The Skew-Torsion Holonomy Theorem 251
Let n > 3 and assume that the theorem holds for dimV < n. We distinguish sev-
eral cases.
Case (a). Assume that there is some
˜
Θ ∈ F such that the derived 2-form
˜
Ω
is nonzero, where
˜
Ω
v,w
=(
˜
Θ
v
.
˜
Θ)
w
. Choose now some v ∈ V so that
˜
Θ
v
=
˜
Ω
v,·
is
nonzero. Note that
˜
Θ
v
is a totally skew 1-form on (Rv)
⊥
with values in the isotropy
algebra g
v
(see Remark 9.4.1). So, by Proposition 9.3.1, the isotropy group H = G
v
at v of the sphere S
n−1
= G ·v satisfies the hypotheses of Corollary 9.2.4 with k ≥ 1,
since
˜
Θ
v
= 0. Then k = 1andsoH acts irreducibly on the tangent space T
v
S
n−1
of
S
n−1
. If the isotropy group H is not transitive on the unit sphere of T
v
S
n−1
=(Rv )
⊥
,
then [(Rv)
⊥
,
˜
Θ
v
,H] is an irreducible nontransitive skew-torsion holonomy system.
Then, by the Weak Skew-Torsion Holonomy Theorem 9.3.2, [(Rv)
⊥
,
˜
Θ
v
,H] must be
symmetric. Moreover, by Proposition 9 .3.3 the Lie algebra h coincides with the linear
span of {
˜
Θ
v
x
: x ∈ (Rv)
⊥
}. Hence we are under the assumptions of Proposition 9.6.9
and so S
n−1
would be isometric to a simple Lie group with a bi-invariant Riemannian
metric which must have rank at least two. This is a contradiction, since in such a
Lie group there are totally geodesic and flat submanifolds of dimension at least 2.
Therefore, H must be transitive on the unit sphere of (Rv)
⊥
.Then[(Rv)
⊥
,
˜
Θ
v
,H]
satisfies the hypotheses of the theorem and dim(Rv)
⊥
= n −1 < n = dimV. So, by
induction, H = SO((Rv)
⊥
), which implies that G = SO(V) since H = G
v
.
Case (b). Assume that
˜
Θ
v
.
˜
Θ = 0forall
˜
Θ ∈ F and v ∈ V, that is, any element
˜
Θ ∈F defines a Lie bracket [u , v]
˜
Θ
=
˜
Θ
u
v on g (and so, ad
˜
Θ
u
=
˜
Θ
u
). In Remark 9.5.3
it is shown that this case can actually occur.
Let G
˜
Θ
denote the Lie subgroup of G associated with the subalgebra {
˜
Θ
v
: v ∈ V}
of g. Choose 0 =
˜
Θ ∈ F . By replacing
˜
Θ with its projection onto a some irreducible
subspace of G
˜
Θ
we may assume that V decomposes as V = V
0
⊕V
⊥
0
into
˜
Θ-invariant
subspaces such that
˜
Θ is trivial on V
0
and irreducible on V
⊥
0
.So
˜
Θ
V
0
= {0} and G
˜
Θ
acts irreducibly on V
⊥
0
.
There are three subcases that require different arguments, depending on whether
d = dimV
0
is 0, 1 or ≥ 2. For these subcases we will use that G acts on V as the
isotropy representation of a simple Riemannian symmetric space (see Lemma 9.4.4).
Let 0 = R be the unique, up to scalar mu ltiples, alge braic Riemannian curvature
tensor on V such that [V,R,G] is a symmetric (transitive) holonomy system, that is,
g(R)=R for all g ∈G. Note that g coincides with the linear span of {R
u,v
: u,v ∈ V}.
We will show that R has constant curvatures and so g = so(V).SinceG preserves
R and acts transitively on the unit sphere of V, we must only show that there exists
v = 0 such that the Jacobi operator J
v
= R
·,v
v : (Rv)
⊥
→ (Rv)
⊥
is a multiple of the
identity transformation.
Before continuing with the proof let us show the following fact: any normal space
ν
z
of the orbit G
˜
Θ
·z at z is R-totally geodesic (or equivalently, curvature-invariant),
that is, R
ν
z
,
ν
z
ν
z
⊂
ν
z
. Recall, by Remark 9.4.3, that the normal space
ν
z
is the fixed set
in V of the 1-parameter subgroup L = {e
t
˜
Θ
z
: t ∈ R} of G.Letg ∈ L and u,x,y ∈
ν
z
.
Since g(R)=R,weget
R
u,x
y = g(R)
u,x
y = g
−1
.R
g.u,g.x
g.y = g
−1
.R
u,x
y.
252 Submanifolds and Holonomy
So, R
u,x
y is a fixed vector of L.ThenR
u,x
y ∈
ν
z
and hence
ν
z
is R-totally geodesic
(that is,
ν
z
is invariant under R).
Subcase (b
1
). Assume that dim V
0
≥2. Define, as before,
ν
v
=
ν
v
(G
˜
Θ
·v) for v ∈
V. Observe that V
0
⊕Rv ⊂
ν
v
and recall that
ν
v
is the set of fixed points of L = {e
t
˜
Θ
z
:
t ∈R}.Thereexist0= v ∈V and
ˆ
Θ ∈F such that the 3-form
ˆ
Θ
·
·,·, w hen restricted
to
ν
v
, is not identically zero. Otherwise, if 0 = u ∈ V
0
is fixed, then
ˆ
Θ
v
u,V
0
= {0}
for all
ˆ
Θ ∈ F and v ∈ V,andsoV
0
⊂
ν
u
(G ·u), which is a contradiction since G is
transitive on the sphere and dim V
0
≥ 2. By perturbating v slightly we may assume
that v /∈V
0
. Moreover, if v
is the orthogonal projection of v onto V
⊥
0
,weget
ν
v
=
ν
v
.
So, we may assume that 0 = v ∈ V
⊥
0
.
Observe, since G is transitive on the sphere, that G ·v is a princ ipal orbit. Since
v ∈
ν
v
, we can now apply Lemma 9.4.2 to conclude that the cohomogeneity of G
ν
v
on
ν
v
is one (that is, it is transitive on the sphere of the normal space). Moreover, there
exists a totally skew 1-form
ˆ
Θ
v
= 0on
ν
v
with values in the Lie algebra
¯
g
ν
v
of
¯
G
ν
v
=
G
ν
v
|
ν
v
(using the notation of Lemma 9.4.2). Since dim
ν
v
< n = dimV we obtain,
by induction, that
¯
G
ν
v
= SO(
ν
v
).Letnowv
0
and w
0
be perpendicular vectors of unit
length that both belong to V
0
and let
λ
= J
v
0
w
0
,w
0
. Choose now an arbitrary vector
z ∈ V that is perpendicular to v
0
.Letv
= gv and g ∈ G
˜
Θ
be such that z ∈
ν
v
(that is,
by choosing v
∈ G
˜
Θ
·v such that the height function x →z,x on this orbit attains
its maximum value). Since
ν
v
= g
ν
v
and G
ν
v
= gG
ν
v
g
−1
,wehaveG
ν
v
= SO(
ν
v
).
Recall that R leaves invariant the subspace
ν
v
(that contains the vectors v
0
,w
0
). Since
¯
G
ν
v
= SO(
ν
v
) preserves R, we conclude that the restriction of R to
ν
v
has constant
sectional curvature
λ
. So, in particular J
v
0
z =
λ
z. Thus J
v
0
: (Rv
0
)
⊥
→ (Rv
0
)
⊥
is a
multiple of the identity. This finishes the proof of this subcase.
Subcase (b
2
). Assume that dim V
0
= 1. Let v ∈ V
0
be of unit length. Observe
that G
˜
Θ
preserves R, fixes v and acts irreducibly on (Rv)
⊥
.ThenG
˜
Θ
commutes with
J
v
and therefore J
v
is a multiple of the identity.
Subcase (b
3
). Assume that dim V
0
= 0, that is, G
˜
Θ
acts irreducibly on V.Inthis
case the principal orbits of G
˜
Θ
are irreducible and full isoparametric submanifolds
of V. In fact, G
˜
Θ
acts as the adjoint r epresentation o f (V, [·,·]
˜
Θ
) and so it acts polarly
(see Theorem 2.3.15). The cohomogeneity of G
˜
Θ
on V is at least 2. Otherwise, the
Lie group associated with (V, [·,·]
˜
Θ
) wouldbeofrank1andtherefore,byRemark
9.3.4, dimV = 3, which is a contradiction since we assume n > 3.
Let M = G
˜
Θ
·v be a principal orbit, where v is of unit length. Let
ξ
belong to the
normal space
ν
v
of M at v such that the shape operator A
ξ
has all of its eigenvalues
λ
1
,...,
λ
g
different from zero and g ≥ 2. Such a
ξ
can be chosen by perturbating
slightly the position vector, since the codimension of M is at least 2 and so M,since
it is full, is not umbilical. Let E
1
,...,E
g
be the eigenspaces of A
ξ
associated with
λ
1
,...,
λ
g
respectively. Let us write
V =
ν
v
⊕E
1
⊕...⊕E
g
The Skew-Torsion Holonomy Theorem 253
and define v
i
= v +
λ
−1
i
ξ
for i ∈{1,...,g}. The normal space at v
i
of the focal orbit
M
i
= G
˜
Θ
·v
i
is
ν
v
i
= C (v
i
)={
ξ
∈V :
˜
Θ
v
i
ξ
= 0} =
ν
v
⊕E
i
,
which is
˜
Θ-invariant since C (v
i
) is a subalgebra of (V,[·,·]
˜
Θ
). Moreover, the restric-
tion of
˜
Θ to
ν
v
i
is nonzero since
ν
v
i
is nonabelian. In fact, it contains properly the
maximal abelian subalgebra
ν
v
. Recall from Remark 9.4.3 that
ν
v
i
is the set of fixed
points of {e
t
˜
Θ
v
i
: t ∈ R} and hence
ν
v
i
is invariant under R. By Lemma 9.4.2,
¯
G
ν
v
i
is
transitive on the unit sphere of
ν
v
i
. Hence, by induction, since dim
ν
v
i
< dimV,we
get
¯
G
ν
v
i
= SO(
ν
v
i
).
As the r estriction R
i
of R to
ν
v
i
is fixed under
¯
G
ν
v
i
= SO(
ν
v
i
), R
i
has constant
curvatures, say equal to
μ
.IfW
i
=(Rv )
⊥
∩
ν
i
,thenJ
v
|
W
i
=
μ
id
W
i
. We omitted the
subscript i for
μ
, since it does not depend on i. In fact, let w ∈
ν
v
be of unit length and
perpendicular to v.Thenw as well as v belong both to any W
i
and
μ
= R
w,v
v,w.
This shows that
μ
is independent of i ∈{1,...,g}. Since, (Rv)
⊥
coincides with the
linear span of
-
i
W
i
, we conclude that J
v
coincides on (Rv)
⊥
with
μ
id
(Rv)
⊥
.
This finishes the proof of Theorem 9.5.1.
The following result was obtained independently by Nagy [232] and by Olmos
and Reggiani [261]. Nagy used an algebraic approach based on the so-called Berger
algebras, classified by Berger [17].
Theorem 9.5.2 (Skew-Torsion Holonomy Theorem [232, 261]) Let [V,Θ,G], Θ =
0, be an irreducible skew-torsion holonomy system with G = SO(V).Then[V, Θ, G]
is symmetric and nontransitive. Moreover,
(i) (V,[·,·]) is an orthogonal simple Lie algebra of rank at least 2 with respect to
the bracket [x, y]=Θ
x
y;
(ii) G = Ad(H), where H is the connected Lie group associated with the Lie alge-
bra (V,[·,·]);
(iii) Θ is unique, up to a scalar multiple.
Proof The proof follows by combining the Weak Skew-Torsion Holonomy Theorem
9.3.2, Proposition 9.3.3 and Theorem 9.5.1.
Remark 9.5.3 Let us consid er in R
4
R ⊕so
3
the bracket given by the product of
the (trivial) bracket on R and the standard bracket on so
3
. Any bracket on R
4
defines
in a natural way a 3-form. Since the space of 3-forms on R
4
is canonically isometric
to R
4
, the group SO
4
acts transitively on the family of 3-forms of unit length. This
implies that any 3-form defines a bracket on R
4
which is orthogonally equivalent,
up to a scalar multiple, to the given one. Let now Θ = 0 be any totally skew 1-form
with values in so
4
.ThenΘ satisfies the equation Θ
x
.Θ = 0forallx ∈ R
4
.However,
[R
4
,Θ,SO
4
] is never a symmetric skew-torsion holonomy system.
254 Submanifolds and Holonomy
9.6 Applications to naturally reductive spaces
In this section we will present some applications of the Skew-Torsion Holonomy
Theorem 9.5.2 to naturally reductive spaces.
Let M = G/H be a simply connected naturally reductive Riemannian homo-
geneus space with naturally reductive decomposition g = h ⊕m, m T
p
M, p ∈ M
with H = G
p
.Let·,· be the G-invariant Riemannian metric on M. We continue
using the notations in Section 9.2. Suppose that the Riemannian metric ·, · is also
naturally reductive with respect to another decomposition, say M = G
/H
,andthe
metric is also naturally reductive with respect to the decomposition g
= h
⊕m
.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 between both con-
nections is
D
v
w = ∇
v
¯w
−
¯
∇
c
v
¯w
= −
1
2
[ ¯v
, ¯w
]
p
= −∇
v
¯w
.
The tensor D
,aswellasD (in the notation of Section 9.2), is totally skew.
We have
D
v
w −D
v
w = −∇
v
( ¯w − ¯w
)=−∇
v
Z,
where Z = ¯w − ¯w
vanishes at p.So,(∇Z)
p
∈
˜
h = Lie(I(M)
p
) (via the isotropy rep-
resentation). In fact,
e
t(∇Z)
p
= d
p
ϕ
Z
t
,
where
ϕ
Z
t
is the flow associated with Z. Observe that D −D
=
¯
∇
c
−∇
c
.
Then, if Θ = D −D
, Θ
·
w = −(∇
·
Z)
p
∈
˜
h, or equivalently,
Θ
w
· =(∇
·
Z)
p
∈
˜
h
since Θ is totally skew. So Θ
w
belongs to the full isotropy algebra for all w ∈ T
p
M.
Let
¯
h = linear span of {g(Θ)
w
: g ∈ I(M)
p
, w ∈ T
p
M}.
As in Section 9.3 we have that
¯
h is an ideal of
˜
h = Lie(I(M)
p
) ⊂ so(T
p
M).
Let
¯
H be the connected Lie subgroup of SO(T
p
M) with Lie algebra
¯
h. Then, by
what has been done for skew-torsion holonomy systems in Section 9.3,
T
p
M = V
0
⊕V
1
⊕...⊕V
k
(orthogonally)
and
I
o
(M)
p
= H
0
×H
1
×...×H
k
,
where H
0
acts only on V
0
and H
i
acts irreducibly on V
i
and trivially on V
j
if i = j,
The Skew-Torsion Holonomy Theorem 255
i ≥ 1. Moreover, such groups satisfy the assumptions of Theorem 9.2.3. Then, if M
is irreducible, we either have
I
o
(M)
p
= H
0
and T
p
M = V
0
or
I
o
(M)
p
= H
1
and T
p
M = V
1
.
If I
o
(M)
p
= H
0
,thenallg(Θ)=0 and in particular Θ = 0andso∇
c
=
¯
∇
c
.
Let us analyze the remaining case I
o
(M)
p
= H
1
. From the Skew-Torsion Holonomy
Theorem, there are only two cases:
(a) H
1
= SO(T
p
M). In this case M has constant curvature. Then M = S
n
or its
symmetric dual M = H
n
. The latter case will be excluded in Proposition 9.6.2, except
for n = 3, in which case the hyperbolic space H
3
is the dual space of the Lie group
S
3
= Spin
3
.
(b) H
1
acts on T
p
M as the adjoint representation of a compact simple Lie group.
Then M is isometric to a comp act simple gr oup with a bi-invariant Riemannian metric
(a classification free and geometric proof of this fact is given in Proposition 9.6.9)
We have proved the following result (see [261, 262]).
Theorem 9.6.1 Let M be a simply connected, irreducible, naturally reductive Rie-
mannian homogeneous space. Assume that M is neither isometric to a sphere nor to
a compact simple Lie group with a bi-invariant Riemannian metric or its symmetric
dual. Then the canonical connection is unique. (In particular, any isometry of M is
affine with respect to the canonical connection).
Proposition 9.6.2 The real hyperbolic space H
n
,n= 3 , admits a unique naturally
reductive decompositon: the Cartan decomposition of H
n
= SO
o
n,1
/SO
n
.
Proof Let G be a connected Lie subgroup of I
o
(H
n
)=SO
o
n,1
that acts transitively
on H
n
andsuchthatH
n
= G/H is a naturally reductive space. If G is semisimple, it
is standard to show that G = SO
o
n,1
. In fact, let K be a maximal compact subgroup
of G.ThenK has a fixed point, say p . We may assume that H is the isotropy group
at p and so H = K since K is maximal. Hence (G, H) is a presentation of H
n
as an
effective Riemannian symmetric pair, and therefore G = SO
o
n,1
(otherwise, H
n
would
have two different presentations as an effective Riemannian symmetric pair). We
will prove for n = 3 that there is only one reductive decompo sition (not necessarily,
a priori, naturally reductive) of the pair (SO
o
n,1
,SO
n
). Using Remark 9.6.3 we can
transfer this question to the sphere and we are done by Remark 9.6.4.
If G is non-semisimple, then G contains a nontrivial normal abelian Lie subgroup
A. It is a well known fact that either A fixes a unique point at infinity or A translates
a unique geodesic. If A translates a unique geodesic, then G leaves this geodesic
invariant since A is a normal subgroup of G and so G cannot be transitive, which is
a contradiction. So, let q
∞
be the unique point at infinity that is fixed by A.Observe
that q
∞
must be fixed by G. In fact, since A is a normal subgroup of G, any element
of G leaves invariant the fixed set {q
∞
} of A at infinity. Let F be the foliation of
H
n
by parallel horospheres centered at q
∞
.ThenG leaves invariant F .Letp ∈ H
n
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