Normal holonomy of orbits and Veronese submanifolds

Abstract

It was conjectured, twenty years ago, the following result that would
generalize the so-called rank rigidity theorem for homogeneous Euclidean
submanifolds: let M^n, n>=2, be a full and irreducible homogeneous submanifold
of the sphere $S^{N-1}\subset R^N$ and such that the normal holonomy group is
not transitive (on the unit sphere of the normal space to the sphere). Then M^n
must be an orbit of an irreducible s-representation (i.e. the isotropy
representation of a semisimple Riemannian symmetric space).
If n=2, then the normal holonomy is always transitive, unless M is a
homogeneous isoparametric hypersurface of the sphere (and so the conjecture is
true in this case). We prove the conjecture when n=3. In this case M^3 must be
either isoparametric or a Veronese submanifold. The proof combines geometric
arguments with (delicate) topological arguments that uses information from two
different fibrations with the same total space (the holonomy tube and the
caustic fibrations).
We also prove the conjecture for n>= 3 when the normal holonomy acts
irreducibly and the codimension is the maximal possible n(n+1)/2. This gives a
characterization of Veronese submanifolds in terms of normal holonomy. We also
extend this last result by replacing the homogeneity assumption by the
assumption of minimality (in the sphere).
Another result of the paper, used for the case n=3, is that the number of
irreducible factors of the local normal holonomy group, for any Euclidean
submanifold M^n, is less or equal than [n/2] (which is the rank of the
orthogonal group SO(n)). This bound is sharp and improves the known bound
n(n-1)/2.

Full text

NORMAL HOLONOMY OF ORBITS AND VERONESE
SUBMANIFOLDS
CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Abstract. It was conjectured, twenty years ago, the following result that
would generalize the so-called rank rigidity theorem for homogeneous Eu-
clidean submanifolds: let Mn,n≥2, be a full and irreducible homogeneous
submanifold of the sphere SN−1⊂RNand such that the normal holonomy
group is not transitive (on the unit sphere of the normal space to the sphere).
Then Mnmust be an orbit of an irreducible s-representation (i.e. the isotropy
representation of a semisimple Riemannian symmetric space).
If n= 2, then the normal holonomy is always transitive, unless Mis a
homogeneous isoparametric hypersurface of the sphere (and so the conjecture
is true in this case). We prove the conjecture when n= 3. In this case M3
must be either isoparametric or a Veronese submanifold. The proof combines
geometric arguments with (delicate) topological arguments that uses informa-
tion from two different fibrations with the same total space (the holonomy
tube and the caustic fibrations).
We also prove the conjecture for n≥3 when the normal holonomy acts
irreducibly and the codimension is the maximal possible 1
2n(n+ 1). This gives
a characterization of Veronese submanifolds in terms of normal holonomy. We
also extend this last result by replacing the homogeneity assumption by the
assumption of minimality (in the sphere).
Another result of the paper, used for the case n= 3, is that the number
of irreducible factors of the local normal holonomy group, for any Euclidean
submanifold Mn, is less or equal than [ n
2] (which is the rank of the orthogonal
group SO(n)). This bound is sharp and improves the known bound 1
2n(n−1).
1. Introduction
The holonomy of the normal connection turns out to be a useful tool in Euclidean
submanifold geometry [BCO]. The most important applications of this tool were
the alternative proof of Thorbergsson theorem [Th], given in [O2], and the rank
rigidity theorems for submanifolds [O3, CO, DO] (see Section 2.1). Moreover, the
extension of Thorbergsson’s result to infinite dimensional geometry, given by [HL],
makes also use of normal holonomy.
It is interesting to remark that normal holonomy is related, in a very subtle way,
to Riemannian holonomy. Namely, by using submanifold geometry, with normal
holonomy ingredients, one can give short and geometric proofs of both Berger ho-
lonomy theorem [B] and Simons holonomy (systems) theorem [S] (see [O5, O6]).
Moreover, by applying this methods, it was proved in [OR] the so-called skew-
torsion holonomy theorem with applications to naturally reductive spaces.
Date: June 11, 2013.
Supported by: FaMAF-Universidad Nacional de C´ordoba and CIEM-Conicet.
MSC (2010): Primary 53C40; Secondary 53C42, 53C39.
Key words: normal holonomy, orbits of s-representations, Veronese submanifolds.
1
arXiv:1306.2225v1 [math.DG] 10 Jun 2013

2 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
The starting point for this theory was the normal holonomy theorem [O1] which
asserts that the (restricted) normal holonomy group representation, of a submani-
fold of a space form, is, up to a trivial factor, an s-representation. (equivalently, the
normal holonomy is a Riemannian non-exceptional holonomy). This implies that
the so-called principal holonomy tubes have flat normal bundle (holonomy tubes
are the image, under the normal exponential map, of the holonomy subbundles of
the normal bundle). Such tubes, despite to the classical spherical tubes, behaves
nicely with respect to products of submanifolds.
But the normal holonomy, which is invariant under conformal transformations of
the ambient space, gives much weaker information in submanifold geometry than
the Riemannian holonomy in Riemannian geometry. For instance, the reducibility
of the normal holonomy representation does not imply that the manifold splits.
So, interesting applications of the normal holonomy can be expected only within a
restrictive class of submanifolds. For instance:
(1) submanifolds with constant principal curvatures,
(2) complex submanifolds of the complex projective space
(3) homogeneous submanifolds.
For the first two classes of submanifolds there are “Berger-type” theorems.
For (1) one has the following reformulation of the Thorbergsson theorem [Th]:
a full and irreducible submanifold with constant principal curvatures, such that the
normal holonomy, as a submanifold of the sphere, is non-transitive must be either
a inhomogeneous isoparametric hypersurface or an orbit of an s-representation.
For (2) we have the following result [CDO]: a complete full and irreducible com-
plex submanifold Mof the complex projective space with non-transitive normal ho-
lonomy is the complex orbit (in the projectivized tangent space) of the isotropy
representation of a Hermitian symmetric space or, equivalently, Mis extrinsically
symmetric . This result is not true without the completeness assumption.
For the class (3) we have the rank rigidity theorem for submanifolds [O3, DO]: if
the normal holonomy of a full and irreducible Euclidean homogeneous submanifold
Mn=K.v,n≥2has a fixed non-null vector, then Mis contained in a sphere.
If the dimension of the fixed set of the normal holonomy has dimension at least 2,
then Mis an orbit of an s-representation (perhaps by enlarging the group K).
But this last result would be only a particular case of a Berger-type result that
it was conjectured twenty years ago in [O3]: if the normal holonomy of a full and
irreducible homogeneous submanifold Mnof the sphere, n≥2, is non-transitive
then Mis an orbit of an s-representation.
For n= 2 the normal holonomy must be always transitive or trivial (see [BCO],
Section 4.5 (c)).
The goal of this article is twofold. On the one hand, to give some progress on this
conjecture. On the other hand, to characterize the classical (Riemannian) Veronese
submanifolds in terms of normal holonomy.
If a submanifold Mnof the sphere has irreducible and non-transitive normal
holonomy, then the first normal space, as a Euclidean submanifold, coincides with
the normal space (see Remark 2.11). This imposes the restriction that the codi-
mension is at most 1
2n(n+ 1). We will prove the above mentioned conjecture in the
case that the normal holonomy acts irreducibly and the (Euclidean) codimension
is the maximal one 1
2n(n+ 1). The proof uses most of the techniques of the theory.
Moreover, the most difficult case is in dimension n= 3 for which we have to use

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 3
also delicate topological arguments involving two different fibrations on a partial
holonomy tube: the holonomy tube fibration and the caustic fibration.
We extend these resuls by replacing the homogeneity by the property that the
submanifold is minimal in a sphere. But the proof of this result is simpler than the
homogeneous case and a general proof works also for n= 3.
We also prove the sharp bound n
2on the number of irreducible factors of the
normal holonomy, which implies, from the above mentioned result, the conjecture
for n= 3 (see Proposition 6.1).
Let us explain our main results which are related to the so-called Veronese sub-
manifolds.
The isotropy representation of the symmetric space Sl(n+ 1)/SO(n+ 1) is nat-
urally identified with the action of SO(n+ 1), by conjugation, on the traceless
symmetric matrices. A Veronese (Riemannian) submanifold Mn, which has par-
allel second fundamental form, is the orbit of a matrix with exactly two eigen-
values, one of which has multiplicity 1. Being Ma submanifold with constant
principal curvatures, the first normal space ν1(M) coincides with the normal space
ν(M). Moreover, ν1(M) has maximal dimension. Namely, the codimension of M
is 1
2n(n+ 1).
The restricted normal holonomy of M, as a submanifold of the sphere, is the
image, under the slice representation, of the (connected) isotropy. Then the nor-
mal holonomy representation of Mis irreducible and it is equivalent to the isotropy
representation of Sl(n)/SO(n). So, the normal holonomy of Mis non-transitive if
and only if n≥3. We have the following geometric characterization of Veronese
submanifolds in terms of normal holonomy, which proves a special case of the con-
jecture on normal holonomy of orbits, when the normal holonomy, of a submanifold
of the sphere, acts irreducibly, not transitively and the codimension is maximal.
Theorem A. Let Mn⊂Sn−1+ 1
2n(n+1),n≥3, be a homogeneous submanifold
of the sphere. Then Mis a (full) Veronese submanifold if and only if the restricted
normal holonomy group of Macts irreducibly and not transitively.
For dimension 3 the conjecture on normal holonomy is true. Namely,
Theorem B. Let M3⊂SN−1be a full irreducible homogeneous 3-dimensional
submanifold of the sphere. Assume that the restricted normal holonomy group of
Mis non-transitive. Then Mis an orbit of an s-representation. Moreover, Mis
either a principal orbit of the isotropy representation of Sl(3)/SO(3) or a Veronese
submanifold.
The irreducibility and fullness condition on Mis always with respect to the
Euclidean ambient space.
We can replace, in Theorem A, the homogeneity condition by the assumption of
minimality in the sphere.
Theorem C. Let Mn,n≥3, be a complete (immersed) submanifold of the
sphere Sn−1+1
2n(n+1). Then Mnis, up to a cover, a (full) Veronese submanifold if
and only if Mis a minimal submanifold and the restricted normal holonomy group
acts irreducibly and not transitively.
The assumptions of homogeneity or minimality, in our main results, cannot be
dropped, since a conformal (arbitrary) diffeomorphism of the sphere transforms M

4 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
into a submanifold with the same normal holonomy but in general not any more
minimal. Last theorem admits a local version.
We will explain the main ideas in the proof of Theorem A, when n≥4.
Let ˜
Abe the traceless shape operator of M=H.v, i.e. ˜
Aξ=Aξ−1
nhH, ξiI d,
where His the mean curvature vector. Let us consider the map ˜
A, from the normal
space ¯νq(M) to sphere into the traceless symmetric endomorphisms Sim0(TqM).
Then ˜
Amaps normal spaces to the Φ(q)-orbits into normal spaces to the SO(n)-
orbits, by conjugation, in Sim0(TqM). By using the results in Section 2, which
are related to Simons theorem, we obtain that ˜
Ais a homothecy which maps the
normal holonomy group Φ(q) into SO(n). This implies that the eigenvalues of ˜
Aξ
do not change if ξis parallel transported along a loop. From the homogeneity, since
the group His always inside the ∇⊥-transvections, we obtain that the eigenvalues
of ˜
Aξ(t)are constant, if ξ(t) is a parallel normal field along a curve. Now we
pass to an appropriate, singular, holonomy tube, Mξ, where Aξhas exactly two
eigenvalues one of them of multiplicity 2. Let ˆ
ξbe the parallel normal field of Mξ
such that Mcoincides with the parallel focal manifold (Mξ)−ˆ
ξto Mξ. One obtains
that the three eigenvalue functions, ˆ
λ1,ˆ
λ2and ˆ
λ3=−1, of the shape operator ˆ
Aˆ
ξ
of Mξhave constant multiplicities. The two horizontal eigendistributions of ˆ
Aˆ
ξ,
let us say E1and E2, have multiplicities 2 and (n−2) respectively. The vertical
distribution is the eigendistribution associates to the constant eigenvalue −1. From
the above mentioned properties of ˜
Aand the tube formulas one obtains that ˆ
λ1
and ˆ
λ2are functionally related (so if one eigenvalue is constant along a curve the
other is also constant). From the Dupin condition, since dim(E1)≥2, ˆ
λ1, and
so ˆ
λ2, as previously remarked, are constant along the integral manifolds of E1. If
n≥4, the the same is true for the distribution E2. So, the eigenvalues of ˆ
Aˆ
ξ
are constant along horizontal curves. But any two points in a holonomy tube can
be joined by a horizontal curve. Then ˆ
Aˆ
ξhas constant eigenvalues and so ˆ
ξis
an isoparametric non-umbilical parallel normal field. Then, by the isoparametric
rank rigidity theorem, the holonomy tube Mξ, and therefore M, is an orbit of an
s-representation. From this we prove, without using classification results, that M
must be a a Veronese submanifold.
If n= 3, the proof is much harder, since the Dupin condition does not apply for
E2, and requires topological arguments, not valid for n > 3, as pointed out before.
2. Preliminaries and basic facts
In this section, as well as in the appendix, for the reader convenience, we recall
the basic notions and results that are needed in this article. We also include in this
part some new results that are auxiliary for our purposes. Some of them have a
small interest in its own right, or the proofs are different from the standard ones.
The general reference for this section is [PT, Te, BCO].
2.1. Orbits of s-representations and Veronese submanifolds.
A submanifold M⊂RNhas constant principal curvatures if the shape operator
Aξ(t)has constant eigenvalues, for any ∇⊥-parallel normal vector field ξ(t) along
any arbitrary (piece-wise differentiable) curve c(t) in M. If, in addition, the normal
bundle of ν(M) is flat, then Mis called isoparametric.

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 5
A submanifold Mwith constant principal curvatures (extrinsically) splits as
M=Rk×M0, where M0is compact and contained in a sphere.
The (extrinsic) homogeneous isoparametric submanifolds are exactly the princi-
pal orbits of polar representations [PT]. The other orbits have constant principal
curvatures (and, in particular, this family of orbits contains the submanifolds with
parallel second fundamental form). But it is not true that all homogeneous subman-
ifolds with constant principal curvatures are orbit of polar representations (there
exists a homogeneous focal parallel manifold to an inhomogeneous isoparametric hy-
persurface of the sphere [FKM]). It turns out, from Dadok’s classification [Da], that
polar representations are orbit-like equivalent to the so-called s-representations, i.e.
the isotropy representations of semisimple simply connected Riemannian symmet-
ric spaces. So, a full and homogeneous (not contained in a proper affine subspace)
Euclidean submanifold Mis isoparametric if and only if it is a principal orbit of an
s-representation. It is interesting to remark that there is a classification free proof
[EH], for cohomogeneity different from 2, of the fact that any polar representation
is orbit-like to an s-representation.
One has the following remarkable result
Theorem 2.1. (Thorbergsson, [Th, O3]). A compact full irreducible isoparametric
Euclidean submanifold of codimension at least 3is homogeneous (and so the orbit
of an irreducible s-representation).
The rank at p, of a Euclidean submanifold M, rankp(M), is the maximal number
of linearly independent parallel normal fields, locally defined around p. The rank
of M, rank(M), is the minimum, over p∈M, of rankp(M). If Mis homogeneous
then rankp(M) = rank(M), independent of p∈M. The submanifold Mis said to
be of higher rank if its rank is at least 2.
One has the following important result.
Theorem 2.2. (Rank Rigidity for Submanifolds, [O3, O4, DO, BCO]) Let Mn,
n≥2, be a Euclidean homogeneous submanifold which is full and irreducible. Then,
(a) If rank(M)≥1, if and only if Mis contained in a sphere.
(b) If rank(M)≥2, then Mis an orbit of an s-representation.
A parallel normal field ξof Mis called isoparametric if the shape operator Aξhas
constant eigenvalues. If the shape operator Aξ, of a parallel isoparametric normal
field, is umbilical, i.e. a multiple λof the identity, then Mis contained in a sphere,
if λ6= 0, or Mis not full, if λ= 0.
One has the following result (see, [BCO], Theorem 5.5.2 and Corollary 5.5.3).
Theorem 2.3. (isoparametric local rank rigidity, [CO]). Let Mnbe a full (local)
and locally irreducible submanifold of SN−1⊂RNwhich admits a non-umbilical
parallel isoparametric normal field. Then Mis an inhomogeneous isoparametric
hypersurface or Mis (an open subset of) an orbit of an s-representation.
One has also a global version of the above result (see [DO] Theorem 1.2 and
[BCO], Section 5.5 (b)).
Theorem 2.4. (isoparametric rank rigidity, [DO]). Let Mnbe a connected, simply
connected and complete Riemannian manifold and let f:M→RNbe an irreducible
isometric immersion. If there exists a non-umbilical isoparametric parallel normal
section then f:M→RNhas constant principal curvatures (an so, if f(M) is not an
isoparametric hypersurface of a sphere, then it is an orbit of an s-representation).

6 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
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NO
Let Kacts (by linear isometries) on RNas an s-representation. Let (G, K) be the
associated simple (simply connected) symmetric pair with Cartan decomposition
g=k⊕p, where p'RN. Let M=K.v be an orbit.
One has that the normal space to Mat vis given by [BCO]
νv(M) = C(v) := {x∈p: [x, v]=0}(*)
where [ ,] is the bracket of g.
An s-representation is always the product of irreducible ones. Then the orbit
M=K.v is a full submanifold if and only if all the components of v, in any
K-irreducible subspace of RN, are not zero.
Let Mbe a full orbit of an s-representation and let p∈M. Then the map
ξ7→ Aξ, from νp(M) into the symmetric endomorphisms of TpM, is injective. In
other words, the first normal space of Mat pcoincides with the normal space (see
[BCO]).
One has the following result from [HO]; see also [BCO], Theorem 4.1.7.
Theorem 2.5. ([HO]) Let Kacts on RNas an s-representation and let M=K.v
be a full orbit. Then the normal holonomy group Φ(v)of Mat vcoincides with
the image of the representation of the isotropy Kvon νv(M) (the so-called slice
representation).
For a Euclidean vector space (V,h,i), let Sim(V) denote the vector space of
(real) symmetric endomorphisms of V. The inner product on Sim(V) is the usual
one, hA, Bi= trace(A.B).
We denote by Sim0(V) the vector space of traceless symmetric endomorphisms.
Corollary 2.6. Let Kacts (by linear isometries) on RNas an s-representation
and let M=K.v, where |v|= 1. Assume that the normal holonomy group Φ(v)
acts irreducibly on ¯νv(M) := {v}⊥∩νv(M). Then Mis a minimal submanifold of
the sphere SN−1⊂RN. Moreover, the map ξ7→ Aξis a homothecy, from ¯νv(M)
onto its image in Sim0(TvM).
Proof. The mean curvature vector H(v) must be fixed by the isotropy, represented
on the normal space. Then, from Theorem 2.5, H(v) must be fixed by Φ(v). Then,
from the assumptions, H(v) must be proportional to v(which is fixed by the normal
holonomy group). Then Mis a minimal submanifold of the sphere.
Let us consider the following inner product ( ,) of ¯νv(M): (ξ, η) = hAξ, Aηi
Then, ( ,) is Φ(v)-invariant. In fact, if φ∈Φ(v), there exists, from Theorem 2.5,
g∈Kvsuch that g|¯νv(M)=φ. Then
(φ(ξ), φ(η)) = (g.ξ, g.η) = hAg.ξ , Ag.η i=hgAξg−1, g Aηg−1i=hAξ, Aηi= (ξ, η)
Since Φ(p) acts irreducibly, then ( ,) is proportional to h,i. Then ξ7→ Aξis a
homothecy.

Recall that the normal holonomy (group) representation, of a submanifold of
a space form, on the normal space, is, up to the fixed set, an s-representation
[O1, BCO].
The proof of the above mentioned result depends on the construction of the
so-called adapted normal curvature tensor R⊥(see [O1] and [BCO], Section 4.3

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 7
c). In fact, if Mis an arbitrary submanifold of a space of constant curvature
then R⊥is an algebraic curvature tensor on the normal space ν(M). Namely, if
p∈Mand R⊥is the normal curvature tensor at p, regarded as a linear map for
Λ2(TpM)→Λ2(νp(M)), the adapted normal curvature tensor is defined by
R⊥=R⊥◦(R⊥)t
where ( )tis the transpose endomorphism. This implies that R⊥has the same
image as R⊥.
From the Ricci identity one has the nice formula, if ξ1, ξ2, ξ3, ξ4∈νp(M),
hR⊥
ξ1,ξ2ξ3, ξ4i=−trace([Aξ1, Aξ2]◦[Aξ3, Aξ4])
=h[Aξ1, Aξ2],[Aξ3, Aξ4]i=−h[[Aξ1, Aξ2], Aξ3], Aξ4i(**)
where Ais the shape operator of M.
Since R⊥(Λ2(νp(M))) = R⊥(Λ2(TpM)), one has that R⊥
ξ1,ξ2belongs to the
normal holonomy algebra at p(since curvature tensors, take values in the holonomy
algebra).
Since the isotropy representation of a semisimple symmetric space coincides with
that of the dual symmetric space, we may always assume that the symmetric space
is compact. Let then (G, K) be a compact simply connected symmetric pair and
let g=k⊕pbe the Cartan decomposition associated to such a pair. The isotropy
representation of Kis naturally identified with the Ad-representation of Kon p.
The Euclidean metric on pis −B, where Bis the Killing form of g. We denote by
a dot the Ad-action of Kon p. Let 0 6=v∈pand let us consider the orbit
M=K.v 'K/Kvwhich is a Euclidean submanifold with constant principal
curvatures (and rank at least 2 if and only if it is not most singular).
Let us consider the restriction h,iof −Bto k. This is an Ad-K invariant positive
definite inner product on k. Let us consider the (normally) reductive decomposition
k=kv⊕m
where kvis the Lie algebra of the isotropy group Kvand mis the orthogonal
complement, with respect to h,i, of k. The restriction of h,ito m'T[e]K/Kv'
TvMinduced a so-called normal homogeneous metric on M, which is in particular
naturally reductive, that we also denote by h,i. Such a Riemannian metric on M
will be called the canonical normal homogeneous metric. In general this metric is
different from the induced metric as a Euclidean submanifold. Namely,
Proposition 2.7. Let Kacts on RNas an irreducible s-representation and let M=
K.v,v6= 0. If the (canonical) normal homogeneous metric on Mcoincides with
the induced metric, then Mhas parallel second fundamental form (or equivalently,
Mis extrinsically symmetric [Fe]).
Proof. We keep the notation previous to this proposition. Let ∇cbe the canonical
connection on Massociated to the reductive decomposition k=kv⊕m. Then
the second fundamental form αof Mis parallel with respect to the connection
¯
∇c=∇c⊕ ∇⊥, i.e. ¯
∇cα= 0 [OSa, BCO]. Let ¯
∇=∇⊕∇⊥, where ∇is the
Levi-Civita connection on Massociated to the induced metric which coincides, by
assumption, with the normal homogeneous metric. Then
(¯
∇xα)(y, z) = α(Dxy, z) + α(y, Dxz)

8 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
where D=∇ − ∇cWe have that Dxy=−Dyx. This is a general fact, for natu-
rally reductive spaces, since the canonical geodesics coincide with the Riemannian
geodesics (see, for instance, [OR]).
Then
(¯
∇xα)(x, x)=2α(Dxx, x) = 0
But, from the Codazzi identity, ( ¯
∇xα)(y, z) is symmetric in all of its three vari-
ables. Then ¯
∇α= 0 and so Mhas parallel second fundamental form.

Corollary 2.8. Let Kacts on RNas an s-representation and let M=K.v,v6= 0.
Assume that Kvacts irreducibly on TvM. Then Mhas parallel second fundamental
form (or, equivalently, Mis extrinsically symmetric [Fe]).
Remark 2.9.A submanifold of the Euclidean space with parallel second fundamen-
tal form is, up to a Euclidean factor, an orbit of an s-representation [Fe] (see also
[BCO]).
Lemma 2.10. Let Mn,¯
Mn⊂SN−1be submanifolds of the sphere with parallel
second fundamental forms (or, equivalently, extrinsically symmetric spaces). As-
sume also that Mis a full submanifold of the Euclidean space RNand that there
exists p∈M∩¯
Mwith TpM=Tp¯
M. Assume, furthermore, that the associated
fundamental forms at p,α, ¯αof Mand ¯
M, respectively, as submanifolds of the
sphere, are proportional (i.e. ¯α=λα,λ6= 0). Then M=¯
M(and so λ= 1) or
M=σ(¯
M), where σis the orthogonal transformation of RNwhich is the identity
on Rp⊕TpMand minus the identity on ¯νp(¯
M)=(Rp⊕TpM)⊥(and so λ=−1).
Proof. Observe, in our assumptions, that the second fundamenal forms of Mand
¯
M, as Euclidean submaniofolds, are not proportional, unless they coincide (since
the shapes operators of Mand ¯
M, coincides in the direction of the position vector
p).
Let us write M=K.p where Kacts as an irreducible s-representation. One has
that the restricted holonomy at p, of the bundle T M ⊕¯ν(M), is the representation,
of the connected isotropy (Kp)0, on TpM⊕¯νp(M). This is a well-known fact that
follows form the following property: if Xbelongs to the Cartan subalgebra asso-
ciated to the symmetric pair (K, Kp), then dlExp(tX)gives the Levi-Civita parallel
transport, when restricted to TpM, along the geodesic γ(t) = Exp(tX).p, and at
the same time, when restricted to ¯νp(M), the normal parallel transport along γ(t).
Since curvature endomorphisms take values in the holonomy algebra, one has
that (Rx,y, R⊥
x,y)∈tp, where tp= Lie(Kp) = Lie((Kp)0)⊂so(TpM)⊕so(¯νp(M))
and R,R⊥are the tangent and normal curvature tensors of Mat p, respectively.
Let RSbe the curvature tensor of the sphere SN−1at p, restricted to TpM.
Then, from the Gauss equation,
Rx,y =Tx,y +RS
x,y
where hTx,yz , wi=hα(x, w), α(y, z)i−hα(x, z), α(y, w)i
For ¯
M=¯
K.p we have similar objects ¯
R, ¯
R⊥,¯
tp,¯
T. From the assumptions one
has that ¯
T=λ2T. So, ¯
Rx,y =λ2Tx,y +RS
x,y (a)
From the assumptions, and Ricci equation, one has that
¯
R⊥
x,y =λ2R⊥
x,y (b)

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 9
Now observe that, for any X∈tp⊂so(TpM)⊕so(¯νp(M)),
X.α =0=X.(λ.α) = X. ¯α(c)
and the same is true for any ¯
X∈¯
tp(the actions of Xand ¯
Xare derivations).
As we observed, (Rx,y, R⊥
x,y)∈tp, and ( ¯
Rx,y,¯
R⊥
x,y)∈¯
tp. Then, form (a), (b) and
(c) one obtains, if λ6=±1 that
(RS
x,y,0).α = 0 = (RS
x,y,0).¯α
.
Since the linear span of {RS
x,y :x, y ∈TpM}is so(TpM), one has that
α(g.x, g.y ) = α(x, y)
for all g∈SO(TpM). Then, from the Gauss equation hAξx, yi=hα(x, y), ξ i,
one obtains that all the shape operators of Mat pcommute with any element of
SO(TpM). Then Mis umbilical at pand hence, since it is homogeneous, at any
point. Then Mis an extrinsic sphere. Since Mis full we conclude that M=SN−1.
Then, since n=N−1, M=¯
M.
Observe that the fullness condition is essential. In fact, if Mand ¯
Mare umbilical
submanifolds of the sphere of different radios, the second fundamental forms at p
are proportional.
If λ= 1, then Mand ¯
Mhave both the same second fundamental form at p.
Since both submanifolds have parallel second fundamental forms, it is well-known
and standard to prove that M=¯
M.
If λ=−1, then we replace ¯
Mby σ(¯
M) and the second fundamental forms of M
and ¯
Mmust coincide. Therefore, M=σ(¯
M).

Remark 2.11.Let us enounce Theorem 4.1 in [O6]: let Mnbe a locally full sub-
manifold either of the Euclidean space or the sphere, such that the local normal
holonomy group at pacts without fixed non- zero vectors. Assume, furthermore,
that no factor of the normal holonomy is transitive on the sphere. Then there are
points in M, arbitrary close to p, where the first normal space coincides with the
normal space. In particular, codim(M)≤1
2n(n+ 1).
This bound on the codimension is correct. But the better and sharp estimate
is codim(M)≤1
2n(n+ 1) −1. In fact, from the proof one has that if the shape
operator, at a generic q∈M,Aξis a multiple of the identity (it needs not to be
zero, as in that proof ), then ξis in the nullity of the adapted normal curvature
tensor R⊥. But this last tensor is not degenerate. This implies that the injective
map A:νq(M)→Sim(TqM) cannot be onto. Then dim(νq(M)) = codim(M)≤
dim(Sim(TqM)) −1 = 1
2n(n+ 1) −1.
If M, in the above assumptions, is is a submanifold of the sphere, then the
codimension of M, as a Euclidean submanifold, is bounded by 1
2n(n+ 1).
2.2. Holonomy systems.
We recall here some facts about holonomy systems that are useful in submanifold
geometry.
Aholonomy system is a triple [V, R, H ], where Vis a Euclidean vector space, His
a connected compact Lie subgroup of SO(V) and R6= 0 is an algebraic Riemannian
curvature tensor on Vthat takes values Rx,y ∈h= Lie(H). The holonomy system
is called:

10 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
-irreducible, if Hacts irreducible on V.
-transitive, if Hacts transitively on the unit sphere of V.
-symmetric, if h(R) = R, for all h∈H.
Observe that a Lie subgroup H⊂SO(V) that acts irreducibly on Vmust be
compact, as it is well-known (since the center of Hmust be one-dimensional).
A holonomy system [V, R, H] is the product (eventually, after enlarging H) of
irreducible holonomy systems (up to a Euclidean factor).
One has the following remarkable result.
Theorem 2.12. (Simons holonomy theorem, [S, O6]).
An irreducible and non-transitive holonomy system [V, R, H]is symmetric. More-
over, Ris, up to a scalar multiple, unique.
Remark 2.13.If [V, R, H] is an irreducible symmetric holonomy system, then h
coincides with the linear span of Rx,y,x, y ∈V. In this case, since hRx,y v, ξ i=
hRv,ξ x, yi, one has that the normal space at vto the orbit H.v is given by
νv(H.v) = {ξ∈V:Rv ,ξ = 0}
From a symmetric holonomy system one can build an involutive algebraic Rie-
mannian symmetric pair g=h⊕V. The bracket [,] is given by:
a) [,]|h×hcoincides with the bracket of h.
b) [X, v] = −[v , X] = X.v, if X∈h⊂so(V) and v∈V.
c) [v, w] = Rv ,w, if v, w ∈V.
This implies the following: if [V, R, H ]is an irreducible and symmetric holonomy
system then Hacts on Vas an irreducible s-representation.
Observe that, in this case, the scalar curvature sc(R) of Ris different from 0
(since this is true for the curvature tensor of an irreducible symmetric space).
Lemma 2.14. Let [V, R, K]be an irreducible and non-transitive holonomy system.
Let T∈SO(V)be such that Rx,y = 0 if and only if RT(x),T (y)= 0. Then T(R) = R.
Proof. Let R0=T(R). If ξ∈νv(K.v) = {ξ∈V:Rv,ξ = 0}, then, from the assump-
tions, R0
v,ξ =T.RT(v),T (ξ).T −1= 0. So, 0 = hR0
v,ξ x, yi=hR0
x,yv , ξi, for all x, y ∈V.
Then the Killing field R0
x,y ∈so(V) of Vis tangent to any orbit K.v. This implies
that R0
x,y ∈˜
h= Lie( ˜
K), where ˜
K={g∈SO(V) : gpreserves any K-orbit}. Ob-
serve that ˜
Kis a (compact) Lie subgroup of SO(V) which is non-transitive (on the
unit sphere of V). Since H⊂˜
Kwe have that [V, R, ˜
K] is also an irreducible and
non-transitive holonomy system. From the Simons holonomy theorem we have that
[V, R, K] and [V, R, ˜
K] are both symmetric. Then hand ˜
hare (linearly) spanned
by Rx,y,x, y ∈V. Then h=˜
hand therefore, K=˜
K.
Since R0takes values in ˜
h=h, then [V, R0, K ] is also an irreducible and non-
transitive holonomy system. Then, from the uniqueness part of Simons theorem,
R0=λR, for some scalar λ6= 0. Since Tis an isometry, it induces an isometry on
the space of tensors. Then λ=±1. But 0 6=sc(R) = sc(R0). Then λ= 1 and
hence R0=R.

Remark 2.15.Let Mn=K.v, where Kacts (by linear isometries) on Rn+1
2n(n+1) as
an s-representation (|v|= 1). Assume that the restricted normal holonomy group

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 11
Φ(v) acts irreducibly on ¯νv(M) = {v}⊥∩νv(M). In this case Mis a minimal
submanifold of the sphere Sn−1+ 1
2n(n+1) (see Corollary 2.6).
Let Abe the shape operator of Mand let Sim0(TpM) be the space of traceless
symmetric endomorphisms of TpM. Then the map A: ¯νv(M)7→ S im0(TvM) is
a linear isomorphism. In fact, it is injective, since the first normal space of M
coincides with the normal space, and dim(¯νv(M)) = dim(Sim0(TvM)). Moreover,
by the second part of Corollary 2.6, Ais a homothecy from ¯νv(M) onto Sim0(TvM),
let us say, of constant β > 0.
Let us consider the following two irreducible and symmetric holonomy systems:
[Sim0(TpM), R, SO(TpM)] and [¯νv(M),R⊥,Φ(v)],
where R⊥is the adapted normal curvature tensor of Mat vand Ris the curvature
tensor of Sl(n)/SO(n) (which is explicitly given by (***) of Section 1.3).
Observe that [¯νv(M),R⊥,Φ(v)] is symmetric since, by Theorem 2.5, the re-
stricted normal holonomy group is given by
Φ(v) = {k|νv(M):k∈(Kv)0}
and R⊥is left fixed by Kv.
Both algebraic curvature tensors are related by the formula (**) of Section 1.1.
This implies that the homothecy Amaps R⊥into R. Then the isometry β−1A
maps R⊥into β4R.
Since in a symmetric irreducible holonomy system the Lie algebra of the group
is (linearly) generated by the curvature endomorphisms, we conclude that Amaps
Φ(v) onto SO(TpM)'SO(n). In particular, the two holonomy systems are equiv-
alent and Φ(v)'SO(n).
2.3. Veronese submanifolds.
Let us consider the isotropy representation of the symmetric space of the non-
compact type X= Sl(n+ 1)/SO(n+ 1) (which coincides with the isotropy repre-
sentation of its compact dual SU(n+ 1)/SO(n+ 1)). The Cartan decomposition of
such a space is
sl(n+ 1) = so(n+ 1) ⊕S im0(n+ 1)
where Sim0(n+ 1) denotes the traceless symmetric (real) (n+ 1) ×(n+ 1)-matrices.
The Ad-representation of SO(n+ 1) on Sim0(n+ 1) coincides with the action, by
conjugation, of SO(n+ 1) on Sim0(n+ 1).
The curvature tensor of Xat [e] is given (up to a positive multiple) by
RA,B C=−[[A, B], C ]
and
hRA,B C, Di=−h[[A, B ], C], Di=h[A, B],[C, D]i(***)
where A, B, C, D ∈Sim0(n+ 1) 'T[e]X.
Let S∈Sim0(n+ 1) with exactly two eigenvalues, one of multiplicity 1 (whose
associated eigenspace we denote by E1) and the other of multiplicity n(whose
associated eigenspace we denote by E2).
The orbit Vn= SO(n+1).S ={kSk−1:k∈SO(n+1)}is called a Veronese-type
orbit (see Appendix).
The following assertions are easy to verify or well-known.

12 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Facts 2.16.
(i) The Veronese-type orbit Vn= SO(n+ 1).S is a full and irreducible
submanifold of Sim0(n+ 1) which has dimension nand codimension 1
2n(n+ 1).
Moreover, Vnis a minimal submanifold of the sphere of radius kSk.
(ii) An orbit of SO(n+ 1) in Sim0(n+ 1) has minimal dimension if and only if
it is of Veronese-type; see Lemma 8.1.
(iii) The normal holonomy group at S, of the Veronese-type orbit Vn, coin-
cides with image of the slice representation of the isotropy group (SO(n+ 1))S=
S(O(E1)×O(E2)) 'S(O(1) ×O(n)). So, from (*), the restricted normal holonomy
representation, on ¯νS(Vn) = {S}⊥∩νS(Vn), is equivalent to the isotropy repre-
sentation of the symmetric space Sl(n)/SO(n) of rank n−1. Then, this normal
holonomy representation is irreducible. Moreover, it is non-transitive (on the unit
sphere of ¯νS(Vn)) if and only if n≥3.
(iv) A Veronese-type orbit Vn= SO(n+ 1).S = SO(n+ 1)/(SO(n+ 1))Sis
intrinsically a real projective space RPn. Moreover, (SO(n+ 1),(SO(n+ 1))S)
is a symmetric pair and so (SO(n+ 1))Sacts irreducibly on TSVn. Then, from
Corollary 2.8, Vnhas parallel second fundamental form (as it is well known).

A submanifold M⊂RNis called a Veronese submanifold if it is extrinsically
isometric to a Veronese-type orbit.
Proposition 2.17. Let Mn=K.v ⊂Rn+1
2n(n+1), where Kacts on Rn+1
2n(n+1)
as an s-representation (n≥2). Assume that the restricted normal holonomy group
Φ(v)of Mat v, restricted to ¯νv(M) = {v}⊥∩νv(M), acts irreducibly (eventually,
in a transitive way). Then,
(i) The normal holonomy representation of Φ(v)on ¯νv(M)is equivalent to the
isotropy representation of the symmetric space Sl(n)/SO(n).
(ii) Mnis a Veronese submanifold.
Proof. Part (i) is a consequence of Remark 2.15.
Since Kacts as an s-representation, then the image under the slice representa-
tion, of the (connected) isotropy group (Kv)0, coincides with the restricted normal
holonomy group Φ(v). But, from part (i), dim(Φ(v)) = dim(SO(n)) Then the
isotropy group Kvhas dimension at least dim(SO(n)) = dim(SO(TvM)).
Observe that the isotropy representation of Kvon TvMis faithful. Otherwise,
Mwould be contained in the proper subspace which consists of the fixed vector of
Kvin RN.
Then, (Kv)0= SO(TvM). So, Kvacts irreducibly on TvM. Then, from Corol-
lary 2.8, Mhas parallel second fundamental form.
Let Vnbe a Veronese submanifold of Rn+1
2n(n+1). We may assume that v∈Vn
and that TvM=TvVn=Rn⊂Rn+1
2n(n+1). For Vnwe have, from Corollary 2.6
and Remark 2.15, that its shape operator ¯
A:{v}⊥∩νv(Vn) = {v}⊥∩νv(M)→
Sim0(TvVn) = Sim0(TvM) is a homothecy which induces an isomorphism from
the normal holonomy group ¯
Φ(v) of Vnonto SO(n).
The same is true, again from Corollary 2.6 and Remark 2.15, for the shape opera-
tor Aof M. Namely, A:{v}⊥∩νv(M)→Sim0(TvVn) = Sim0(TvM) = Sim0(Rn)
is a homothecy which induces an isomorphism from the (restricted) normal holo-
nomy group Φ(v) of Monto SO(n). Then the map A−1◦¯
Ais a homothecy with

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 13
constant, let us say, β > 0, of the space {v}⊥∩νp(M). Let h=β−1A−1◦¯
A. Then
his a linear isometry of {v}⊥∩νv(M).
Let now gbe the linear isometry of Rn+1
2n(n+1) defined by the following prop-
erties:
(i) g(v) = v.
(ii) g|{v}⊥∩νv(M)=h−1.
(iii) g|TvM= Id.
Then Vnand g(M) have proportional second fundamental forms and satisfy all
the other assumptions of Lemma 2.10. Then, by this lemma, g(M), and hence M,
is a Veronese submanifold.

2.4. Coxeter groups and holonomy systems.
The goal of this section is to prove Proposition 2.21 that will be important for
proving our main theorems. In order to prove this proposition we need some basic
results, related to Coxeter groups, that we have not found through the mathematical
literature. So, and also for the sake of self-completeness, we include the proofs.
Lemma 2.18. Let Cbe a Coxeter group acting irreducibly, by linear isometries, on
the Euclidean n-dimensional vector space (V,h,i). Let H1, ..., Hrbe the family of
(different) reflection hyperplanes, associated to the symmetries of C(that generates
C). Let us define the group G={g∈End(V) : gpermutes H1, ..., Hrand det(g) =
±1}.Then Gis finite.
Proof. Let Prbe the (finite) group of bijections of the set {1, ..., r}. Let ρ:G→Pr
be the group morphism defined by ρ(g)(i) = j, if g(Hi) = Hj. The group Gis finite
if and only if ker(ρ) is finite. Let us prove that ker(ρ) is finite. If g∈ker(ρ) then
it induces the trivial permutation on the family H1, ..., Hr. Then, its transpose gt,
with respect to h,i, induces the trivial permutation on the set of lines L1, ..., Lr,
where Liis the line which is perpendicular to Hi,i= 1, ..., r (and hence, any
vector in any line L1, ..., Lris an eigenvector of gt. Let us define, for i6=j, the
2-dimensional subspace Vi,j := linear span of (Li∪Lj). This subspace is called
generic if there exists k∈ {1, ..., r},i6=k6=jsuch that Lk⊂Vi,j . In other
words, Vi,j is generic if there are at least three different lines of {L1, ..., Lr}which
are contained in Vi,j. We have, if Vi,j is generic, that gt:Vi,j →Vi,j is a scalar
multiple of the identity Idi,j of Vi,j . In fact, any vector in Li∪Lj∪Lkis an
eigenvector of (gt)|Vi,j. Then, since dim(Vi,j ) = 2, (gt)|Vi,j =λIdi,j , for some
λ∈R. Let us define the following equivalence relation ∼on the set {1, ..., r}:i∼i0
if there exist i1, ..., il∈ {1, ..., r}with i1=i,il=i0and such that Vis,is+1 is generic,
for s= 1, ..., l −1. Let i∈ {1, ..., r}be fixed. By the previous observations one
has that there must exist λ∈Rsuch that for any j∈[i] (the equivalence class of
i) and for any vj∈Lj,gt(vj) = λvj. In order to prove this lemma, it suffices to
show that there is only one equivalence class on {1, ..., r}. In fact, if [i] = {1, ..., r},
then gt=λId, since L1, ..., Lrspan V(because of its othogonal complement is
point-wise fixed by C). So g=λI d. But det(g) = ±1. Then λn=±1 and hence
λ=±1. So, g=±Id and therefore there are at most two elements in ker(ρ).
Let i∈ {1, ..., r}be fixed. Let us show that [i] = {1, ..., r}. If j /∈[i] then Ljis
perpendicular to any Lk, for all k∈[i]. In fact, assume that this is not true for
some k∈[i]. Let sj∈Cbe the symmetry across the hyperplane Hj. Then sj(Lk)
is a line, which belongs to {L1, ..., Lr}, that is contained in Vk,j and it is different

14 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
from both Lkand Lj. Then j∼kand therefore j∼i. A contradiction. Then, if
j /∈[i], Lk⊂Hj, for all k∈[i]. So, sjacts trivially on V[i], the subspace spanned
by Sk∈[i]Lk. Observe that sjcommutes with sk, for all k∈[i]. Let now V0be
the maximal subspace of Vsuch that it is point-wise fixed by all the symmetries
sjwith j /∈[i]. Observe that this space is not the null subspace, since V[i]⊂V0. If
there exists j /∈[i], then V0must be a proper subspace of V, since sj6= Id. On the
other hand, if k∈[i], then sk(V0)⊂V0, since skcommutes with all the symmetries
sj,j /∈[i]. Then V0is a proper and non-trivial subspace of Vwhich is invariant
under the irreducible Coxeter group C. A contradiction. So, [i] = {1, ..., r}.

Lemma 2.19. . We are under the assumptions and notation of the above lemma.
Then Gacts by isometries.
Proof. By the above lemma, Gis finite. By averaging the inner product h,iover
the elements of G, we obtain a G-invariant inner product ( ,) on V. Since C⊂G,
then ( ,) is C-invariant. Since Cacts irreducible, h,imust be proportional to ( ,).
Then Gacts by isometries on (V,h,i). 
Corollary 2.20. Let (Vi,h,ii)be a Euclidean vector spaces and let Cibe a Cox-
eter group acting irreducibly, by linear isometries, on (Vi,h,ii),i= 1,2. Let
h:V1→V2be a linear map such that it induces a bijection from the family of
reflection hyperplanes of C1into the family of reflection hyperplanes of C2. Then
his a homothetical map.
Proof. Let ( ,) = h∗(h,i2) and let C∗=h∗(C2) = h−1C2h. Observe that the
determinant of any element of C2is ±1, since it is an isometry of (V2,h,i2). So,
any element in C∗has determinant ±1. From the assumptions, we obtain that the
family of reflection hyperplanes of the irreducible Coxeter group C∗of (V1,(,))
coincides with the family H1, ..., Hrof reflection hyperplanes of C1. Then any
element of C∗induces a permutation in this family of hyperplanes. Then, by
Lemma 2.19, C∗acts by isometries on (V1,h,i1). Since C∗acts irreducibly, one
has that h,i1is proportional to ( ,). This implies that his a homothecy

Proposition 2.21. Let (V, R, K )and (V0, R0, K0)be irreducible, non-transitive
(and hence symmetric) holonomy systems. Let h:V→V0be a linear isomorphism
such that, for any K-orbit K.v in V,h(νv(K.v)) = νh(v)(K0.h(v)), where νdenotes
the normal space. Then his a homothecy and h−1
∗(K0) = K.
Proof. Observe that the groups Kand K0act as irreducible s-representations. We
have that K.v is a maximal dimensional orbit if and only if K0.h(v) is so.
Recall that, for s-representations, an orbit is maximal dimensional if and only if
it is principal.
Let K.v be a principal K-orbit. This orbit is an irreducible (homogeneous)
isoparametric submanifold of V. There is an irreducible Coxeter group C, associated
to this isoparametric submanifold, that acts on the normal space νv(K.v) [Te, PT,
BCO]. If H1, ..., Hrare the reflection hyperplanes of the symmetries of C, then
r
[
i=1
Hi={z∈νv(K.v) : K.z is a singular orbit}(a)

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 15
If v0=h(v) one has the similar objects K0.v0,νv0(K0v0), C0and H0
1, ..., H0
sand
s
[
i=1
H0
i={z0∈νv0(K0.v0) : K0.z0is a singular orbit}(b)
Moreover, from (a) and (b), one has that hmaps, bijectively, the family H1, ..., Hr
onto the family H0
1, ..., H0
s. Then, s=rand so we may assume that h(Hi) = H0
i,
i= 1, ..., s.
Then, from Corollary 2.20, one has that
h:νw(K.w)→νw0(K0.w0)
is a homothecy, for any principal K-vector w, where w0=h(w). Denote by λ(w)>0
the homothecy constant of this map.
Observe, since w∈νw(K.w) and w0∈νw0(K0.w0), that
hh(w0), h(w0)i0=λ(w)hw, wi
where h,iand h,i0are the inner products on Vand V0, respectively.
Let v0be a fixed K-principal vector and let M=K.v0.
Let T M =E1⊕...⊕Er, where E1, ..., Erare the (autoparallel) eigendistributions
of T M associated to the commuting family of shape operators Aξof the isopara-
metric submanifold M⊂V. Associated to any Eithere is a parallel normal field
ηi, a so-called curvature normal, such that, for any normal field ξ,
Aξ|Ei=hξ, ηiiIdEi
Let, for q∈M,Si(q) denote the integral manifold of Eiby q. Such integral
manifold is a so-called curvature sphere. If x∈Si(q) then
νx(M)∩νq(M)=(ηi(q))⊥
where the orthogonal complement is inside νq(M). Observe that this intersection is
non-trivial, since the codimension of Min Vis at least 2. This implies λ(x) = λ(q).
Since the eigendistributions span T M , one has that moving along different curvature
sphere one can reach, from v0, any other point of M. Then λ(x) = λ(v0), for all
x∈M.
Observe now that, for any y∈V, there exists ¯x∈Msuch that y∈ν¯x(M).
In fact, such an ¯xcan be chosen as a point where the function, from Minto R,
x→ hx, yiattains a maximum.
Then hh(y), h(y)i0
hy, yi=hh( ¯x), h(¯x)i0
h¯x, ¯xi=λ(x) = λ(v0),
for all 0 6=y∈V. Then his a homothecy of constant λ:= λ(v0). This proves the
first assertion.
Let g∈K0and let T=h−1◦g◦h. Since his a homothecy, T∈SO(V).
Then, from the assumptions and Remark 2.13 one has that Tsatisfies hypothesis
of Lemma 2.14. Then, by this lemma, T(R) = R. This implies, since the Lie
algebra of Kis generated by {Rx,y}, that Tbelongs to N(K), the normalizer of K
in O(V). Moreover, Tmust belong to the connected component N0(K) (because
of Tcan be deformed to the identity, since K0is connected). But N0(K) = K,
since Kacts as an s-representation (see [BCO] Lemma 6.2.2). Then T∈K, thus
h−1
∗(K0) = K.


16 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Remark 2.22.The above proposition is not true if the holonomy systems are tran-
sitive. In fact, let (V, R, K) and (V0, R0, K 0) be the (symmetric) holonomy systems
associated to the rank 1 symmetric spaces S2n= SO(2n+ 1)/SO(2n) and CPn=
SU(n+1)/S(U(1) ×U(n)), respectively. In this case dim(V) = dim(V0) = 2n. Then
any linear isomorphism from Vinto V0, satisfies the assumption of Proposition 2.21,
since the normal spaces of non-trivial Kor K0-orbits are lines.
3. non-transitive normal holonomy
Let Mn=H.v ⊂Sn−1+ 1
2n(n+1) be a homogeneous submanifold of the sphere.
Assume that the (restricted) normal holonomy group, as a submanifold of the
sphere, acts irreducibly and it is not transitive (on the unit normal sphere).
From now on, we will regard Mnas a submanifold of the Euclidean space
Rn+1
2n(n+1) . Let ν(M) be the normal bundle and let Φ(v) be the restricted nor-
mal holonomy group at v(regarding Mas a Euclidean submanifold). Observe that
Φ(v) acts trivially on R.v and that Φ(v), restricted to ¯νv(M) := {v}⊥∩νv(M), is
naturally identified with the (restricted) normal holonomy group of Mat v, as a
submanifold of the sphere.
Observe that the irreducibility of the normal holonomy group representation on
{v}⊥∩νv(M) implies that rank(M) = 1. Namely, vis the only vector of νv(M)
which is fixed by Φ(v). This implies that Mis a full and irreducible submanifold
of the Euclidean space. In fact, if Mis not full then any non-zero constant normal
vector is a parallel normal field which is not a multiple of the position vector.
Then rank(M)≥2. A contradiction. If Mis reducible it must be a product of
submanifolds contained in spheres. Then rank(M)≥2. Also a contradiction.
One has, from Remark 2.11, that the first normal space ν1(M) coincides with
the normal space νM , regarding Mas a Euclidean submanifold. This means, that
the linear map, from νv(M) into Sim(TvM), ξ7→ Aξis injective, where Ais the
shape operator of M. Since dim(νv(M)) = 1
2n(n+ 1) = dim(S im(TvM)), then
A:νv(M)→Sim(TvM)is a linear isomorphism.
Let R⊥
ξ1,ξ2be the adapted normal curvature tensor (see Section 1). This tensor
is given by
hR⊥
ξ1,ξ2ξ3, ξ4i=−trace([Aξ1, Aξ2]◦[Aξ3, Aξ4])
=h[Aξ1, Aξ2],[Aξ3, Aξ4]i=−h[[Aξ1, Aξ2], Aξ3], Aξ4i
Observe that the right hand side of the above equality is, with the usual identifica-
tions, the Riemannian curvature tensor h˜
RAξ1,Aξ2Aξ3, Aξ4iof the symmetric space
Gl(n)/SO(n).
Observe that such a symmetric space is isometric to the following product:
Gl(n)/SO(n) = R×Sl(n)/SO(n)
The tangent space of the second factor is canonically identified with the traceless
symmetric matrices Sim0(n).
Let us consider the so-called traceless shape operator ˜
Aof M. Namely,
˜
Aξ:= Aξ−1
ntrace(Aξ)Id = Aξ−1
nhξ, H iId
where His the mean curvature vector.

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 17
Observe that
hR⊥
ξ1,ξ2ξ3, ξ4i=h[˜
Aξ1,˜
Aξ2],[˜
Aξ3,˜
Aξ4]i
=h˜
RAξ1,Aξ2Aξ3, Aξ4i=hR˜
Aξ1,˜
Aξ2
˜
Aξ3,˜
Aξ4i(****)
where Ris the curvature tensor at [e] of the symmetric space Sl(TvM)/SO(TvM)
(see formula (***) of Section 1.3).
If ¯νv(M) = {v}⊥∩νv(M), we have the following two symmetric non-transitive
irreducible holonomy systems: [ ¯νv,R⊥,Φ(v)] and [Sim0(TvM), R, SO(TvM)].
Recall that for a symmetric irreducible holonomy system [V,¯
R, K], from Remark
2.13, the normal space to an orbit K.v is given by νv(K.v) = {ξ∈V:¯
Rv,ξ = 0}
Then, from (****), we have that the map ˜
Ais a liner isomorphism that maps
normal spaces to Φ(v)-orbits into normal spaces to SO(TvM)-orbits. Then, by
Proposition 2.21, ˜
Ais a homothecy and ˜
A: ¯νv(M)→Sim0(TvM) transforms Φ(v)
into SO(TvM). Then Φ(v) is isomorphic to SO(TvM). Therefore, we have the
following result:
Lemma 3.1. Let Mn=K.v ⊂Sn−1+ 1
2n(n+1) be a homogeneous submanifold.
Assume that the restricted normal holonomy group of Macts irreducibly and it
is non-transitive. Then the representation of the normal holonomy group Φ(v)on
¯νv(M)is (orthogonally) equivalent to the isotropy representation of the symmetric
space Sl(n)/SO(n)'Sl(TvM)/SO(TvM). Moreover, the traceless shape operator
˜
A: ¯νv(M)→Sim0(TvM)is a homothecy that transforms, equivariantly, Φ(v)into
SO(TvM).(In particular, dim(Φ(v)) = 1
2n(n−1) = dim(SO(n))). 
Proposition 3.2. Let Mn=K.v ⊂Sn−1+ 1
2n(n+1) be a homogeneous submanifold.
Assume that the restricted normal holonomy group of Macts irreducibly and it
is non-transitive. Then, for any ξ(t)parallel normal section along a curve, the
traceless shape operator ˜
Aξ(t)has constant eigenvalues.
Proof. Note that Mmust be full and irreducible as a Euclidean submanifold (see
the beginning of this section). Let p∈Mbe arbitrary and let Kpbe the isotropy
subgroup of Kat p. Let us decompose
Lie(K) = m⊕Lie(Kp)
where mis a complementary subspace of Lie(Kp) Let Br(0) be an open ball, cen-
tered at the origin, of radius rof msuch that Exp : Br(0) →Mis a diffeomorphism
onto its image U= Exp(Br(0)), which is a neighbourhood of p(the inner product
on Lie(K) is irrelevant).
Let β: [0,1] →Ube an arbitrary piece-wise differentiable curve with β(0) = p.
Since β(1) ∈U, there exits X∈msuch that β(1) = Exp(X).p. Let γ: [0,1] →M
be defined by γ(t) = Exp(tX).p. Let us denote, for k∈K, by lkthe linear isometry
v7→ k.v of V. Let τ⊥
tdenote the ∇⊥-parallel transport along γ|[0,t]. Then, from
remarks 6.2.8 and 6.2.9 of [BCO],
τ⊥
t= (dlExp(tX))|νp(M)◦e−tAX(A)
where AXbelongs to the normal holonomy algebra Lie(Φ(p)) and it is defined by
AX=d
dt|t=0τ⊥
−t◦(dlExp(tX))|νp(M)

18 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Let τ⊥
βbe the ∇⊥-parallel transport along βand φ=τ⊥
−1◦τ⊥
β. Then φbelongs to
Φ(p), the restricted normal holonomy group at p. In fact, φcoincides with the ∇⊥-
parallel transport along the null-homotopic, since it is contained in U, loop β∗˜γ,
obtained from gluing the curve βtogether with the curve ˜γ, where ˜γ(t) = γ(1 −t).
We have that τ⊥
β=τ1◦φand so, by (A),
τ⊥
β= ((dlExp(X))|νp(M)◦e−AX)◦φ= (dlExp(X))|νp(M)◦¯
φ
where ¯
φ= e−AX◦φbelongs to Φ(p). Then, for any ξ∈νp(M),
˜
Aτ⊥
β(ξ)=˜
AdlExp(X)(¯
φ(ξ)) = dlExp(X)◦˜
A¯
φ(ξ)◦(dlExp(X))−1
= Exp(X).˜
A¯
φ(ξ).(Exp(X))−1
Then, from the paragraph just before Lemma 3.1, we have that there exists g∈
SO(Tp(M)) such that ˜
A¯
φ(ξ)=g. ˜
A¯
φ(ξ).g−1. Then
˜
Aτ⊥
β(ξ)= (Exp(X).g).˜
Aξ.(Exp(X).g)−1
This shows that the eigenvalues of ˜
Aτ⊥
β(ξ)are the same as the eigenvalues of ˜
Aξ.
The curve βwas assumed to be contained in U. Since pis arbitrary, one obtains
that the eigenvalue of ˜
Aξ(t)are locally constant for any ξ(t) parallel normal field
along a curve c(t). This implies that the eigenvalues of ˜
Aξ(t)are constant.

The following lemma is well known and the proof is similar to the case of hyper-
surfaces of a space form.
Lemma 3.3. (Dupin Condition). Let Mbe a submanifold of a space of constant
curvature and let ξbe a parallel normal field such that the eigenvalues of the shape
operator Aξhave constant multiplicities. Let λ:M→Rbe an eigenvalue function
of Aξsuch that its associated (and integrable from Codazzi identity) eigendistribu-
tion Ehas dimension at least 2. Then λis constant along any integral manifold of
E(or equivalently, dλ(E)=0).
Theorem 3.4. Let Mn⊂Sn−1+ 1
2n(n+1) be a homogeneous submanifold, where
n > 3. Assume that the restricted normal holonomy group acts irreducibly and not
transitively. Then Mis a Veronese submanifold.
Proof. Note that Mmust be full and irreducible as a Euclidean submanifold (see
the beginning of this section).
We will regard Mas a submanifold of the Euclidean space Rn+1
2n(n+1). Then,
as we have observed at the beginning of this section, A:νp(M)→Sim(TpM) is
an isomorphism (p∈Mis arbitrary). Now choose ξ∈νp(M) such that Aξhas
exactly two eigenvalues λ1(p), λ2(p) with multiplicities m1, m2≥2 (this is not
possible if n≤3). In particular, we assume that m1= 2 and m2=n−2. We
may assume that ξis small enough such that the holonomy tube [BCO] Mξis an
immersed Euclidean submanifold (see Remark 3.5). We may also assume that ξis
perpendicular to the position (normal) vector p, since Ap=−Id.
There is a natural projection π:Mξ→M,π(c(1) + ¯
ξ(1)) = c(1). Moreover,
ˆ
ξdefines a parallel normal field to Mξ, where ˆ
ξ(q) = q−π(q). In this way M
is a parallel focal manifold to Mξ. Namely, M= (Mξ)−ˆ
ξ. Observe that the
holonomy tube Mξis not a maximal one and so it has not a flat normal bundle

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 19
(this would have been the case, in our situation, where all of the eigenvalues of Aξ
have multiplicity one). Let ¯
ξ(t) be a parallel normal field along an arbitrary curve
c(t) with c(0) = p,¯
ξ(0) = ξ. Then, from Proposition 3.2, the eigenvalues of the
traceless shape operator ˜
A¯
ξ(t)are constant and hence the same as the eigenvalues
of ˜
Aξwhich are ˜
λ1=λ1(p)−1
n(2λ1(p)+(n−2)λ2(p)), with multiplicity 2 and
˜
λ2=λ2(p)−1
n(2λ1(p)+(n−2)λ2(p)), with multiplicity n−2.
Let Hbe the mean curvature vector field on M. Then the eigenvalues of the
shape operator A¯
ξ(1) can be written as
λi(c(1)) = ˜
λi+1
nh¯
ξ(1), H(c(1))ii= 1,2
with multiplicities 2 for and n−2, respectively (independent of c(1) ∈M).
From the tube formula [BCO], one has that the eigenvalues functions ˆ
λ1and
ˆ
λ2of the shape operator ˆ
Aˆ
ξof the holonomy tube, restricted to the horizontal
subspace Hqof the holonomy tube Mξ, at a point q=c(1) + ¯
ξ(1) are:
ˆ
λ1(q) = ˜
λ1+1
nh¯
ξ(1), H(c(1))i
1−˜
λ1−1
nh¯
ξ(1), H(c(1))i
and
ˆ
λ2(q) = ˜
λ2+1
nh¯
ξ(1), H(c(1))i
1−˜
λ2−1
nh¯
ξ(1), H(c(1))i
or, equivalently,
ˆ
λ1(q) = ˜
λ1+1
nhˆ
ξ(q), H(π(q))i
1−˜
λ1−1
nhˆ
ξ(q), H(π(q))i
and
ˆ
λ2(q) = ˜
λ2+1
nhˆ
ξ(q), H(π(q))i
1−˜
λ2−1
nhˆ
ξ(q), H(π(q))i
with (constant) multiplicities 2 and n−2, respectively. Observe that ˆ
Aˆ
ξ(q), re-
stricted to the vertical distribution (tangent to the orbits in Mξof the normal
holonomy group of Mat projected points) is minus the identity. So, ˆ
Aˆ
ξ(q)has a
third eigenvalue ˆ
λ3(q) = −1 with constant multiplicity m3= dim(Mξ)−dim(M).
The real injective function sf
7→ s
1+stransforms ˆ
λi(q) into ˜
λi+1
nhˆ
ξ(q), H(π(q))i
(i= 1,2). Then,
ˆ
λ1(q) = ˆ
λ1(q0)⇐⇒ ˆ
λ2(q) = ˆ
λ2(q0) (I)
In fact, any of both equalities implies 1
nhˆ
ξ(q), H(π(q))i=1
nhˆ
ξ(q0), H(π(q0))i. This,
by the above equalities, implies (I).
Let now E1and E2be the (horizontal) eigendistributions associated to eigenvalue
functions ˆ
λ1and ˆ
λ2of the shape operator ˆ
Aˆ
ξ. Observe that dim(E1) = 2 and
dim(E2) = n−2≥2.
Up to here everything is valid, except the last inequality, also for n= 3.(II)
(This will be used in next section where we deal with the case n= 3).

20 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
If γ(t) is a curve that lies in E1then, from the Dupin Condition (see Lemma
3.3) we have that ˆ
λ1is constant along γ. So, by (I), ˆ
λ2is also constant along γ.
The same is true if γlies in E2. This implies that 0 = v(ˆ
λ1) = v(ˆ
λ2) = v(ˆ
λ3) for
any vector vthat lies in H. Then the eigenvalues of the shape operator ˆ
Aˆ
ξare
constant along any horizontal curve. Since any two points, in a holonomy tube, can
be joined by a horizontal curve we conclude that the (three) eigenvalues of ˆ
Aˆ
ξare
constant on Mξ.
Then ˆ
ξis a parallel isoparametric (non-umbilical) normal section. Observe that
Mξis a full irreducible Euclidean submanifold, since Mis so. Moreover, Mξis
complete with the induced metric (see Remark 3.5). Then, by [BCO],[DO], Mξ
must be a submanifold with constant principal curvatures. Since M= (Mξ)−ˆ
ξ, we
have that Mis also a submanifold with constant principal curvatures. Any principal
holonomy tube of Mhas codimension at least 3 in the Euclidean space, since the
normal holonomy of M, as a submanifold of the sphere, is non-transitive. Then,
by the theorem of Thorbergsson [Th, O2, BCO], Mis an orbit of an (irreducible)
s-representation.
The fact that Mis a Veronese submanifold follows from Proposition 2.17.

Remark 3.5.Let Mn=H.v be a full irreducible homogeneous submanifold of RN
which is (properly) contained in the sphere SN−1. We are not assuming that Mis
compact (in which case the assertions of this remark are trivial).
By making use of the homogeneity of Mone obtains that there exists ε > 0 such
that: if ξ∈ν(M) with 0 <kξk< ε then any of the eigenvalues λof the shape
operator Aξsatisfies |λ|<1−a, for some 0 < a < 1.
Let us assume that rank(M) = 1, i.e., Mis not a submanifold of higher rank
(otherwise, Mwould be an orbit of an s-representation and hence compact).
Let ξ∈νv(M) with 0 <kξk< ε and let us consider the normal holonomy
subbundle by ξ[BCO] of the normal bundle π:ν(M)→M.
Holξ(M) = {η∈ν(M) : ηH
∼ξ}
where His the horizontal distribution of ν(M) and ηH
∼ξif ηand ξcan be joined
by a horizontal curve. Equivalently, ηH
∼ξyηis the ∇⊥-parallel transport of ξ
along some curve.
One has that the fibres of π: Holξ(M)→Mare compact. In fact, π−1({π(η)}) =
Φ(π(η)).η, where Φ denotes the normal holonomy group. Observe that such a
group is compact, since its connected component acts as an s-representation (see
the discussion inside the proof of Theorem 4.1, Case (2), (c)).
Let us consider the normal exponential map expν:ν(M)→RN, given by
expν(η) = π(η) + η. Let η∈νp(M) and identify, as usual, via d π,TpM' Hη. The
vertical distribution νη=Tηνp(M) is canonically identified to νp(M). With this
identification one has the well-known expression for the differential of the normal
exponential map:
d(expν)|Hη= (I−Aη),d(expν)|νη=Idνp(M)(C)

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 21
Then expν: Holξ(M)→RNis an immersion. The image of this map is the
so-called holonomy tube Mξof Mby ξ. It is given by
Mξ={c(1) + ¯
ξ(1) : ¯
ξ(t) is ∇⊥-parallel along c(t) where c(0) = p, ¯
ξ(0) = ξ}
Many times, and in particular in the proof of Theorem 3.4, for the sake of
simplifying the notation, the immersed submanifold expν: Holξ(M)→RNwill be
also denoted by Mξ.
One has that the Euclidean submanifold expν: Holξ(M)→RN, with the induced
metric h,i, is a complete Riemannian manifold. In fact, let ( ,) be the Sasaki metric
on Holξ(M). In such a metric the horizontal distribution is perpendicular to the
vertical one. Moreover, πis a Riemannian submersion and the metric in the vertical
space Φ(p).η is that induced from the metric on the normal space νp(M). Since
Mis complete and the fibres are compact, then ( ,) is complete. Then, from (C),
a2(,)≤ h ,i. This implies that the induced metric is also complete.
4. The proof of the conjecture in dimension 3
Theorem 4.1. Let M3=H.p be a 3-dimensional homogeneous submanifold of the
sphere SN−1which is full and irreducible (as a submanifold of the Euclidean space
RN). Assume that the normal holonomy group of Mis non-transitive. Then Mis
an orbit of an s-representation.
Proof. Assume that Mis not isoparametric (in which case it must be an orbit
of an s-representation). Then, by Lemma 4.2, the normal holonomy of M, as a
submanifold of the sphere acts irreducibly and N=9=3+1
23(3 + 1). We have
also that the first normal bundle, which coincides with the normal bundle, has
maximal codimension.
Keeping the notation and general constructions in the proof of Theorem 3.4,
we have that everything is still valid up to (II). The only difference is that the
eigenvalue ˆ
λ2has multiplicity 1. So, we have the Dupin condition only for the
eigendistribution E1but not for the 1-dimensional eigendistribution E2.
Let ¯
M=Mξ/E1be the quotient of the (partial) holonomy tube Mξby the
(maximal) integral manifolds of the 2-dimensional integrable distribution E1.
Observe that the (partial) holonomy tube Mξhas dimension 5. In fact, from
Lemma 4.2, any focal orbit of the restricted normal holonomy group Φ(p)'SO(3)
has dimension 2 (and it is isometric to the Veronese V2).
By [BCO], Theorem 6.2.4, part (2) one has that H⊂SO(9) acts by (extrinsic)
isometries on Mξ. Moreover, the projection π:Mξ→Mis H-equivariant.
If H.(p+ξ) = Mξ, then Mξis a full and irreducible homogeneous Euclidean
submanifold which is of higher rank. Then, in this case, by the rank rigidity theorem
for submanifolds, Mξis an orbit of an s-representation. Hence M= (Mξ)−ˆ
ξis an
orbit of an s-representation.
So, we may assume that H.(p+ξ)(Mξ. Let h= Lie(H). Let us consider
the subspace h.(p+ξ) of Tp+ξMξ. This subspace has dimension at least 3, since
dπ(h.(p+ξ)) = h.p =TpM. The horizontal subspace H(p+ξ)of Tp+ξMξhas dimen-
sion 3. Since Tp+ξMξhas dimension 5, dim(H(p+ξ)∩h.(p+ξ)) ≥1.
Case (1): E1(x)+(Hx∩h.x) = Hx,for some x∈Mξ

22 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
We may assume that x=p+ξ. Observe that if the above equality holds at
(p+ξ) then it also holds for qin some open neighbourhood Uof (p+ξ) in Mξ.
Recall, continuing with the notation in the proof of Theorem 3.4, that the eigen-
values functions (which are differentiable) of the shape operator ˆ
Aˆ
ξat qare: ˆ
λ1(q)
with multiplicity 2, ˆ
λ2(q) with multiplicity 1 and ˆ
λ3(q) = −1 with multiplicity 2
(whose associated eigenspace is the vertical distribution νq).
On one hand, from the Dupin condition, since dim(E1) = 2, and the equivalence
(I) in the proof of the above mentioned theorem, we have that
0 = v(ˆ
λ1) = v(ˆ
λ2) = v(ˆ
λ3)
for any v∈E1(q). Or, briefly,
{0}=E1(q)(ˆ
λ1) = E1(q)(ˆ
λ2) = E1(q)(ˆ
λ3)
On the other hand, if X∈h,
0 = (X.q)(ˆ
λ1) = (X.q)(ˆ
λ2)=(X.q).(ˆ
λ3)
In fact, this follows from the fact that the parallel normal field ˆ
ξof Mξis H-invariant
and that ˆ
Ah.ˆ
ξ(q)=h. ˆ
Aˆ
ξ(q).h−1, for all h∈H.
Then, from the assumptions of this case,
{0}=Hq(ˆ
λ1) = Hq(ˆ
λ2) = Hq(ˆ
λ3) (III)
for any q∈U.
Since Mis (extrinsically) homogeneous, the local normal holonomy groups have
all the same dimension. Then the local normal holonomy group at any x∈M
coincides with the restricted normal holonomy group Φ(x).
The ∇⊥-parallel transport along short loops, based at p∈M, produces a neigh-
bourhood Ω of ein the local normal holonomy group, see [CO, DO]). This implies,
from (III), that the eigenvalues of ˆ
Aˆ
ξ(p+ω.ξ)are the same as the eigenvalues ˆ
λ1(p+ξ),
ˆ
λ2(p+ξ), ˆ
λ3(p+ξ) = −1 of ˆ
Aˆ
ξ(p+ξ), for all ω∈Ω. From this it is standard to show
that the eigenvalues of ˆ
Aˆ
ξ(p+φ.ξ)are the same of those of ˆ
Aˆ
ξ(p+ξ), for all φ∈Φ(p).
Therefore, the eigenvalues of ˆ
Aˆ
ξare constant on p+ Φ(p).ξ =π−1({p}). Since H
acts transitively on M, then H.π−1({p}) = Mξ. This implies, since ˆ
ξis H-invariant,
that the eigenvalues of ˆ
Aˆ
ξare constant on Mξ.
Observe that the parallel normal field ˆ
ξis not umbilical, since ˆ
Aˆ
ξhas three
distinct (constant) eigenvalues. Then, from [DO] (see Theorem 5.5.8 of [BCO]),
Mξhas constant principal curvatures. So, M= (Mξ)−ˆ
ξhas constant principal
curvatures. If ˜
Mis a principal holonomy tube of M, then ˜
Mis isoparametric [HOT].
Observe that ˜
Mis not a hypersurface of a sphere (since the normal holonomy
group, in the Euclidean space, is not transitive on the orthogonal complement of
the position vector), then by the theorem of Thorbergsson [Th, O2] ˜
Mis an orbit
of an s-representation. Then Mis an orbit of an s-representation, since it is a focal
(parallel) manifold to ˜
M.
Case (2): E1(x)+(Hx∩h.x)(Hx,for all x∈Mξ
or equivalently, (Hx∩h.x)⊂E1(x), since dim(E1(x)) = 2 and dim(Hx) = 3.

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 23
This case splits into several sub-cases, depending on how big is the group H.
Namely, depending on dim(H)≥3 = dim(M). The most difficult case is the
generic one where dim(H) = 3. For this case we will have to use topological
arguments.
Note that dim(H)≤6. In fact, Hacts effectively on M, since Mis a full
submanifold. Otherwise, if h∈Hacts trivially on Mthen it acts trivially on the
(affine) span of Mwhich is R9. But the dimension of the isometry group of an
n-dimensional Riemannian manifold is bounded by 1
2(n+ 1)n(the dimension of the
isometry group of an n-dimensional space of constant curvature). In our case, since
n= 3, dim(H)≤6.
Observe that Hcannot be abelian. In fact if His abelian, since the dimension
of the ambient space N= 9 is odd, the (connected) subgroup H⊂SO(9) must
fix a vector, let us say v6= 0. So, no H-orbit H.q is a full submanifold, since it is
contained in q+{v}⊥. A contradiction, since M=H.p is full.
Observe that dim(H) cannot be 5. In fact, if dim(H) = 5 then the isotropy Hp
has dimension 2 and so it is abelian. We regard Hp⊂SO(TpM)'SO(3), via the
isotropy representation. But the rank of SO(3) is 1 and so it has no abelian two
dimensional subgroups. A contradiction.
(a) dim(H) = 6.
In this case we must have that (Hp)0= SO(3), since dim(Hp) = 3. Since SO(3)
is simple, the slice representation sr of (Hp)0on the normal space νp(M) must be
either trivial or its image has dimension 3. In the first case we obtain that all shape
operators Aµof Mat pare a multiple of the identity, since they commute all with
(Hp)0. Note that Aµ=Ah.µ =h.Aµ.h−1. So M=M3is an umbilical submanifold
of S8⊂R9. So, Mis not full. A contradiction.
Let us deal with the case that the image of the slice representation has dimen-
sion 3. By [BCO], Corollary 6.2.6 sr((Hp)0)⊂Φ(p) where Φ(p) is the restricted
normal holonomy group of Mas a Euclidean submanifold. Since dim(Φ(p)) = 3,
we conclude that sr((Hp)0) = Φ(p). Then, any holonomy tube of Mis an Horbit.
In particular the principal ones, which have flat normal bundle. But the holonomy
tubes are full and irreducible Euclidean submanifolds, which have codimension at
least 3 (since Φ(p) acts on the 6-dimensional normal space νp(M) with cohomo-
geneity 3). Then, by the theorem of Thorbergsson [Th, O2], any holonomy tube
is an orbit of an s-representation and so Mis an orbit of an s-representation. By
Proposition 2.17 one has that M=M3is a Veronese submanifold.
(b) dim(H) = 4.
In this case the isotropy Hphas dimension 1. If the slice representation sr of
(Hp)0is trivial, then, as in (a), all shape operators at pcommute with (Hp)0'S1.
A contradiction, since the family of shape operators is Sim(TpM).
Let us then restrict to the case that the slice representation is not trivial. For
this we have to use a result of [OS] (see [BCO], Theorem 6.2.7). In fact, we need the
following weaker version, which was the main step in the proof of Simons holonomy
theorem given in [O6]. Namely, Proposition 2.4 of [O6]: for a full and irreducible
H-homogeneous Euclidean submanifold Mn,n≥2, the projections, on the normal
space νp(M), of the (Euclidean) Killing fields given by the elements of h=Lie(H),
belong to the normal holonomy algebra g.

24 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Then, in our situation, since dim(h) = 4 and dim(g) = 3, there must exist
06=X∈hsuch that it projects trivially on the normal space. Such an Xcannot be
in the isotropy algebra, since we assume that the slice representation of (Hp)0'S1
is non-trivial. This implies that 0 6=X.p ∈TpM.
Let us consider the H-invariant parallel normal field ˆ
ξof Mξ. Recall that
(Mξ)−ˆ
ξ=M(and so Mis a parallel focal manifold of Mξ).
Since Xprojects trivially on νp(M), X.q ∈ Hq, for all q∈(p+ Φ(p).ξ) =
((π)−1({p}))q⊂Mξ.
Recall that we are in Case (2). Then, X.q ∈E1(q), for all q∈(p+ Φ(p).ξ). Let
us consider the curve γ(t) = Exp(tX).p of M3. One has that γ0(0) = X.p 6= 0. Let
q∈(p+ Φ(p).ξ) and let ψ(t) be the normal parallel transport of (q−p)∈νp(M)
along γ(t). Then ψ(t) = ˆ
ξ(γ(t)+ψ(t)), as it is well known, from the construction of
holonomy tubes [HOT, BCO] (observe that Mξ=Mq−p). From the tube formula
of [BCO], Lemma 4.4.7 (the notation in this lemma permutes our objects),
A(q−p)=ˆ
A(q−p)|H.((I d −ˆ
A(q−p))|H)−1
one has that E1(q) is an eigenspace of the shape operator A(q−p)of M.
On the one hand, since π(q) = q−ˆ
ξ(q),
dπ(E1(q)) = (Id +ˆ
Aˆ
ξ)(E1(q)) ⊂E1(q)
On the other hand, since ˆ
ξis H-invariant and ˆ
ξ(q)=(q−p),
dπ(X.q) = d
dt|0(Exp(tX).q −ˆ
ξ(Exp(tX).q))
=d
dt|0(Exp(tX).q −Exp(tX).(q−p)) = X.p
Therefore, X.p belongs to an eigenspace of any shape operator Aq−pof M, such
that q∈(p+ Φ(p).ξ) (recall that we have assumed, without loss of generality, that
ξis perpendicular to the position vector p).
Observe that Φ(p).ξ spans {p}⊥, since Φ(p) acts irreducibly on {p}⊥. So X.p is
an eigenvector of any shape operator Aη, where hη , pi= 0.
Since Ap=−Id, we conclude that X.p is an eigenvector of all shape operators of
Mat p. This is a contradiction, since the family of shape operators at pcoincides
with Sim(TpM).
(c) dim(H) = 3.
Since we have excluded the case where His abelian, then Hmust be simple,
with universal cover the (compact) group Spin(3) 'S3. This case is the generic
one where the isotropy is finite. Note that Mmust be compact.
Also note that the (full) normal holonomy group ˜
Φ(p) of Mis compact. In fact,
(˜
Φ(p))0coincides with the restricted normal holonomy group Φ(p). Moreover, ˜
Φ(p)
is included in the compact group N(Φ(p)), the normalizer of Φ(p) in O(νp(M)).
Observe that (N(Φ(p)))0= Φ(p), since Φ(p) acts as an s-representation (see [BCO]
Lemma 6.2.2). Then ˜
Φ(p) has a finite number of connected components, as well as
˜
Φ(p).ξ. This implies that Mξis compact.
Let us construct the so-called caustic fibration. The eigenvalues functions of ˆ
Aˆ
ξ
are bounded on Mξ. Since Mis contained in a sphere, Mξis contained in a (differ-
ent) sphere. If ηis the position vector field of Mξ, then ηis an umbilical parallel

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 25
normal field. In fact, ˆ
Aη=−Id. By adding, eventually, to the parallel normal field
ˆ
ξa (big) constant multiple of −ηwe obtain a new parallel and H-invariant normal
field, such that its associated shape operator has the same eigendistributions as ˆ
Aˆ
ξ
and all of the three eigenvalues functions are everywhere positive and so nowhere
vanishing. Just for the sake of simplifying the notation, we also denote this per-
turbed normal field by ˆ
ξ. The eigenvalues of ˆ
Aˆ
ξare also denoted by ˆ
λ1,ˆ
λ2,ˆ
λ3,
which differ from the original ones by a (same) constant c.
The caustic map ρ, from Mξinto R9,qρ
7→ q+ (ˆ
λ1(q))−1ˆ
ξ(q) has constant rank.
In fact, ker(dρ) = E1has constant dimension 2, since from the Dupin condition,
ˆ
λ1is constant along any integral manifold Q1(q) of E1. Observe that ˆ
λ2is also
constant along Q1(q), due to equivalence (I) in the proof of Theorem 3.4 (and the
same is true, of course, for the third eigenvalue ˆ
λ3≡ −1 + c).
Let ¯
M=Mξ/E1be the quotient of Mξby the family E1of (maximal) integral
manifolds of E1. From Lemma 4.3 we have that ¯
Mis a compact 3-manifold im-
mersed in R9, via the projection ¯ρ, of the caustic map ρ, to the compact quotient
manifold ¯
M. Moreover, ¯π:Mξ→¯
Mis a fibration, where ¯π:Mξ→¯
Mis the
projection. The distribution E1is H-invariant, since ˆ
ξis so. So, the action of H
on Mξprojects down to an action on ¯
M. So, ¯πis H-equivariant.
Observe that ρis H-equivariant, since ˆ
ξis H-invariant. Then, since ¯πis H-
equivariant, the immersion ¯ρ:¯
M→R9is H-equivariant.
We have the following two H-equivariant fibrations on Mξ:
0→Φ(p).ξ →Mξ
˜π
→˜
M→0(holonomy tube fibration)
0→Q→Mξ
¯π
→¯
M→0(caustic fibration)
where Qis any integral manifold of E1and ˜
Mis the quotient manifold Mξover
the connected component of the fibres of π:Mξ→M, which are orbits of the
restricted normal holonomy groups Φ(p), p∈M. We have that ˜
Mis a finite cover
of M.
Recall that we are under the assumptions of Case (2)
We will derive a topological contradiction. This is by using that the holonomy
tube Mξis the total space of above two different fibrations.
On the one hand the holonomy tube has a finite fundamental group π1(Mξ). This
follows from the long exact sequence of homotopies, associated to the holonomy
tube fibration. In fact, the fibres are (real) projective 2-spaces (which have a finite
fundamental group). Moreover, the base space ˜
Mhas also a finite fundamental
group, since it is an orbit, with finite isotropy, of the group Spin(3) 'S3. Since
the fibres of the caustic fibration are connected and the total space Mξhas finite
fundamental group, then the caustic (base) manifold ¯
Mhas a finite fundamental
group.
On the other hand, from Lemma 4.4 we have that the fundamental group of the
caustic manifold ¯
Mis not finite (this is by showing that Hacts with cohomogeneity
1 and without singular orbits on ¯
M).
A contradiction. So we can never be under the assumptions of Case (2) if
H'Spin(3).

26 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
This finishes the proof that Mis an orbit of an s-representation. 
Lemma 4.2. We are in the assumptions of Theorem 4.1. Then, if rank(M) =
1, the (restricted) normal holonomy group Φ(p), as a submanifold of the sphere,
acts irreducibly and N= 9. Moreover, the (restricted) normal holonomy acts as
the action of SO(3), by conjugation, on the traceless 3×3-symmetric matrices.
Furthermore, the traceless shape operator ˜
Aof Mat pis SO(3)-equivariant.
Proof. Let us regard M3as a submanifold of the Euclidean space RN. If Mis
not of higher rank one has, from Proposition 6.1, that the (restricted) normal
holonomy group Φ(p) acts irreducibly on ¯ν(p) (the orthogonal complement of the
position vector p). Since Φ(p) is non-transitive (on the unit sphere of ¯νp(M)),
the first normal space, as a submanifold of the Euclidean space, coincides with
the normal space (see Remark 2.11). Then, the codimension k=N−3 satisfies
k≤6 = 1
23(3 + 1). Then the normal holonomy group representation coincides with
the isotropy representation of an irreducible symmetric space of rank at least 2 and
dimension at most 5. Then, by Remark 4.6, the normal holonomy representation
is equivalent to the isotropy representation of Sl(3)/SO(3). So the codimension of
M, in the sphere, is 5 and hence N= 9. The equivariance follows form Lemma 3.1.

Lemma 4.3. (Caustic fibration lemma). Let ˆ
Mbe a compact immersed submani-
fold of RNwhich is contained in the sphere SN−1. Let ˆ
ξbe a parallel normal field
to ˆ
Msuch that the eigenvalues of the shape operator Aˆ
ξhave constant multiplic-
ities on ˆ
M. Let ˆ
λ:ˆ
M→Rbe an eigenvalue function of Aˆ
ξwhose associated
(integrable) eigendistribution Ehas (constant) dimension at least 2. Let Ebe the
family of (maximal) integral manifolds of E. Assume that the eigenvalue function ˆ
λ
never vanishes (this can always be assumed by adding to ˆ
ξan appropriate constant
multiple of the umbilical position vector). Then
(i) Any integral manifold Q∈ E is compact.
(ii) The quotient space ¯
M=ˆ
M/Eis a (compact) manifold and the projection
π:ˆ
M→¯
Mis a fibration (in particular, a submersion).
(iii) The caustic map ρ:ˆ
M→RN,ρ(q) = q+ (ˆ
λ(q))−1ˆ
ξ(q), projects down to
an immersion ¯ρ:¯
M→RN(i.e. ρ= ¯ρ◦π).
Proof. From the Dupin condition, see Lemma 3.3, one has that ˆ
λis constant along
any integral manifold Qof E.
Consider the caustic map ρ(q) = q+ (λ(q))−1ˆ
ξ(q) (see the proof of Theorem 4.1,
Case (2),(c)). Then ker(dρ) = Eand so dρhas constant rank. From the local form
of a map with constant rank and the compactness of ˆ
Mone has that there exists
a finite open cover V1, ..., Vdof ˆ
Msuch that, for any i= 1, ..., d and q, q 0∈Vi, the
following equivalence holds:
ρ(q) = ρ(q0)⇐⇒ qand q0belong both to a same integral manifold of E.
This implies that any (maximal) integral manifold Qof Emust be a closed
subset of ˆ
Mand hence compact. Moreover, the above equivalence implies that the
foliation Eis a regular foliation in the sense of Palais [P].
In order to prove that the quotient is a manifold we need to prove that this
quotient is Hausdorff. But this can be done as follows: let E⊥be the distribution

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 27
which is perpendicular, with respect to the metric, induced by the ambient space,
on ˆ
M. Let us define a new Riemannian metric h,ion ˆ
Mby changing the induced
metric ( ,) on the distribution E⊥in such a way that ρis locally a Riemannian
submersion onto its image. Namely,
.hE, E ⊥i= 0.
.h,icoincides with ( ,) when restricted to E
.d|qρis a linear isometry from (E⊥)qonto its image.
Such a metric is a bundle-like metric in the sense of Reinhart [Re] Since ˆ
Mis
compact, h,iis a complete Riemannian metric. Then, [Re],Corollary 3, pp. 129,
the quotient space ¯
Mis Hausdorff and πis a fibration (cf. [DO], Proposition 2.4,
pp. 83)
Then one has that the map ρprojects down to an immersion ¯ρ:¯
M→RNand
ρ= ¯ρ◦π.

Lemma 4.4. We keep the assumptions of Theorem 4.1. Moreover, we are in
the assumptions and notation of Case (2)(c), inside the proof of this theorem (in
particular, H'Spin(3), up to a cover).
(i) All orbits of the action of Hon ¯
Mhave dimension 2.
(ii) The universal cover ˜
Mof ¯
Msplits off a line and hence the fundamental group
of ¯
Min not finite (since ¯
Mis compact).
Proof. The action of Hon Mξprojects down to ¯
M, since ˆ
ξis H-invariant and so
any eigendistribution of ˆ
Aˆ
ξis H-invariant. Let q∈Mξ. Then the 3-dimensional
subspace h.q ⊂TqMξintersects the 3-dimensional horizontal subspace Hqin a
non-trivial subspace, since dim(Mξ) = 5). Since we are in Case (2),
{0} 6= (h.q ∩ Hq)⊂E1(q)
Let H¯qbe the isotropy group of Hat the point ¯q= ¯π(q)∈¯
M. Let h¯q= Lie(H¯q).
Then one has that
h¯q={X∈h:X.q0∈E1(q0)}
independent of q0∈S1(q) = (¯π)−1(¯π({q})), since a Killing field that is tangent to
an integral manifold S1(q) of E1, at some point, must be always tangent to it (since
the action projects down to the quotient).
If dim(h¯q) = 3. Then h¯q=h. Then Hleaves invariant the 2-dimensional integral
manifold S1(q) of E1by q. Then the isotropy Hqhas positive dimension. But
Hq⊂Hπ(q), where Hπ(q)is the isotropy group of Hat the point π(q)∈M3=H.p.
A contradiction, since dim(H) = 3.
Observe that dim(h¯q)6= 2. In fact, if this dimension is 2, then h¯qis an ideal of
the 3-dimensional (compact type) Lie algebra h. A contradiction, since his simple.
We have used that a Lie subalgebra of codimension 1 of a Lie algebra which admits
a bi-invariant metric must be an ideal. (Also, this 2-dimensional Lie subalgebra
should be abelian, in contradiction with rank(h) = 1).
Then dim(h¯q) = 1 for all q∈Mξ. This implies that all H-orbits in ¯
Mhave
dimension 2. Since Hacts with cohomogeneity 1 on ¯
Mthen, the universal cover of
¯
Mcannot be compact. Otherwise, as it is well-known, there would exist a singular
orbit (after lifting the action to the universal cover).

28 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
For the sake of self-completeness we will show the argument of this assertion.
We define an auxiliary Riemannian metric on ¯
M, by changing, along the H-
orbits, the metric h,iinduced by the immersion ¯ρ.
Since Hacts with cohomogeneity 1 on ¯
M,Hacts locally polarly. In particular,
the one dimensional distribution Don ¯
M, perpendicular to the H-orbits, is an
autoparallel distribution. If ¯q∈¯
Mthen we put on the orbit H.¯qthe normal
homogeneous metric. That is, the metric associated to the reductive decomposition
h=h¯q⊕(h¯q)⊥
where the orthogonal complement is taken with respect to a (fixed) bi-invariant
metric on h.
We define h,i0by:
a) h,i0
|D =h,i|D
b) hU,Di0= 0, where Uis the distribution given by the tangent spaces of the
H-orbits on ¯
M.
c) h,i0
|U¯qcoincides with the normal homogeneous metric of H.¯q, for any ¯qon ¯
M
Since ¯
Mis compact, the metric h,i0is complete. Let h,i0also denote the lift
of the Riemannian metric h,i0to the universal cover ˜
Mof ¯
M. Then ( ˜
M , h,i0)
is a complete Riemannian manifold. Let us denote by ˜
Uand ˜
Dthe lifts to ˜
M
of the distributions Uand D, respectively. Let us also lift the H-action on ¯
M
to ˜
MThen, since ˜
Mis simply connected, the one dimension distribution ˜
Dis
parallelizable. Namely, there exists a nowhere vanishing vector field ˜
Xof ˜
Msuch
that R.˜
X=˜
D. Let ˜
Z=1
k˜
Xk˜
X, where the norm is with the metric h,i0. Then, the
flow φt, associated to ˜
Z, is by isometries. So ˜
Zis a Killing field. Then h∇.˜
Z, .i0
is skew-symmetric. So, in particular, h∇v˜
Z, v i0= 0, for any vector vthat lies in
˜
U. But, if ˜
A˜
Zis the shape operator of the orbit H.x,x∈˜
M, then h˜
A˜
Zv, vi0=
h∇v˜
Z, v i0= 0. Then ˜
U=˜
D⊥is an autoparallel distribution. The distribution ˜
Dis
also autoparallel, since the Killing fields induced by Hare always perpendicular to
it. But two complementary perpendicular autoparallel distribution must be parallel.
Then, by the de Rham decomposition theorem, ˜
Mis a Riemannian product. Since
one of the parallel distributions is one dimensional then ˜
M=R×M0.

Remark 4.5.In this paper, for dealing with homogeneous submanifolds of dimension
3, we need to know which are the compact Lie groups Gof dimension at most 4.
For the sake of self-completeness we will briefly show, without using classification
results, which are these compact Lie groups G(up to covering spaces).
We will use the following fact that it is well-known and standard to show: a
codimension 1subgroup, of a Lie group with a bi-invariant metric, must be a normal
subgroup.
(i) dim(G)≤2.
In this case, from the above fact, one has that Gmust be abelian.
(ii) dim(G) = 3.
If rank(G)≥2 then, from the above fact, Gmust be abelian. If rank(G) = 1,
then Gis, up to a cover, Spin(3). This well-known result follows from a topological
argument that proves that a rank 1 simply connected compact group is isomorphic
to Spin(3) (a proof can be found in Remark 2.6 of [OR]).

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 29
(iii) dim(G) = 4.
If Gis neither simple nor abelian, then, from the previous cases, we have that a
finite cover of Gsplits as S1×Spin(3).
If Gis simple then rank(G)≤2. Otherwise Gwould have a codimension 1 (abelian)
subgroup (which must be normal).
If Gis simple, then rank(G)>1. Otherwise, G= Spin(3) which has dimension 3.
Let Gbe simple and rank(G) = 2. Then the Ad-representation of Gon g= Lie(G)
must have a focal (non-trivial) orbit G.v. Such an orbit must have codimension
3. The 3-dimensional normal space νv(G.v) is Lie triple system, since it coincides
with the commutator of v. Then νv(G.v) is an ideal of g. A contradiction.
Remark 4.6.Let X=G/K be an irreducible simply connected symmetric space of
the non-compact type and rank at least 2, where Gis the connected component of
the full isometry group of X. Assume that the dimension of Xis at most 5. Then,
X'SL(3)/SO(3).
We will next outline a classification free proof of this fact.
Observe, since rank(X)≥2, that the isotropy representation of Kon TpXhas
a non-trivial focal orbit M=K.v (p= [e]). Such an orbit Mmust have dimension
2. In fact, Mcannot have dimension 3. Otherwise, a principal K-orbit must have
dimension 4 and so Kwould act transitively on the sphere. Observe also that the
dimension of Mcannot be 1. In fact, since Kacts irreducibly on TpX, then Kacts
effectively on any non trivial orbit. If dim(M) = 1, then dim(K) = 1. Then, since
dim(X)>2, Kdoes not act irreducibly on TpX. A contradiction.
Observe that the isotropy Kvof the focal orbit M2=K.v at vmust have
positive dimension (and so dim(Kv) = 1). Moreover, since Mis not a principal
orbit, the image under the slice representation of Kvis not trivial. So, by Corollary
2.5, the restricted normal holonomy group Φ(v) of Mat vis not trivial. Then
Φ(v) must act irreducibly on the 2-dimensional space ¯νv(M) = {v}⊥∩νv(M).
Observe that the codimension of M2is 3 = 1
22(2 + 1) Then, by Proposition 2.17,
Mis a Veronese submanifold, i.e. orthogonally equivalent to a Veronese-type orbit
V2of SO(3) on Sim0(3) (the action is by conjugation). So, may assume that
Sim0(3) = TpXand that M=V2. Then both Kand SO(3) are Lie subgroups of
˜
K={g∈SO(Sim0(3)) : g.M =M}. Observe that ˜
Kis not transitive on the unit
sphere of Sim0(3) since the codimension of Mis 3. Let R0and Rbe the curvature
tensors at p= [e] of Xand SL(3)/SO(3). Then we have the following irreducible
non-transitive holonomy systems: [Sim0(3), R, ˜
K] and [Sim0(3), R0,˜
K].
Then by the holonomy theorem of Simons 2.12, Ris unique up to scalar multiple
and ˜
K=K= SO(3), since its Lie algebra is spaned by R. This implies that the
symmetric space Xis homothetical to SL(3)/SO(3).
Remark 4.7.Let M3=K.v ⊂RNbe a 3-dimensional full and irreducible homo-
geneous (Euclidean) submanifold. Assume that rank(M)≥2. In this case, by the
rank rigidity theorem, Mis an orbit of an s-representation. So, we may assume,
that K-acts as an s-representation.
Let ξbe a K-invariant parallel normal field to Mwhich is not umbilical. If
the shape operator Aξhas two different (constant) eigenvalues then its associated
eigendistributions, let us say E1and E2are autoparallel distributions that are
invariant under the shape operators of M(recall that Aξcommutes with any other

30 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
shape operator due to Ricci equality). Then, by the so-called Moore’s lemma [BCO],
Lemma 2.7.1, Mis product of submanifolds. A contradiction.
If Aξhas three eigenvalues, then the multiplicities of any of them are 1. Since Aξ
commutes with any other shape operator, all shape operators of Mmust commute.
Then, by the Ricci identity, Mhas flat normal bundle. Then Mis isoparametric,
since it is an orbit of an s-representation.
Therefore, a full irreducible and homogeneous Euclidean 3-dimensional subman-
ifold M3, of higher rank, must be isoparametric with exactly three curvature nor-
mals. This implies that the irreducible Coxeter group associated to M[Te, PT] has
exactly three reflection hyperplanes. This is only possible if the dimension of the
normal space is 2. Otherwise, the curvature normals must be mutually perpendic-
ular and hence Mwould be a product of circles.
This implies that N= 5 and that Mis an isoparametric hypersurface of the
sphere S4. Moreover, from Remark 4.6, Mis a principal orbit of the isotropy
representation of Sl(3)/SO(3). 
Proof of Theorem A. If Mnis a (full) Veronese submanifold, n≥3, then
the normal holonomy, as a submanifold of the sphere, acts irreducibly and non-
transitively (see Facts 2.16, (iii)).
For the converse observe that Mmust be a full and irreducible Euclidean sub-
manifold, since the normal holonomy group (as a submanifold of the sphere) acts
irreducibly (see the beginning of Section 3). Then, from Theorem 3.4, Theorem 4.1
and Proposition 2.17, Mis a Veronese submanifold. 
Proof of Theorem B. From Theorem 4.1 Mis an orbit of an s-representation.
Assume that rank(M) = 1. Then, by Lemma 4.2, the (restricted) normal holonomy
group of M, as a submanifold of the sphere, acts irreducibly and N= 9 = 3+ 1
23(3+
1). Then, by Proposition 2.17, Mis a Veronese submanifold.
If Mis of higher rank, then, by Remark 4.7, Mis a principal orbit of the isotropy
representation of Sl(3)/SO(3). 
5. minimal submanifolds with non-transitive normal holonomy
In this section we prove Theorem C of the Introduction.
We use many of the ideas used for the homogeneous case, when n > 3. But now
the situation is much more simple, for n= 3.
Proof of Theorem C. Observe that Mmust be full and irreducible as a Euclidean
submanifold (since the normal holonomy group, as a submanifold of the sphere,
acts irreducibly; see Section 3). Note, by the minimality, that the traceless shape
opertator coincides with the shape operator (of vectors which are perpendicular to
the position vector).
We keep the notation in the proof of Theorem 3.4.
Let p∈Mbe such that the adapted normal curvature tensor R⊥(p)6= 0,
or equivalently, R⊥(p)6= 0. Let us consider the irreducible and non-transitive
holonomy systems [¯νp(M),R⊥(p),Φ(p)] and [Sim0(TpM), R, SO(TpM)].
We have, from formula (****) of Section 3 and Proposition 2.21, that the shape
operator at p,Ap: ¯νp(M)→Sim0(TpM) is a homothecy and ApΦ(p)(Ap)−1=

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 31
SO(TpM). This implies, if φ∈Φ(p), that the eigenvalues of Ap
ηcoincide with the
eigenvalues of Ap
φ(η).
Let Ube a contractible neighbourhood of pin Msuch that R⊥never vanishes
on U.
Let now p0∈Ube arbitrary and let γ: [0,1] →Ube a piece-wise differentiable
curve from pto p0. Let τtbe the ∇⊥-parallel transport along γ[0,t].
We have that τtΦ(p)(τt)−1= Φ(γ(t)).
Let us choose ξ∈¯νp(M) such that Ap
ξ∈Sim0(TpM) has exactly two eigenvalues
λ1=1
2of multiplicity 2 and λ2=−1
(n−2) of multiplicity (n−2).
Recall that the shape operator Aγ(t): ¯νγ(t)→Sim0(Tγ(t)) maps Φ(γ(t)) into
SO(Tγ(t)). Then, the homothecy gt:= Aγ(t)◦τt◦(Ap)−1:Sim0(TpM)→Sim0(Tγ(t)M)
maps the group SO(TpM) into SO(Tγ(t)M). Then gtmaps the isotropy subgroup
SO(TpM)Ap
ξ'S(O(2) ×O(n−2)) into the isotropy subgroup SO(Tγ(t)M)B(t),
where B(t) = Aγ(t)
τt(ξ). This implies, as it is not difficult to see, that B(t) has two
eigenvalues, let us say λt
1of multiplicity 2 and λt
2of multiplicity n−2. Since
B(t)∈Sim0(Tγ(t)), λt
2=−2
n−2λt
1.
Then the two eigenvalues of B(t) are constant up to the multiplication by a(t) =
λt
16= 0. Note, if γis a loop by p, that τ1∈Φ(p). Then, as we have previously
observed, the eigenvalues of Ap
ξare the same as those of B(1). Then a(t) depends
only on γ(t). So there is a non-vanishing f:U→Rsuch that a(t) = f(γ(t)). It is
standard to show that fmust be C∞. Note that f(p) = 1
2.
Let us consider (eventually, by making Usmaller) the holonomy tube Uξ. We
use the notation in the proof to Theorem 3.4. We will modify the arguments in
this proof.
We have the parallel normal field ˆ
ξof Uξ. The eigenvalues of the shape operator
ˆ
Aˆ
ξat q∈Uξare given by
ˆ
λ1(q) = f(π(q))
1−f(π(q))
associated to the (horizontal) eigendistribution E1of dimension 2
ˆ
λ2(q) = −f(π(q))
n−2
1 + f(π(q))
n−2
associated to the (horizontal) eigendistribution E2of dimension n−2.
The third eigenvalue of ˆ
Aˆ
ξ, is ˆ
λ3=−1, associated to the vertical distribution ν,
tangent to the normal holonomy orbits.
By the Dupin condition, d(ˆ
λ1)(E1) = 0 which implies that
d(f◦π)(E1) = 0 (J)
If n > 3 this is also true for the eigendistribution E2, since it has dimension at least
2. But we will not assume this and the proof will also work for n= 3.
From the tube formula, as we have observed in the proof of Theorem 4.1, Case
(2), (b), dπ(E1(q)) = E1(q), as linear subspaces. Moreover, E1(q) is an eigenspace
of Aq−π(q)=Aˆ
ξ(q), where Ais the shape operator of M(we drop the supra-index
π(q) of A). Let now q∈Uξwith π(q) = pand let Vbe the subspace of TpM
which is generated by E1(q0), with q0∈Φ(p).q = (π−1({p}))q. If V=TpM,

32 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
then, from formula (J), df(TpM) = {0}. If Vis properly contained in TpM, then
let 0 6=v∈V⊥. We will derive, in this case, a contradiction. In fact, since
any shape operator Aq0−phas only two eigenvalues and vis perpendicular to the
eigenspace E1(q0) of Aq0−p, then vis an eigenvector of this shape operator, for any
q0∈Φ(p).q. Observe that the linear span of Φ(p).q is ¯νp(M), since q06= 0 and Φ(p)
acts irreducibly on this normal space. Then vis a common eigenvector for all shape
operators Aη,η∈νp(M). But A: ¯νp(M)→Sim0(TpM) is an isomorphism. This
is a contradiction. Then df(TpM) = {0}and the same is valid for all p0∈U. Then
f=f(p) = 1
2is constant on U.
Then the eigenvalues ˆ
λ1,ˆ
λ2,ˆ
λ3are constant on Uξ. Then ˆ
ξis a (non-umbilical)
parallel normal isoparametric field of Uξ. Then, by [CO] (see [BCO], Theorem
5.5.2), Uξand hence Uhas constant principal curvatures. But this is true provided
one shows that Uξis full and locally irreducible around some point q∈π−1({p}).
Let us show that the local normal holonomy group of Mat pcoincides with the re-
stricted normal holonomy group. In fact, the holonomy system [¯νp(M),R⊥(p),Φ(p)]
is irreducible and non-transitive. Then, by the holonomy theorem of Simons [S],
it is symmetric. Moreover, Lie(Φ(p)) is linearly generated by the endomorphisms
{R⊥
ξ,η (p)}. This implies that the local normal holonomy at pcoincides with Φ(p).
Then the local rank of M, as submanifold of the Euclidean space, is 1. This implies
that Mis full and locally irreducible around p. Hence Uξis full and irreducible
around any point q∈π−1({p}). Then Uis a submanifold with constant principal
curvatures.
Since the normal holonomy of Mis not transitive on the unit sphere, of the
normal space to the sphere, any principal holonomy tube (which is isoparametric)
has codimension at least 3 in the Euclidean space. Then, by the theorem of Thor-
bergsson [Th], Uis locally an orbit of an s-representation. Then kR⊥kis constant
on U. From this one obtains that kR⊥kis constant on Ω, where Ω is a connected
component of the open subset {p∈M:R⊥(p)6= 0}. But if p0∈Mis a limit
point of Ω then, R⊥(p0)6= 0. This implies that Ω can be enlarged unless p0∈Ω.
This shows that the open subset Ω is also closed in M. Then Mhas constant
principal curvatures. Hence, the image of M(under the isometric immersion), is
an embedded submanifold with constant principal curvatures. Moreover, it is an
orbit of an s-representation. From Proposition 2.17, the image of Mis a Veronese
submanifold.
The converse is true by Facts 2.16, (i) and (iii). 
Remark 5.1.We keep the notation of the proof of Theorem C. The fact that f
is constant can also be proved in the following way. Let p∈Mbe such that
R⊥(p)6= 0. Then, since the shape operator Amaps Φ(p)-orbits into SO(TpM)-
orbits of Sim0(TpM), one obtains that the second fundamental form is λ-isotropic.
That is, the length of α(X, X ) is λ(p) independent of Xin the unit sphere of
TpM, where αis the second fundamental form. The function λmust be a constant
multiple of f. Then, by Proposition 4.1. of [IO], λ, and hence f, must be constant
(n≥3).

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 33
6. The number of factors of the normal holonomy
In this section we will prove a sharp linear bound, depending on the dimension
nof the submanifold, of the number of irreducible factors of the local normal holo-
nomy representations. This improves, substantially, the quadratic bound 1
2n(n−1)
given in Theorem 4.5.1 of [BCO].
Proposition 6.1. Let Mnbe a submanifold of the Euclidean space RN. Assume
that at any point of Mthe local normal holonomy group and the restricted normal
holonomy group coincide (or, equivalently, the dimensions of the local normal holo-
nomy groups are constant on M). Let p∈Mand let rbe the number of irreducible
(non-abelian) subspaces of the representation of the restricted normal holonomy
group Φ(p)on νp(M). Then r≤n
2. Moreover, this bound is sharp for all n∈N
(also in the class of irreducible submanifolds).
Proof. Let us decompose νp(M) = ν0
p(M)⊕ν1
p(M)⊕... ⊕νr
p(M), where Φ(p) acts
trivially on ν0
p(M) and irreducibly on νi
p(M), for i= 1, ..., r. From the assumptions
we obtain that νi
p(M) extends to a ∇⊥-parallel subbundle νiof the normal bundle
ν(M), i= 0, ..., r (eventually, by making Msmaller around p). Note that we have
the decomposition ν(M) = ν0(M)⊕ν1(M)⊕... ⊕νr(M). Moreover, we obtain
from the assumptions, for any q∈M, that the local normal holonomy group Φ(q)
acts trivially on ν0
q(M) and irreducibly on νi
q(M), for any i= 1, ..., r.
Let R⊥
ξ,ξ0be the adapted normal curvature tensor (see Section 1.1). From the
expression of R⊥in terms of shape operators A, one has that R⊥
ξ,ξ0= 0 if and only
if [Aξ, Aξ0] = 0.
Observe, if i6=j, that R⊥
ξi,ξ0
j= 0 if ξi, ξ 0
jare normal sections that lie in νi(M)
and νj(M), respectively.
There must exist q∈M, arbitrary close to p, such that R⊥
νi
q,νi
q6={0}, for all
i= 1, ..., r. In fact, there exists q1∈M, arbitrary close to psuch that R⊥
ν1
q1,ν1
q1
6=
{0}(otherwise, ν1(M) would be flat). The above inequality must be true in a
neighbourhood V1of q1. Now choose q2∈V1such that R⊥
ν2
q2,ν2
q2
6={0}. Continuing
with this procedure we find q:= qrwith the desired properties.
Let us show that for any i= 1, ...., r there exist ξi, ξ0
ien νi
q(M) such that [Aξi, Aξ0
i]
does not belong to the algebra of endomorphisms generated by {Aηi}, where ηi∈
νq(M) has no component in νi
q(M). In fact, if this is not true, then, for any ξi, ξ0
i
in νi
q(M), [Aξi, Aξ0
i] commutes with Aξi(since the shape operators of elements of
the subspaces νj
q(M) commute with con Aξi, if j6=i). Then
h[[Aξi, Aξ0
i], Aξi], Aξ0
ii=0=−h[Aξi, Aξ0
i],[Aξi, Aξ0
i]i
and hence [Aξi, Aξ0
i] = 0. A contradiction, since R⊥
νi
q,νi
q6={0}. This proves our
assertion.
Observe that [Aξ1, Aξ0
1], ..., [Aξr, Aξ0
r] are linearly independent and commuting
skew-symmetric endomorphisms of TqM. Then r≤rank(SO(TpM)) = [ n
2] (the
integer part of n
2). This proves the inequality.
Let us see that it is sharp. For M2⊂Sk1−1,¯
M3⊂Sk2−1be a surface and a 3-
dimensional submanifold and such that the normal holonomies have one irreducible
factor (for example, the Veronese V2and V3). Let n > 3 and write n= 2dis nis
even or n= 2d+ 3 if nis odd.

34 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
Let Mnbe the product of dtimes M2or Mnbe the product of dtimes M2
by ¯
M3. Such submanifolds are contained in the product of Euclidean ambient
spaces. Moreover, the number of irreducible factors of the normal holonomy group
(representation) of Mnis exactly the upper bound [ n
2]. Moreover, since Mnis
contained in a sphere, we can apply to Mna conformal transformation of the
sphere (the normal holonomy group is a conformal invariant) in such a way that
Mnis an irreducible (Riemannian) submanifold of the Euclidean space.

7. Further comments
Remark 7.1.There is a beautiful result of Little and Phol [LP] which characterizes
Veronese submanifolds Mn, modulo projective diffeomorphisms, by the two-piece
property and the fact that the codimension is the maximal one 1
2n(n+ 1) (for
submanifolds with the two-piece property). Note that a tight submanifold has the
two-piece property. This result generalizes the well-know result of Kuiper for n= 2.
A projective transformation, in general, does not preserve the normal holonomy
(unless it induces a conformal transformation of the ambient sphere).
Remark 7.2.A natural question that arises, since the normal holonomy group is a
conformal invariant, is the following: is a compact submanifold Mn⊂Sn−1+ 1
2n(n+1),
with irreducible and non-transitive (restricted) normal holonomy, equivalent, mod-
ulo conformal transformations of the sphere, to a Veronese submanifold?.
Remark 7.3.The symmetric space X= SU(4)/SO(4), dual to Sl(4)/SO(4), is
isometric to the Grassmannian SO(6)/SO(3) ×SO(3). In this last model, T[e]X=
R3×3and the isotropy representation is given by (g, h).T =gT h−1, (g, h)∈SO(3)×
SO(3). The Veronese submanifold V3is given by
SO(3) ×SO(3).Id = SO(3) × {Id}.Id = SO(3) ⊂R3×3
Thus V3is also an orbit of the smaller group SO(3) 'SO(3) × {Id}. The other
orbits SO(3).A, where Ais invertible and near Id, must be full and irreducible
submanifolds of R3×3, since V3is so. Note that the action of SO(3) on R3×3is
reducible. In fact, it is the sum of three times the standard representation of SO(3)
on R3. The orbit, SO(3).A is not minimal in the sphere, for Ageneric. So, the
normal holonomy holonomy group of this orbit must be transitive on the unit sphere
(of the normal space to the sphere).
Observe that the linear isomorphism rA−1of R3×3,rA−1(T) = T A−1, transforms
SO(3).A into V3. In particular, since V3is a tight submanifold, that orbit is so.
Hence, as it is well known, SO(3).A is a taut submanifold, since it lies in a sphere
(see [CR, G]).

NORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 35
8. Appendix
8.1. The Veronese embedding.
We recall here some basic definitions and facts about the well-known Veronese
submanifolds.
Let Sn,n≥2, be the unit sphere of the Euclidean space Rn+1 and let Rn+1 ⊗sRn+1
be space of symmetric 2-tensors of Rn+1. Let h:Rn+1⊗sRn+1 →Sim(n+ 1) the
usual isomorphism onto the symmetric matrices of Rn+1. Namely, let e1, ..., en+1
be the canonical basis of Rn+1. Then, h(ei⊗ej+ej⊗ei) is the matrix whose
coefficients ak,l are all zero except:
ai,j =aj,i = 1,if i6=j;ai,i = 2,if i=j
The Veronese map Q:Sn→Sim(n+ 1) is defined by
Q(v) = h(v⊗v)
Observe that (Q(v))i,j =vivj, where v= (v1, ..., vn+1). Let h,ibe the inner
product on Sim(n+ 1) given by hA, Bi=1
2trace(AB). Then Q is an isometric
immersion. Observe that trace(Q(v)) = 1, for all v∈Sn. So, the image of Qis
contained in the affine hyperplane of Sim(n+ 1), given by the linear equation
h · , Idi=1
2
Let ˜ρ:Sn→Sim0(n+1) be defined by ˜ρ(v) = Q(v)−1
n+1 Id, where Sim0(n+ 1)
are the symmetric traceless matrices. The map ˜ρis called the Veronese Riemannian
immersion of the sphere Sninto Sim0(n+ 1). One has that ˜ρ, (as well as Q) is
O(n+ 1)-equivariant. Namely, if g∈O(n+ 1), then
˜ρ(g.v) = g. ˜ρ(v).g−1
In fact, if we regard v∈Rn+1 as a column vector, then
˜ρ(v) = v.vt−1
n+ 1Id
From the above formula it follows easily the O(n+ 1)-equivariance of ˜ρ. It
is also not difficult to verify, as it is well known, that ˜ρ(v) = ˜ρ(w) if and only if
w=±v. Therefore, ˜ρprojects down to an isometric O(n+1)-equivariant embedding
ρ:RPn→Sim0(n+ 1), the so-called Veronese Riemannian embedding.
Let us consider the simple symmetric pair (Sl(n+ 1),SO(n+ 1)) of the non-
compact type. The Cartan decomposition associated to such a pair is
sl(n+ 1) = so(n+ 1) ⊕S im0(n+ 1)
Then the (irreducible) isotropy representation of X= Sl(n+ 1)/SO(n+ 1) is natu-
rally identified with the action, by conjugation, of SO(n+1) on S im0(n+ 1). Then,
the image of the Veronese embedding, is the orbit
M= SO(n+ 1).S
where S∈Sim0(n+1) is the diagonal matrix with exactly two eigenvalues. Namely,
1−1
n+1 and −1
n+1 . The first one, with multiplicity 1, is associated to the eigenspace
Re1and the second one, with multiplicity n, is associated to the eigenspace (Re1)⊥.
Let S0∈Sim0(n+1) with exactly two eigenvalues λ1of multiplicity 1 and λ2with
multiplicity n. Assume that kS0k=kSk(i.e. Sand S0have the same length). It is
easy to verify that either λ1= 1 −1
n+1 , λ2=−1
n+1 or λ1=−1 + 1
n+1 , λ2=1
n+1 .

36 CARLOS OLMOS AND RICHAR FERNANDO RIA ˜
NO-RIA ˜
NO
In the first case one has that S0∈S0(n+ 1).S =ρ(RPn). In the second case,
−S0∈S0(n+ 1).S.
Observe that S0and −S0cannot be both in the image of the Veronese embedding,
since the respective eigenvalues of multiplicity 1 are different. In general, if ¯
S∈
Sim0(n+ 1) has two different eigenvalues, one of multiplicity 1 and the other of
multiplicity n, then ¯
S=λS, for some 0 6=λ∈R. The orbit SO(n+ 1).¯
Sis
called a Veronese-type orbit (see Section 1.1). Observe that there are exactly two
Veronese-type orbits in any given sphere, centered at 0, of Sim0(n+ 1). Moreover,
any of these two Veronese-type orbits is isometric to the other, via the isometry
−IdSim0(n+1) of Sim0(n+ 1).
We have the following well-known fact.
Lemma 8.1. Let SO(r)acts by conjugation on Sim0(r), the traceless symmetric
r×r-matrices, and let M=SO(r).A be an orbit, A6= 0. Then r−1≤dim(M).
Moreover, the equality holds if and only if Mis an orbit of Veronese-type.
Proof. Let us assume that Mhas minimal dimension. We will first prove that Ahas
exactly two eigenvalues. If not, let λ1, ..., λdbe the different eigenvalues of Awith
associated eigenspaces E1, ..., Ed(d≥3). Then the isotropy subgroup SO(r)A=
S(SO(E1)×... ×SO(Ed)) has less dimension than S(SO(E1)×SO(E2⊕... ⊕Ed)),
which is the isotropy group of some 6=A0∈Sim0(r) with two different eigen-
values whose associated eigenspaces are E1and E2⊕... ⊕Ed. Then dim(M)>
dim(SO(r).A0). A contradiction. Therefore, d= 2. (Observe that d= 1 implies
that A= 0, since it is traceless).
Let now k= dim(E1) and so r−k= dim(E2).
We have the well known formula for the dimension of the Grassmannians,
dim(M) = dim(SO(r)) −dim(SO(k)) −dim(SO(r−k)) = k(r−k)
But the quadratic q(x) = x(r−x), x∈[0, r], is increasing in the interval [0, r/2)
and it is decreasing in (r/2, r]. So, the minimum of q, restricted to the finite set
{1, ..., r −1}is attained at both, x= 1 and x=r−1. Then k= 1 or k=r−1, in
which case Mis a Veronese-type orbit (of dimension r−1).

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Facultad de Matem´
atica, Astronom
´
ıa y F
´
ısica, Universidad Nacional de C´
ordoba,
Ciudad Universitaria, 5000 C´
ordoba, Argentina
E-mail address:olmos@famaf.unc.edu.ar riano@famaf.unc.edu.ar