are centered at the same point q
∞
at infinity. Let G be the
(identity component of) the subgroup of I
o
(H
n
)=SO
o
n,1
leaving F invariant. Then
G acts tr ansitively on H
n
since it contains the solvable subgroup of SO
o
n,1
fixing
the point q
∞
.Letp ∈ H
n
and v ∈ T
p
M be perpendicular to the horosphere in F
containing p. Then the isotropy group G
p
, acting via the isotropy representation,
fixes v. Moreover , G
p
restricted to (Rv)
⊥
coincides with SO((Rv)
⊥
) SO
n−1
.If
Corollary 9.2.4 holds, then H
n
would be reducible (in this case it would split off a
line), which is a contradiction.
9.3 Totally skew one-forms with values in a Lie algebra
We will extend here Simons’ definition of holonomy systems to algebraic 1-
forms that are totally skew an d have values in a Lie algebra.
Let V be a Euclidean vector space and G be a connected Lie subgroup of SO(V).
Let Θ : V → g ⊂ so (V) be linear and such that Θ
x
y,z is an algebraic 3-form on
V. Such a triple [V,Θ,G] is called a skew-torsion holonomy system. A skew-torsion
holonomy system is said to be
- irreducible,ifG acts irreducibly on V;
- transitive,ifG acts transitively on the unit sphere of V;
- symmetric,ifg(Θ)=Θ for all g ∈ G,whereg(Θ)
v
= g
−1
Θ
g.v
g.
Let [V,Θ
α
,G],
α
∈ I, be a family of skew-torsion holonomy systems and define
F = {g(Θ
α
) : g ∈G,
α
∈I}.Letg
be the linear sp an of the set {Θ
x
: Θ ∈F , x ∈V}.
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