186 Submanifolds and Holonomy
the p roduct of the connected component of the extrinsic group of isometries of each
factor. The corollary also holds for homogeneous curves. In fact, if M = G ·p with G
connected and dim M = 1, then any element g in the isotropy group G
p
acts trivially
on T
p
M.Sinceg is an intrinsic isometry of M,theng is the identity transformation
on M. Then the fixed point set of g in R
n
is an affine subspace containing M. Thus g
is the identity if M is full.
5.2.2 Computation of the normal holonomy of orbits
The holonomy group Hol(M) of a locally irreducible homogeneous Riemannian
manifold M = G/H can be computed from G. Indeed, Kostant [188] proved that if M
is not Ricci-flat, then the Lie algebra of the holonomy group of M is (algebraically)
generated by the skewsymmetric endomorphisms given by the Nomizu operators
∇X
∗
, X ∈ g (see Section 2.3). Actually, the assumption that M is not Ricci-flat can
be dropped, since Alekseevski˘ı and Kimelfeld [9] proved that a homogeneous non-
flat Riemannian manifold cannot be Ricci-flat (see also [14, page 553]).
Let us now turn to the case of a full irreducible orbit M = G ·p of a representation
G ⊂ I(R
n
). We have the following analogous result of Olmos and Salvai for the
computation of the normal holonomy group in terms of G.
Theorem 5.2.7 (Olmos-Salvai [264]) Let G be a Lie subgroup of I(R
n
) and let
M = G · p be full and irreducible as a submanifold of R
n
with dimM ≥ 2.Then
the Lie algebra of the normal holonomy group Φ
p
is (algebraically) generated by
the orthogonal projections of the Killing vector fields on R
n
induced by G onto the
affine subspace p +
ν
p
M . Moreover, Φ
p
=
¯
G
p
Φ
∗
p
where Φ
∗
p
is the restricted norma l
holonomy group and
¯
G
p
= {d
p
g|
ν
p
M
: g ∈ G
p
} is the isotropy group regarded as the
subgroup of O(
ν
p
M) via the slice representation.
We introduce first some notations. Let X belong to the Lie algebra g of G and
consider the curve
γ
X
p
: [0, 1] → M , s → Exp(sX)p.
Observe that
γ
tX
p
(1)=
γ
X
p
(t).Wethendefine the operator
A
X
:
ν
p
M →
ν
p
M
ξ
→ (X .
ξ
)
⊥
=
d
dt
t=0
d
p
Exp(tX)(
ξ
)
⊥
=
D
⊥
dt
t=0
d
p
Exp(tX)(
ξ
),
where (·)
⊥
denotes the orthogonal projection onto
ν
p
M.
Remark 5.2.8
ξ
→ A
X
ξ
is the Killing vector field on the normal space
ν
p
M ob-
tained by projectin g the Killing vector field defined by X .
Let
τ
⊥
p,X
denote ∇
⊥
-parallel transport along
γ
X
p
.(So
τ
⊥
p,tX
is the parallel transport
Rank Rigidity of Submanifolds and Normal Holonomy of Orbits 187
along
γ
X
p
from 0 to t.) Then we have
A
X
ξ
=
d
dt
t=0
(
τ
⊥
p,tX
)
−1
◦dExp(tX)(
ξ
). (5.8)
(We omit the p oint in the differential dExp(tX) here and in the sequel for the sake of
simplicity.) Note that A
X
is skewsymmetric with respect to the induced inner prod-
uct on
ν
p
M,thatis,A
X
ξ
,
η
+
ξ
,A
X
η
= 0. Therefore we have A
X
∈ so(
ν
p
M).
Moreover, A
X
belongs to the normalizer of the normal holonomy algebra, since
(
τ
⊥
p,tX
)
−1
◦d(ExptX) belongs to the normalizer of the normal holonomy group.
Remark 5.2.9 Let M = G · p be a full and irreducible submanifold o f R
n
with
dimM ≥ 2andlet
ξ
∈ (
ν
p
M)
0
.ThendExp(tX)(
ξ
) is parallel in the normal con-
nection for any X ∈ g (see Remark 5.2.1). Then (
τ
⊥
p,tX
)
−1
◦dExp(tX )(
ξ
)=
ξ
and so
A
X
ξ
= 0. Since A
X
belongs to the normalizer of the normal holonomy algebra at p,
it follows from Lemma 5.2.2 that A
X
belongs to the normal holonomy algebra at p.
Let g ∈ G.Theng ◦
γ
X
p
=
γ
X
gp
and hence
dg◦
τ
⊥
p,tX
◦dg
−1
|
ν
p
M
=
τ
⊥
gp,tX
(we omit the point in the differential dg also here and in the sequel for the sake of
simplicity). From this it is not hard to see that
(
τ
⊥
p,tX
)
−1
◦dExp(tX)|
ν
p
M
is a one-parameter group of linear isometries of
ν
p
M. From Equation (5.8) we have
dExp(−tX) ◦
τ
⊥
p,tX
= e
−tA
X
, and therefore
τ
⊥
p,tX
= Exp(tX) ◦e
−tA
X
(5.9)
is an explicit formula for computin g ∇
⊥
-parallel transport along
γ
X
p
from 0 to t.
5.2.3 Parallel transport along broken Killing lines
We continue using the notations in Section 5.2.2. Let p ∈ M, g ∈ G, Y ∈ g and
consider the curve
γ
Y
gp
(t)=Exp(tY )gp. This is the integral curve of the Killing vector
field Y
∗
,whereY
∗
q
=
d
dt
t=0
Exp(tY )q, with initial condition
γ
Y
gp
(0)=gp.Wehave
γ
Y
gp
= g ◦
γ
Ad(g
−1
)Y
p
and so, by Equation (4.8) on page 159,
τ
⊥
gp,Y
◦dg = dg◦
τ
⊥
p,Ad(g
−1
)Y
. (5.10)
Let X
1
,...,X
r
∈g and g
i
= Exp(X
i−1
)...Exp(X
1
) ∈ G for i = 1,...,r + 1(g
1
= id).
Consider the broken Killing line
β
obtained b y gluing together the integral curves
β
i
=
γ
X
i
g
i
p
, t ∈ [0,1], i = 1,...,r, n amely
β
=
β
1
∗...∗
β
r
and
β
i
(t)=Exp(tX
i
)Exp(X
i−1
)...Exp(X
1
)p.
188 Submanifolds and Holonomy
Using Equations (5.9) and (5.10) it is straightforward to compute ∇
⊥
-parallel trans-
ports along
β
(which is the composition of the parallel transports along the curves
β
i
), namely
τ
⊥
β
= dg
r+1
◦e
−A
Z
r
e
−A
Z
r−1
...e
−A
Z
1
(5.11)
where Z
i
= Ad(g
−1
i
)X
i
.
A broken Killing line whose pieces are integral curves of Killing vector fields
induced by G is called G-broken Killing line. The following lemma asserts that ∇
⊥
-
parallel transport along loops which are G-broken Killing lines gives the normal
holonomy g roup. The proof will only be sketched since it requires a background on
connections on principal bundles.
Lemma 5.2.10 Let M = G · p be an extrinsically homogeneous submanifold of R
n
.
Let
˜
Φ
p
be the group obtained by parallel transport along loops based at p that are
G-broken Killing lines. Then
˜
Φ
p
coincides with the normal holonomy group Φ
p
at p.
Proof (sketch) Let P be the set of loops based at p that are G-broken Killing lines.
Let c
0
,c
1
∈ P be homotopic loops that are homotopic via a family of loops based
at p. It is standard to show that there exists a piecewise smooth homotopy c
s
∈ P ,
s ∈[0, 1].Thens →
τ
⊥
c
s
is a piecewise smooth curve that lies in
˜
Φ
p
, a Lie subgroup of
O(
ν
p
M) (see [178, vol. I, Appendix 5]). Let
φ
s
and
ψ
t
be the flows associated with
two arbitrary Killing vector fields X ,Y induced by G. Then, for any q ∈ M, h(s,t)=
(
φ
s
◦
ψ
t
)q is a parametrized surface whose coordinate lines are integral curves of
Killing vector fields induced by G. The same argument as in Section 3.1.6 shows
that R(X
q
,Y
q
) belongs to the Lie algebra of
˜
Φ
q
.
Let B(
ν
M) be the orthonormal frame bundle of the normal bundle
ν
M. Then the
tangent spaces to the orbits of the groups
˜
Φ
q
define an integrable subdistribution V
of the vertical distribution of B(
ν
M). The distribution V is invariant under the flow
of the horizontal lift of any Killing vector field (since
˜
Φ conjugates under parallel
transpor t along integral curves of Killing vector fields induced by G). Let H be the
horizontal distribution of B(
ν
M). The horizontal part of the bracket of any two basic
vector fields is the curvature (applied to the corresponding basis). But the curvature
lies in V .ThisimpliesthatD = H ⊕V is integrable. Thus D contains the distribu-
tion given by the holonomy subbundles, so
˜
Φ
p
contains the normal holonomy group
at p. The lemma follows because the inclusion
˜
Φ
p
⊂ Φ
p
is trivial.
We return to the operators A
X
.Letk be the smallest subalgebra of so(
ν
p
M) which
contains A
X
for all X ∈g.LetK be the connected Lie subgroup of SO(
ν
p
M) with Lie
algebra k. From the construction of K we obtain that ¯gK ¯g
−1
= K,where ¯g = g|
ν
p
M
for g ∈ G
p
.So
¯
G
p
K is a group.
Remark 5.2.11 Let X belong to the Lie algebra of the isotropy group G
p
.Then
Exp(tX)p = p and so d
p
Exp(tX)|
ν
p
M
= e
tA
X
by Equation (5.9). This shows that
¯
g
p
⊂ k,where
¯
g
p
is the Lie algebra of
¯
G
p
= {g|
ν
p
M
: g ∈ G
p
}.
We can now provide a proof of Theorem 5.2.7.
Rank Rigidity of Submanifolds and Normal Holonomy of Orbits 189
Proof of Theorem 5.2.7 The group
¯
G
p
K contains the normal holonomy group Φ
p
by equation (5.11) and Lemma 5.2.10. The opposite inclusion follows from Remark
5.2.9 and Corollary 5.2.6, so Φ
p
=
¯
G
p
K altogether. Moreover, by Remark 5.2.11, the
normal holonomy algebra at p coincides with k.Thisfinishes the proof.
Remark 5.2.12 Suppose M = K · p is an orbit of an s-representation and take the
reductive decomposition k = k
p
⊕m with m = k
⊥
p
. From Lemma 3.1.5 we know
[k
p
,m]
⊥
= 0. This means that A
X
= 0ifX ∈ m. Thus, Theorem 5.2.7 gives an al-
ternative proof of Theorem 3.1.7. Namely, the normal holonomy representation of M
coincides with the slice representation, that is, the (effectivized) action of the isotropy
group K
p
on the normal space
ν
p
M.
Remark 5.2.13 Theorem 5.2.7 is used in [141] for the geometric characterization
of orthogonal representations with copolarity one (and for their classification in the
irreducible case, see Remark 2.3.13 for the definition of copolarity).
We finish the chapter by formulating the following conjecture:
Conjecture 5.2.14 (Olmos [257]) Let M be a full homogeneous submanifold of the
sphere S
n−1
with dimM ≥ 2 and which is not an orbit o f an s-representation. Then
the normal holonomy group of M acts transitively on the sphere of the normal space
(in particular the normal holonomy group acts irreducibly).
For dim M = 2 the conjecture is true, since the normal holonomy group must
always be transitive on unit normal vectors by Theorem 3.5.2. For dimM = 3the
conjecture is also true by a result of Olmos and Ria˜no-Ria˜no ( [263, Theorem B]),
but the arguments are very involved, including delicate topological considerations.
The conjecture is also true if the normal holonomy group of M acts irreducibly and
the codimension is the maximal possible one, namely
1
2
n(n+1)−1 =
1
2
(n+2)(n−1)
with n = dimM ( [263, Theorem A]). This last result is also true if one replaces the
homogeneity condition by the assumption that M is a minimal submanifold of the
sphere (see [263, Theorem C]).
5.3 Exercises
Exercise 5.3.1 Let M be an isoparametric hypersurface of the sphere S
n
and let M
i
be a focal submanifold. Prove that (
ν
M
i
)
0
= {0}.
Exercise 5.3.2 Let g : [a,b]×[c,d] →M be a p iecewise differentiable map with vari-
ables s,t and M be an immersed submanifold o f a Riemannian manifold N. Assume
that R
⊥
(
∂
g
∂
s
,
∂
g
∂
t
)=0. Let, for i ∈{1, 2}, c
i
: [0,1] → [a, b] ×[c, d] be two piecewise
differentiable curves with c
1
(0)=c
2
(0) and c
1
(1)=c
2
(1). Prove that
τ
⊥
g◦c
1
=
τ
⊥
g◦c
2
,
190 Submanifolds and Holonomy
where
τ
⊥
is the parallel transport in the normal connection of M. Prove a similar
result for the Levi-Civita parallel transport. [Remark: These are indeed special cases
of flat connections induced on pull b ack vector bundles (cf. [279]).]
Exercise 5.3.3 Let f : M → R
n
be a connected submanifold. Then rank
loc
f
(M)
q
is
constant on M if and only if rank
loc
f
(M)=rank
f
(M).
Exercise 5.3.4 Prove equation (5.1).
Exercise 5.3.5 Let f : M →R
n
be an immersed simply connected submanifold with
rank
f
(M) ≥ 1 and assume that the number of curvature normals is constant on M.
Prove that the curvatur e normals are globally defined smooth normal vector fields
(assuming the local version). [Hint: Let H be the subset of
ν
M consisting of all
curvature normals at any point. Prove that H is a differentiable manifold and that
the projection
π
:
ν
M → M restricted to H is a covering map. Thus
π
restricted to
any connected component of H is a diffeomorphism. The inverse map is a globally
defined curvature normal field.]
Exercise 5.3.6 Prove that Corollary 5.2.6 is not true if M is not full.
Exercise 5.3.7 Let M be a compact full submanifold of R
n
with parallel second fun-
damental form. Then any Killing vector field on M extends uniquely to a Killing
vector field on R
n
.[Hint:SinceM is locally symmetric, any bounded Killing vec-
tor field lies in the Lie algebra of the transvections. Let p ∈ M be a fixed point and
let g be a transvection of M.Set
τ
γ
= d
p
g. Then the isometry ˜g of R
n
defined by
˜g(p)=g(p ), d
p
˜g|
T
p
M
=
τ
γ
and d
p
˜g|
ν
p
M
=
τ
⊥
γ
leaves M invariant.]
Exercise 5.3.8 (based on an unpublished proof of Ferus’ Theorem by Hulett and Ol-
mos). Let M be a compact full submanifold of R
n
.Forv ∈R
n
let h
v
be the restriction
to M of the height function in the direction of v (that is, h
v
(x)=x,v with x ∈ M).
Let X
v
be the gradient of h
v
(M is en dowed with the induced metric). Prove:
(i) The second fundamental form of M is parallel if and only if [X
v
,X
w
] is a Killing
vector field on M for all v, w ∈ R
n
.
(ii) Assume that the second fundamental form of M is p a rallel. Let k be the Lie
algebra of Killing vector fields on R
n
that are tangent to M.Define on the
vector space k ⊕R
n
the following bracket: [X ,v]=X .v if X ∈ k and v ∈ R
n
;
[X,Y ] is the bracket of k if X ,Y ∈ k; [v,w] is the extension to R
n
of the Killing
vector field [X
v
,X
w
] (see Exercise 5.3.7).
(a) Prove that k ⊕R
n
is an orthogonal involutive Lie algebra (see [347]).
(b) M is (orthogonally) equivalent to an orbit of an s-representation.
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