The Normal Holonomy Theorem 131
3.5.4 Normal holonomy of surfaces
Using the bound of Theorem 3.5.1 on the number of normal holonomy repre-
sentation components and properties of holonomy systems we will now prove the
following result.
Theorem 3.5.2 Let M be a surface in a Euclidean space with the property that
around any point it is not contained in a sphere or in a proper affine subspace. Then
the local normal holonomy group is either trivial or it acts transitively on the unit
sphere of the normal space.
Proof Assume that the local normal holonomy g roup Φ
loc
p
is not trivial and let p ∈
M. We will show that Φ
loc
p
acts transitively on the unit sphere of the normal space
ν
p
M.
Note first that there are no parallel nontrivial umbilical normal sections, o r else M
would be contained either in a sphere or in an affine subspace. Furthermore, the factor
V
0
(that is, the fixed point set of the normal holonomy group) is trivial, for otherwise
there should exist a nonumbilical parallel normal vector field
ξ
(around arbitrary
close points to p). This is impossible, because it would imply that the normal bundle
is flat. In fact, A
ξ
would commute with all other shape operators and by the Ricci
equations all shape operators would commute, since dimM = 2, and so R
⊥
= 0.
The bound given in Theorem 3.5.1 forces the local normal holonomy group to act
irreducibly. We now claim that Φ
loc
p
is transitive on the unit sphere of
ν
p
M. Suppose
that this is not the case. Then there exists a point q arbitrarily close to p such that
R
⊥
q
= 0andso[
ν
q
M,R
⊥
q
,Φ
loc
q
] is an irreducible nontransitive holonomy system.
This holonomy system is symmetric by Theorem 3.3.7. In particular, the first normal
space N
1
q
coincides with
ν
q
M. This is because, otherwise, there would exist
ξ
∈
ν
q
M
with A
ξ
= 0andsoR
⊥
q
(
ξ
,·)=0, contradicting irreducibility. The map
ξ
→ A
ξ
is
injective and so we have that dim
ν
q
M ≤ 3 (note that the dimension of the space
of 2 ×2 symmetric matrices is 3). Now, it is not difficult to see that an irreducible
symmetric space of dimension at most 3 must be of rank one (Exercise 3.6.18, or
else use the classification of symmetric spaces, cf. Appendix). This means the normal
holonomy group is transitive on the unit sphere of the normal space.
Note that, for a surface contained in a sphere but not contained in a proper affine
subspace (or, equivalently in a smaller dimensional sphere), there is an analogous
result (Exercise 3.6.16).
3.5.5 Computing the normal holonomy group
The description of the Lie algebr a L (Φ
p
) given by the Ambrose-Singer Theo-
rem is not very explicit since the normal holonomy algebra depends also on parallel
transport
τ
⊥
γ
. Thus it is not very useful for explicit computations. In some cases, like
homogeneous submanifolds, one can compute the normal holonomy group by taking
the covariant derivatives of the normal curvature tensor.