Polar actions on compact Euclidean hypersurfaces
Polar actions on compact Euclidean hypersurfaces.
Ion Moutinho & Ruy Tojeiro
Abstract: Given an isometric immersion f : Mn → Rn+1 of a compact Riemannian manifold
of dimension n ≥ 3 into Euclidean space of dimension n+ 1, we prove that the identity com-
ponent Iso0(Mn) of the isometry group Iso(Mn) of Mn admits an orthogonal representation
Φ: Iso0(Mn) → SO(n + 1) such that f ◦ g = Φ(g) ◦ f for every g ∈ Iso0(Mn). If G is a
closed connected subgroup of Iso(Mn) acting locally polarly on Mn, we prove that Φ(G) acts
polarly on Rn+1, and we obtain that f(Mn) is given as Φ(G)(L), where L is a hypersurface of
a section which is invariant under the Weyl group of the Φ(G)-action. We also find several suf-
ficient conditions for such an f to be a rotation hypersurface. Finally, we show that compact
Euclidean rotation hypersurfaces of dimension n ≥ 3 are characterized by their underlying
warped product structure.
MSC 2000: 53 A07, 53 C40, 53 C42.
Key words: polar actions, rotation hypersurfaces, isoparametric submanifolds, rigidity of
hypersurfaces, warped products.
1 Introduction
Let G be a connected subgroup of the isometry group Iso(Mn) of a compact Riemannian
manifold Mn of dimension n ≥ 3, which we always assume to be connected. Given an
isometric immersion f : Mn → RN into Euclidean space of dimension N , in general one
can not expect G to be realizable as a group of rigid motions of RN that leave f(Mn)
invariant. Nevertheless, a fundamental fact for us is that in codimension N −n = 1 this
is indeed the case.
Theorem 1 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface. Then the identity
component Iso0(Mn) of the isometry group of Mn admits an orthogonal representation
Φ: Iso0(Mn) → SO(n+ 1) such that f ◦ g = Φ(g) ◦ f for all g ∈ Iso0(Mn).
Theorem 1 may be regarded as a generalization of a classical result of Kobayashi
[Ko], who proved that a compact homogeneous hypersurface of Euclidean space must
http://arxiv.org/abs/0704.1807v1
be a round sphere. In fact, the crucial step in the proof of Kobayashi’s theorem is to
show that the isometry group of the hypersurface can be realized as a closed subgroup
of O(n+ 1). The idea of the proof of Theorem 1 actually appears already in [MPST],
where Euclidean G-hypersurfaces of cohomogeneity one, i.e., with principal orbits of
codimension one, are considered.
We apply Theorem 1 to study compact Euclidean hypersurfaces f : Mn → Rn+1,
n ≥ 3, on which a connected closed subgroup G of Iso(Mn) acts locally polarly. Recall
that an isometric action of a compact Lie group G on a Riemannian manifoldMn is said
to be locally polar if the distribution of normal spaces to principal orbits on the regular
part of Mn is integrable. If Mn is complete, this implies the existence of a connected
complete immersed submanifold Σ of Mn that intersects orthogonally all G-orbits (cf.
[HLO]). Such a submanifold is called a section, and it is always a totally geodesic
submanifold of Mn. In particular, any isometric action of cohomogeneity one is locally
polar. The action is said to be polar if there exists a closed and embedded section.
Clearly, for orthogonal representations there is no distinction between polar and locally
polar actions, for in this case sections are just affine subspaces.
It was shown in [BCO], Proposition 3.2.9 that if a closed subgroup of SO(N) acts
polarly on RN and leaves invariant a submanifold f : Mn → RN , then its restricted action
on Mn is locally polar. Our next result states that any locally polar isometric action
of a compact connected Lie group on a compact Euclidean hypersurface of dimension
n ≥ 3 arises in this way.
Theorem 2 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface and let G be a closed
connected subgroup of Iso(Mn) acting locally polarly onMn with cohomogeneity k. Then
there exists an orthogonal representation Ψ: G→ SO(n+1) such that Ψ(G) acts polarly
on Rn+1 with cohomogeneity k + 1 and f ◦ g = Ψ(g) ◦ f for every g ∈ G.
A natural problem that emerges is how to explicitly construct all compact hy-
persurfaces f : Mn → Rn+1 that are invariant under a polar action of a closed sub-
group G ⊂ SO(n + 1). This is accomplished by the following result. Recall that
the Weyl group of the G-action is defined as W = N(Σ)/Z(Σ), where Σ is a section,
N(Σ) = {g ∈ G | gΣ = Σ} and Z(Σ) = {g ∈ G | gp = p, ∀ p ∈ Σ} is the intersection of
the isotropy subgroups Gp, p ∈ Σ.
Theorem 3 Let G ⊂ SO(n + 1) be a closed subgroup that acts polarly on Rn+1, let
Σ be a section and let L be a compact immersed hypersurface of Σ which is invariant
under the Weyl group of the G-action. Then G(L) is a compact G-invariant immersed
hypersurface of Rn+1. Conversely, any compact hypersurface f : Mn → Rn+1 that is
invariant under a polar action of a closed subgroup of SO(n+ 1) can be constructed in
this way.
The simplest examples of hypersurfaces f : Mn → Rn+1 that are invariant under a
polar action on Rn+1 of a closed subgroup of SO(n+ 1) are the rotation hypersurfaces.
These are invariant under a polar representation which is the sum of a trivial represen-
tation on a subspace Rk ⊂ Rn+1 and one which acts transitively on the unit sphere of
its orthogonal complement Rn−k+1. In this case any subspace Rk+1 containing Rk is a
section, and the Weyl group of the action is just the group Z2 generated by the reflection
with respect to the “axis” Rk. Thus, the condition that a hypersurface Lk ⊂ Rk+1 be
invariant under the Weyl group reduces in this case to Lk being symmetric with respect
to Rk. We now give several sufficient conditions for a compact Euclidean hypersurface
as in Theorem 2 to be a rotation hypersurface.
Corollary 4 Under the assumptions of Theorem 2, any of the following additional con-
ditions implies that f is a rotation hypersurface:
(i) there exists a totally geodesic (in Mn) G-principal orbit;
(ii) k = n− 1;
(iii) the G-principal orbits are umbilical in Mn;
(iv) there exists a G-principal orbit with nonzero constant sectional curvatures;
(v) there exists a G-principal orbit with positive sectional curvatures.
Moreover, G is isomorphic to one of the closed subgroups of SO(n − k + 1) that act
transitively on Sn−k.
For a list of all closed subgroups of SO(n) that act transitively on the sphere, see,
e.g., [EH], p. 392. Corollary 4 generalizes similar results in [PS], [AMN] and [MPST]
for compact Euclidean G-hypersurfaces of cohomogeneity one. In part (v), weakening
the assumption to non-negativity of the sectional curvatures of some principal G-orbit
implies f to be a multi-rotational hypersurface in the sense of [DN]:
Corollary 5 Under the assumptions of Theorem 2, suppose further that there exists
a G-principal orbit Gp with nonnegative sectional curvatures. Then there exist an or-
thogonal decomposition Rn+1 =
ni into G̃-invariant subspaces, where G̃ = Ψ(G),
and connected Lie subgroups G1, . . . , Gk of G̃ such that Gi acts on R
ni, the action being
transitive on Sni−1 ⊂ Rni, and the action of Ḡ = G1 × . . .×Gk on R
n+1 given by
(g1 . . . gk)(v0, v1, . . . , vk) = (v0, g1v1, . . . , gkvk)
is orbit equivalent to the action of G̃. In particular, if Gp is flat then ni = 2 and Gi is
isomorphic to SO(2) for i = 1, . . . , k.
Finally, we apply some of the previous results to a problem that at first sight has no
relation to isometric actions whatsoever.
Let f : Mn → Rn+1 be a rotation hypersurface as described in the paragraph following
Theorem 3. Then the open and dense subset of Mn that is mapped by f onto the
complement of the axis Rk is isometric to the warped product Lk ×ρN
n−k, where Nn−k
is the orbit of some fixed point f(p) ∈ Lk under the action of G̃, and the warping
function ρ : Lk → R+ is a constant multiple of the distance to R
k. Recall that a warped
product N1 ×ρ N2 of Riemannian manifolds (N1, 〈 , 〉N1) and (N2, 〈 , 〉N2) with warping
function ρ: N1 → R+ is the product manifold N1 ×N2 endowed with the metric
〈 , 〉 = π∗1〈 , 〉N1 + (ρ ◦ π1)
2π∗2〈 , 〉N2,
where πi: N1 ×N2 → Ni, 1 ≤ i ≤ 2, denote the canonical projections.
We prove that, conversely, compact Euclidean rotation hypersurfaces of dimension
n ≥ 3 are characterized by their warped product structure.
Theorem 6 Let f : Mn → Rn+1, n ≥ 3, be a compact hypersurface. If there exists
an isometry onto an open and dense subset U ⊂ Mn of a warped product Lk ×ρ N
with Nn−k connected and complete (in particular if Mn is isometric to a warped product
Lk ×ρ N
n−k) then f is a rotation hypersurface.
Theorem 6 can be seen as a global version in the hypersurface case of the local
classification in [DT] of isometric immersions in codimension ℓ ≤ 2 of warped products
Lk ×ρ N
n−k, n− k ≥ 2, into Euclidean space.
Acknowledgment. We are grateful to C. Gorodski for helpful discussions.
2 Proof of Theorem 1
The proof of Theorem 1 relies on a result of Sacksteder [Sa] (see also the proof in [Da],
Theorem 6.14, which is based on an unpublished manuscript by D. Ferus) according to
which a compact hypersurface f : Mn → Rn+1, n ≥ 3, is rigid whenever the subset of
totally geodesic points of f does not disconnect Mn. Recall that f is rigid if any other
isometric immersion f̃ : Mn → Rn+1 differs from it by a rigid motion of Rn+1. The proof
of Sacksteder’s theorem actually shows more than the preceding statement. Namely, let
f, f̃ : Mn → Rn+1 be isometric immersions into Rn+1 of a compact Riemannian manifold
Mn, n ≥ 3, and let φ: T⊥Mnf → T
be the vector bundle isometry between the
normal bundles of f and f̃ defined as follows. Given a unit vector ξx ∈ T
x ∈M , let φ(ξx) be the unique unit vector in T
such that {f∗X1, . . . , f∗Xn, ξx} and
{f̃∗X1, . . . , f̃∗Xn, φ(ξx)} determine the same orientation in R
n+1, where {X1, . . . , Xn} is
any ordered basis of TxM
n. Then it is shown that
(x) = ±φ(x) ◦ αf(x) (1)
at each point x ∈Mn, where αf and αf̃ denote the second fundamental forms of f and f̃ ,
respectively, with values in the normal bundle. The proof is based on a careful analysis of
the distributions on Mn determined by the relative nullity subspaces ∆(x) = ker αf (x)
and ∆̃(x) = ker α
(x) of f and f̃ , respectively. Rigidity of f under the assumption
that the subset of totally geodesic points of f does not disconnect Mn then follows
immediately from the Fundamental Theorem of Hypersurfaces (cf. [Da], Theorem 1.1).
Proof of Theorem 1. Given g ∈ Iso0(Mn), let αf◦g denote the second fundamental form
of f ◦ g. We claim that
αf◦g(x) = φg(x) ◦ αf(x) (2)
for every g ∈ Iso0(Mn) and x ∈ Mn, where φg denotes the vector bundle isometry
between T⊥Mnf and T
⊥Mnf◦g defined as in the preceding paragraph. On one hand,
αf◦g(x)(X, Y ) = αf(gx)(g∗X, g∗Y ) (3)
for every g ∈ Iso0(Mn), x ∈ Mn and X, Y ∈ TxM
n. In particular, this implies that for
any fixed x ∈ Mn the map Θx: Iso
0(Mn) → Sym(TxM
n × TxM
n → T⊥x M
f ) into the
vector space of symmetric bilinear maps of TxM
n × TxM
n into T⊥x M
f , given by
Θx(g)(X, Y ) = φg(x)
−1(αf◦g(x)(X, Y )) = φg(x)
−1(αf (gx)(g∗X, g∗Y ))
for any X, Y ∈ TxM
n, is continuous. On the other hand, by the preceding remarks on
Sacksteder’s theorem, either αf◦g(x) = φg(x) ◦αf (x) or αf◦g(x) = −φg(x) ◦αf (x). Thus
Θx is a continuous map taking values in {αf(x),−αf (x)}, hence it must be constant
because Iso0(Mn) is connected. Since Θx(id) = αf(x), our claim follows.
We conclude that for each g ∈ Iso0(Mn) there exists a rigid motion g̃ ∈ Iso(Rn+1)
such that f ◦g = g̃◦f . It now follows from standard arguments that g 7→ g̃ defines a Lie-
group homomorphism Φ: Iso0(Mn) → Iso(Rn+1), whose image must lie in SO(n + 1)
because it is compact and connected.
Remarks 7 (i) Theorem 1 is also true for compact hypersurfaces of dimension n ≥ 3
of hyperbolic space, as well as for complete hypersurfaces of dimension n ≥ 4 of the
sphere. It also holds for complete hypersurfaces of dimension n ≥ 3 of both Euclidean
and hyperbolic spaces, under the additional assumption that they do not carry a com-
plete leaf of dimension (n− 1) or (n− 2) of their relative nullity distributions. In fact,
the proof of Theorem 1 carries over in exactly the same way for these cases, because so
does equation (1) (cf. [Da], p. 96-100).
(ii) Clearly, Theorem 1 does not hold for isometric immersions f : Mn → Rn+ℓ of codi-
mension ℓ ≥ 2. Namely, counterexamples can be easily constructed, for instance,
by means of compositions f = h ◦ g of isometric immersions g: Mn → Rn+1 and
h: V → Rn+ℓ, with V ⊂ Rn+1 an open subset containing g(Mn).
The next result gives a sufficient condition for rigidity of a compact hypersurface
f : Mn → Rn+1 as in Theorem 1 in terms of G̃ = Φ(Iso0(Mn)).
Proposition 8 Under the assumptions of Theorem 1, suppose that G̃ = Φ(Iso0(Mn))
does not have a fixed vector. Then f is free of totally geodesic points. In particular, f
is rigid.
Proof: Suppose that the subset B of totally geodesic points of f is nonempty. Since
αf◦g(x) = φg(x) ◦ αf(x) for every g ∈ G = Iso
0(Mn) and x ∈ Mn by (2), B coincides
with the set of totally geodesic points of f ◦ g for every g ∈ G. In view of (3), this
amounts to saying that B is G-invariant. Thus, if Gp is the orbit of a point p ∈ B
then Gp ⊂ B. Since Gp is connected, it follows from [DG], Lemma 3.14 that f(Gp) is
contained in a hyperplane H that is tangent to f along Gp. Therefore, a unit vector
v orthogonal to H spans T⊥gpM
f for every gp ∈ Gp. Since g̃T
f = T
f for every
g̃ = Φ(g) ∈ G̃, because f is equivariant with respect to Φ, the connectedness of G̃
implies that it must fix v.
Proposition 8 implies, for instance, that if a closed connected subgroup of Iso(Mn)
acts onMn with cohomogeneity one then either f is a rotation hypersurface over a plane
curve or it is free of totally geodesic points, and in particular it is rigid (see [MPST],
Theorem 1). As another consequence we have:
Corollary 9 Let f : M3 → R4 be a compact hypersurface. If f has a totally geodesic
point (in particular, if it is not rigid) then either M3 has finite isometry group or f is
a rotation hypersurface.
Proof: Let Φ: Iso0(M3) → SO(4) be the orthogonal representation given by Theorem 1.
By Proposition 8, if f has a totally geodesic point then G̃ = Φ(Iso0(M3)) has a fixed
vector v, hence it can be regarded as a subgroup of SO(3). Therefore, either the re-
stricted action of G̃ on {v}⊥ has also a fixed vector or it is transitive on the sphere.
In the first case, either Iso0(M3) is trivial, that is, Iso(M3) is finite, or G̃ fixes a two
dimensional subspace R2 of R4, in which case f is a rotation hypersurface over a surface
in a half-space R3+ with R
2 as boundary. In the latter case, f is a rotation hypersurface
over a plane curve in a half-space R2+ having span{v} as boundary.
3 Proof of Theorem 2
For the proof of Theorem 2, we recall from Theorem 1 that there exists an orthogonal
representation Φ: Iso0(Mn) → SO(n + 1) such that f ◦ g = Φ(g) ◦ f for every g ∈
Iso0(Mn). Since G is connected we have G ⊂ Iso0(Mn), hence it suffices to prove that
G̃ = Φ(G) acts polarly on Rn+1 with cohomogeneity k + 1 and then set Ψ = Φ|G.
We claim that there exists a principal orbit Gp such that the position vector f is
nowhere tangent to f(Mn) along Gp, that is, f(g(p)) 6∈ f∗(g(p))Tg(p)M
n for any g ∈ G.
In order to prove our claim we need the following observation.
Lemma 10 Let f : Mn → Rn+1 be a hypersurface. Assume that the position vector
is tangent to f(Mn) on an open subset U ⊂ Mn. Then the index of relative nullity
νf(x) = dim ∆f (x) of f is positive at any point x ∈ U .
Proof: Let Z be a vector field on U such that f∗(p)Z(p) = f(p) for any p ∈ U and let η
be a local unit normal vector field to f . Differentiating 〈η, f〉 = 0 yields 〈AX,Z〉 = 0 for
any tangent vector field X on U , where A denotes the shape operator of f with respect
to η. Thus AZ = 0.
Going back to the proof of the claim, since Mn is a compact Riemannian manifold
isometrically immersed in Euclidean space as a hypersurface, there exists an open subset
V ⊂ Mn where the sectional curvatures of Mn are strictly positive. In particular, the
index of relative nullity of f vanishes at every x ∈ V . If the position vector were tangent
to f(Mn) at every regular point of V , it would be tangent to f(Mn) everywhere on V ,
because the set of regular points is dense onMn. This is in contradiction with Lemma 10
and shows that there must exist a regular point p ∈ V such that f(p) 6∈ f∗(p)TpM
Since f is equivariant, we must have in fact that f(g(p)) 6∈ f∗(g(p))Tg(p)M
n for any
g ∈ G and the claim is proved.
Now let Gp be a principal orbit such that the position vector f is nowhere tangent
to f(Mn) along Gp. Then the normal bundle of f |Gp splits (possibly non-orthogonally)
as span{f}⊕ f∗T
⊥Gp. Let ξ be an equivariant normal vector field to Gp in Mn and let
η = f∗ξ. Then, denoting Φ(g) by g̃ and identifying g̃ with its derivative at any point,
because it is linear, we have
g̃η(p) = (f ◦ g)∗(p)ξ(p) = f∗(gp)g∗(p)ξ(p) = f∗(gp)ξ(gp) = η(gp) (4)
for any g ∈ G. In particular, 〈η(gp), f(gp)〉 = 〈g̃η(p), g̃f(p)〉 = 〈η(p), f(p)〉 for every
g ∈ G, that is, 〈η, f〉 is constant on Gp. It follows that
X〈η, f〉 = 0, for any X ∈ TGp, (5)
and hence
〈∇̃Xη, f〉 = X〈η, f〉 − 〈ξ,X〉 = 0, (6)
where ∇̃ denotes the derivative in Rn+1. On the other hand, since G acts locally polarly
on Mn, we have that ξ is parallel in the normal connection of Gp in M . Therefore
(∇̃Xη)f∗T⊥Gp = f∗(∇Xξ)T⊥Gp = 0, (7)
where ∇ is the Levi-Civita connection of Mn; here, writing a vector subbundle as a
subscript of a vector field indicates taking its orthogonal projection onto that subbundle.
It follows from (6) and (7) that η is parallel in the normal connection of f |Gp. On the
other hand, the position vector f is clearly also an equivariant normal vector field of
f |Gp which is parallel in the normal connection.
Thus, we have shown that there exist equivariant normal vector fields to G̃(f(p)) =
f(Gp) that form a basis of the normal spaces at each point and which are parallel in
the normal connection. The statement now follows from the next known result, a proof
of which is included for completeness.
Lemma 11 Let G̃ ⊂ SO(n+ 1) have an orbit G̃(q) along which there exist equivariant
normal vector fields that form a basis of the normal spaces at each point and which are
parallel in the normal connection. Then G̃ acts polarly.
Proof: Since there exist equivariant normal vector fields to G̃(q) that form a basis of
the normal spaces at each point, the isotropy group acts trivially on each normal space,
hence G̃(q) is a principal orbit. We now show that the normal space T⊥q G̃(q) is a section,
for which it suffices to show that any Killing vector field X induced by G̃ is everywhere
orthogonal to T⊥q G̃(q). Given ξq ∈ T
q G̃(q), let ξ be an equivariant normal vector field
to G̃(q) extending ξq, which is also parallel in the normal connection. Then, denoting
by φt the flow of X and setting c(t) = φt(q) we have
|t=0φt(ξq)
T⊥q G̃(q)
|t=0ξ(c(t))
T⊥q G̃(q)
= ∇⊥X(q)ξ = 0,
where
denotes the covariant derivative in Euclidean space along c(t).
Remark 12 A closed subgroup G ⊂ SO(n + ℓ), ℓ ≥ 2, that acts non-polarly on Rn+ℓ
may leave invariant a compact submanifold f : Mn → Rn+ℓ and induce a locally polar
action on Mn. For instance, consider a compact submanifold f : Mn → Rn+2 that is
invariant by the action of a closed subgroup G ⊂ SO(n+2) that acts non-polarly on Rn+2
with cohomogeneity three. Then the induced action of G onMn has cohomogeneity one,
whence is locally polar. Moreover, taking f as a compact hypersurface of Sn+1 shows
also that Theorem 2 is no longer true if Rn+1 is replaced by Sn+1.
Theorem 2 yields the following obstruction for the existence of an isometric immer-
sion in codimension one into Euclidean space of a compact Riemannian manifold acted
on locally polarly by a closed connected Lie subgroup of its isometry group.
Corollary 13 Let Mn be a compact Riemannian manifold of dimension n ≥ 3 acted
on locally polarly by a closed connected subgroup G of its isometry group. If G has an
exceptional orbit then Mn can not be isometrically immersed in Euclidean space as a
hypersurface.
Proof: Let f : Mn → Rn+1 be an isometric immersion of a compact Riemannian manifold
acted on locally polarly by a closed connected subgroup G of its isometry group. We
will prove that G can not have any exceptional orbit. By Theorem 2 there exists an
orthogonal representation Ψ: G → SO(n + 1) such that G̃ = Ψ(G) acts polarly on
n+1 with cohomogeneity k + 1 and f ◦ g = Ψ(g) ◦ f for every g ∈ G. Let Gp be
a nonsingular orbit. Then Gp has maximal dimension among all G-orbits, and hence
G̃f(p) = f(Gp) has maximal dimension among all G̃-orbits. Since polar representations
are known to admit no exceptional orbits (cf. [BCO], Corollary 5.4.3), it follows that
G̃f(p) is a principal orbit. Then, for any g in the isotropy subgroup Gp we have that
g̃ = Ψ(g) ∈ G̃f(p), thus for any ξp ∈ T
p Gp we obtain
f∗(p)ξp = g̃∗f∗(p)ξp = (g̃ ◦ f)∗(p)ξp = (f ◦ g)∗(p)ξp = f∗(gp)g∗ξp = f∗(p)g∗ξp.
Since f∗(p) is injective, then g∗ξp = ξp. This shows that the slice representation, that
is, the action of the isotropy group Gp on the normal space T
p G(p) to the orbit G(p)
at p, is trivial. Thus Gp is a principal orbit.
4 Isoparametric submanifolds
We now recall some results on isoparametric submanifolds and derive a few additional
facts on them that will be needed for the proofs of Theorem 3 and Corollaries 4 and 5.
Given an isometric immersion f : Mn → RN with flat normal bundle, it is well-
known (cf. [St]) that for each point x ∈Mn there exist an integer s = s(x) ∈ {1, . . . , n}
and a uniquely determined subset Hx = {η1, . . . , ηs} of T
f such that TxM
n is the
orthogonal sum of the nontrivial subspaces
Eηi(x) = {X ∈ TxM
n : α(X, Y ) = 〈X, Y 〉 ηi, for all Y ∈ TxM
n}, 1 ≤ i ≤ s.
Therefore, the second fundamental form of f has the simple representation
α(X, Y ) =
〈Xi, Yi〉 ηi, (8)
or equivalently,
AξX =
〈ξ, ηi〉Xi, (9)
where X 7→ Xi denotes orthogonal projection onto Eηi . Each ηi ∈ Hx is called a
principal normal of f at x. The Gauss equation takes the form
R(X, Y ) =
i,j=1
〈ηi, ηj〉Xi ∧ Yj, (10)
where (Xi ∧ Yj)Z = 〈Z, Yj〉Xi − 〈Z,Xi〉Yj.
Lemma 14 Let f : Mn → RN be an isometric immersion with flat normal bundle of a
Riemannian manifold with constant sectional curvature c. Let η1, . . . , ηs be the distinct
principal normals of f at x ∈ Mn. Then
(i) There exists at most one ηi such that |ηi|
2 = c.
(ii) For all i, j, k ∈ {1, . . . , s} with i 6= j 6= k 6= i the vectors ηi − ηk and ηj − ηk are
linearly independent.
Proof: It follows from (10) that 〈ηi, ηj〉 = c for all i, j ∈ {1, . . . , s} with i 6= j. If
2 = |ηj|
2 = c, this gives |ηi − ηj |
2 = 0.
(iii) Assume that there exist λ 6= 0 and i, j, k ∈ {1, . . . , s} with i 6= j 6= k 6= i such that
ηi − ηk = λ(ηj − ηk). Then
2 − c = 〈ηi − ηk, ηi〉 = λ〈ηj − ηk, ηi〉 = 0,
and similarly |ηj|
2 = c, in contradiction with (i).
Let f : Mn → RN be an isometric immersion with flat normal bundle and let Hx =
{η1, . . . , ηs} be the set of principal normals of f at x ∈M
n. If the mapMn → {1, . . . , n}
given by x 7→ #Hx has a constant value s on an open subset U ⊂M
n, there exist smooth
normal vector fields η1, . . . , ηs on U such that Hx = {η1(x), . . . , ηs(x)} for any x ∈ U .
Furthermore, each Eηi = (Eηi(x))x∈U is a C
∞- subbundle of TU for 1 ≤ i ≤ s. The
following result is contained in [DN], Lemma 2.3.
Lemma 15. Let f : Mn → Rn+p be an isometric immersion with flat normal bundle and
a constant number s of principal normals η1, . . . , ηs everywhere. Assume that for a fixed
i ∈ {1, . . . , s} all principal normals ηj, j 6= i, are parallel in the normal connection along
Eηi and that the vectors ηi− ηj and ηi− ηℓ are everywhere pairwise linearly independent
for any pair of indices 1 ≤ j 6= ℓ ≤ s with j, ℓ 6= i. Then E⊥ηi is totally geodesic.
Proof: The Codazzi equation yields
〈∇XℓXj , Xi〉(ηi − ηj) = 〈∇XℓXj , Xi〉(ηi − ηℓ), i 6= j 6= k 6= i, (11)
∇⊥Xiηj = 〈∇XjXj , Xi〉(ηj − ηi), i 6= j, (12)
for all unit vectors Xi ∈ Eηi , Xj ∈ Eηj and Xℓ ∈ Eηℓ .
An isometric immersion f : Mn → RN is called isoparametric if it has flat normal
bundle and the principal curvatures of f with respect to every parallel normal vector
field along any curve in Mn are constant (with constant multiplicities). The following
facts on isoparametric submanifolds are due to Strübing [St].
Theorem 16 Let f : Mn → RN be an isoparametric isometric immersion. Then
(i) The number of principal normals is constant on Mn.
(ii) The first normal spaces, i.e, the subspaces of the normal spaces spanned by the im-
age of the second fundamental form, determine a parallel subbundle of the normal
bundle.
(iii) The subbundles Eηi, 1 ≤ i ≤ s, are totally geodesic and the principal normals
η1, . . . , ηs are parallel in the normal connection.
(iv) The subbundles Eηi, 1 ≤ i ≤ s, are parallel if and only if f has parallel second
fundamental form.
(v) If the principal normals η1, . . . , ηs satisfy 〈ηi, ηj〉 ≥ 0 everywhere then f has parallel
second fundamental form and 〈ηi, ηj〉 = 0 everywhere.
The next result will be used in the proofs of Corollaries 4 and 5.
Proposition 17 Let f : Mn → RN , n ≥ 2, be a compact isoparametric submanifold.
(i) If Mn has constant sectional curvature c, then either c > 0 and f(Mn) is a round
sphere or c = 0 and f(Mn) is an extrinsic product of circles.
(ii) If Mn has nonnegative sectional curvatures, then f(Mn) is an extrinsic product
of round spheres or circles. In particular, if Mn has positive sectional curvatures
then f(Mn) is a round sphere.
Proof: By Theorem 16, the number of distinct principal normals η1, . . . , ηs of f is con-
stant on Mn and all of them are parallel in the normal connection. Moreover, the
subbundles Eηi , 1 ≤ i ≤ s, are totally geodesic. If M
n has constant sectional cur-
vature c, then it follows from Lemmas 14 and 15 that also E⊥ηi is totally geodesic for
1 ≤ i ≤ s, and hence the sectional curvatures along planes spanned by vectors in dif-
ferent subbundles vanish. Therefore c = 0, unless s = 1 and f is umbilic, in which case
c > 0 and f(Mn) is a round sphere. Furthermore, if c = 0 and Eηi has rank at least 2
for some 1 ≤ i ≤ s then the sectional curvature along a plane tangent to Eηi is |ηi|
2 = 0,
in contradiction with the compactness of Mn. Hence Eηi has rank 1 for 1 ≤ i ≤ s. We
conclude that the universal covering of Mn is isometric to Rn, and that f ◦ π splits as
a product of circles by Moore’s Lemma [Mo], where π: Rn → Mn is the covering map.
Assume now that Mn has nonnegative sectional curvatures. It follows from (10)
that 〈ηi, ηj〉 ≥ 0 for 1 ≤ i 6= j ≤ s, whence f has parallel second fundamental form
and all subbundles Eηi , 1 ≤ i ≤ s, are parallel by parts (iv) and (v) of Theorem 16.
We obtain from the de Rham decomposition theorem that the universal covering of Mn
splits isometrically as Mn11 × · · ·×M
s , where each factor M
i is either R if ni = 1 or a
sphere Snii of curvature |ηi|
2 if ni ≥ 2. Moreover, if π: M
1 × · · · ×M
s → M
n denotes
the covering map, then Moore’s Lemma implies that f ◦π splits as f ◦π = f1×· · ·× fs,
where fi(M
i ) is a round sphere or circle for 1 ≤ i ≤ s.
To every compact isoparametric submanifold f : Mn → RN one can associate a finite
group, its Weyl group, as follows. Let η1, . . . , ηg denote the principal normal vector fields
of f . For p ∈ Mn, let Hj(p), 1 ≤ j ≤ g, be the focal hyperplane of T
n given by the
equation 〈ηj(p), 〉 = 1. Then one can show that the reflection on the affine normal space
p+T⊥p M
n with respect to each affine focal hyperplane p+Hi(p) leaves
j=1(p+Hj(p))
invariant, and thus the set of all such reflections generate a finite group, the Weyl group
of f at p. Moreover, the Weyl groups of f at different points are conjugate by the parallel
transport with respect to the normal connection, hence a well-defined Weyl group W
can be associated to f . We refer to [PT2] for details. In the proof of Theorem 3 we will
need the following property of the Weyl group of an isoparametric submanifold.
Proposition 18 Let f : Mn → RN be a compact isoparametric submanifold and let
W (p) be its Weyl group at p ∈ Mn. Assume that W (p) leaves invariant an affine
hyperplane H orthogonal to ξ ∈ T⊥p M . Then f(M
n) is contained in the affine hyperplane
of RN through p orthogonal to ξ.
Proof: It follows from the assumption that H is orthogonal to every focal hyperplane
p + Hj(p), 1 ≤ j ≤ g, of f at p. For q ∈ H, let Q =
g∈W (p)
gq ∈ H. Then Q
is a fixed point of W (p), hence it lies in the intersection
(p + Hj(p)) of all affine
focal hyperplanes of f at p. We obtain that the line through Q orthogonal to H lies in
j=1(p +Hj(p)). Therefore 〈ηj, Q+ λξ〉 = 1 for every λ ∈ R, 1 ≤ j ≤ g, which implies
that 〈ηj , ξ〉 = 0, for every 1 ≤ j ≤ g. Now extend ξ to a parallel vector field along M
with respect to the normal connection. Since the principal normal vector fields η1, . . . , ηg
of f are parallel with respect to the normal connection by Theorem 16-(iii), it follows
that 〈ηj, ξ〉 = 0 everywhere, and hence the shape operator Aξ of f with respect to ξ is
identically zero by (9). Then ξ is constant in RN and the conclusion follows.
A rich source of isoparametric submanifolds is provided by the following result of
Palais and Terng (see [PT1], Theorem 6.5).
Proposition 19 If a closed subgroup G ⊂ SO(N) acts polarly on RN then any of its
principal orbits is an isoparametric submanifold of RN .
To conclude this section, we point out that if G ⊂ SO(N) acts polarly on RN and Σ
is a section, then the the Weyl group W = N(Σ)/Z(Σ) of the G-action coincides with
the Weyl group W (p) just defined of any principal orbit Gp, p ∈ Σ, as an isoparametric
submanifold of RN (cf. [PT2]).
5 Proof of Theorem 3
We first prove the converse. Let G ⊂ SO(n+1) act polarly on Rn+1, let Σ be a section
of the G-action and let Mn ⊂ Rn+1 be a G-invariant immersed hypersurface. It suffices
to prove that Σ is transversal to Mn, for then L = Σ ∩Mn is a compact hypersurface
of Σ that is invariant under the Weyl group W of the action and Mn = G(L).
Assume, on the contrary, that transversality does not hold. Then there exists p ∈
Σ ∩Mn with TpΣ ⊂ TpM
n. Fix v ∈ TpΣ in a principal orbit of the slice representation
at p and let γ: (−ǫ, ǫ) → Mn be a smooth curve with γ(0) = p and γ′(0) = v. Since Mn
is G-invariant, it contains {gγ(t) | g ∈ Gp, t ∈ (−ǫ, ǫ)}. Therefore, TpM
n contains Gpv,
and hence Rv⊕TvGpv. Recall that TpΣ is a section for the slice representation at p (see
[PT1], Theorem 4.6). Therefore TvGpv is a subspace of T
p G(p) orthogonal to TpΣ with
dimTvGpv = dim T
p G(p)− dimΣ. (13)
Moreover, again by the G-invariance of Mn, we have that G(p) ⊂ Mn, and hence
TpG(p) ⊂ TpM
n. Using (13), we conclude that
dimTpM
n ≥ dimG(p) + dimΣ + dimTvGpv
= dimG(p) + dimT⊥p G(p) = n + 1,
a contradiction.
In order to prove the direct statement, it suffices to show that at each point p ∈ L
which is a singular point of the G-action the subset
H = {γ′(0) | γ: (−ǫ, ǫ) → Rn+1 is a curve in Rn+1 with γ(−ǫ, ǫ) ⊂Mn and γ(0) = p}
is an n-dimensional subspace of Rn+1, the tangent space of Mn at p. Clearly, we have
H = TpG(p)
gTpL.
We use again that TpΣ is a section of the slice representation at p and that, in addition,
the Weyl group for the slice representation is W (Σ)p = W ∩Gp ([PT1], Theorem 4.6).
Since L is invariant under W by assumption, it follows that TpL is invariant under
W (Σ)p. Let ξ ∈ Σ be a unit vector normal to L at p. Then, for any principal vector
v ∈ TpL ⊂ TpΣ of the slice representation at p it follows from Proposition 18 that Gpv
lies in the affine hyperplane π of RN through p orthogonal to ξ. Therefore gTpL ⊂ π
for every g ∈ Gp. Since TpG(p) is orthogonal to Σ, we conclude that H ⊂ π, and hence
H = π by dimension reasons.
Remark 20 Theorems 1 and 3 yield as a special case the main theorem in [MPST]:
any compact hypersurface of Rn+1 with cohomogeneity one under the action of a closed
connected subgroup of its isometry group is given as G(γ), where G ⊂ SO(n + 1) acts
on Rn+1 with cohomogeneity two (hence polarly) and γ is a smooth curve in a (two-
dimensional) section Σ which is invariant under the Weyl group W of the G-action. We
take the opportunity to point out that starting with a smooth curve β in a Weyl chamber
σ of Σ (which is identified with the orbit space of the G-action) which is orthogonal to
the boundary ∂σ of σ is not enough to ensure smoothness of γ = W (β), or equivalently,
of G(β), as claimed in [MPST]. One should require, in addition, that after expressing
β locally as a graph z = f(x) over its tangent line at a point of intersection with ∂σ,
the function f be even, that is, all of its derivatives of odd order vanish and not only
the first one.
6 Proofs of Corollaries 4 and 5.
For the proof of Corollary 4 we need another known property of polar representations,
a simple proof of which is included for the sake of completeness
Lemma 21 Let G̃ ⊂ SO(n+ 1) act polarly and have a principal orbit G̃(q) that is not
full, i.e., the linear span V of G̃(q) is a proper subspace of Rn+1. Then G̃ acts trivially
on V ⊥.
Proof: Let v ∈ V ⊥. Then v belongs to the normal spaces of G̃(q) at any point, hence
admits a unique extension to an equivariant normal vector field ξ along G̃(q). Moreover,
the shape operator Aξ of G̃(q) with respect to ξ is identically zero, for Av is clearly zero
and ξ is equivariant. Now, since G̃ acts polarly, the vector field ξ is parallel in the
normal connection. It follows that ξ is a constant vector in Rn+1, which means that G̃
fixes v.
Proof of Corollary 4. We know from Theorem 2 that there exists an orthogonal represen-
tation Ψ: G→ SO(n+1) such that G̃ = Ψ(G) acts polarly on Rn+1 and f ◦g = Ψ(g)◦f
for every g ∈ G. We claim that any of the conditions in the statement implies that f
immerses some G-principal orbit Gp in Rn+1 as a round sphere (circle, if k = n − 1).
Assuming the claim, it follows from Lemma 21 that G̃ fixes the orthogonal complement
V ⊥ of the linear span V of G̃p, hence f is a rotation hypersurface with V ⊥ as axis.
Moreover, if k 6= n − 1 then f |Gp must be an embedding by a standard covering map
argument. From this and f ◦ g = Ψ(g) ◦ f for every g ∈ G it follows that any g in the
kernel of Ψ must fix any point of Gp. Since Gp is a principal orbit, this easily implies
that g = id. Therefore Ψ is an isomorphism of G onto G̃.
We now prove the claim. By Proposition 19 we have that f immerses Gp as an
isoparametric submanifold. If condition (i) holds then the first normal spaces of f |Gp
in Rn+1 are one-dimensional. By Theorem 16-(ii) and a well-known result on reduction
of codimension of isometric immersions (cf. [Da], Proposition 4.1), we obtain that
f(Gp) = G̃(f(p)) is contained as a hypersurface in some affine hyperplane H ⊂ Rn+1,
and hence f(Gp) must be a round hypersphere of H (a circle if k = n − 1). Moreover,
condition (i) is automatic if k = n − 1, for in this case the principal orbit of maximal
length must be a geodesic. Now, if (iii) holds, let Gp be a principal orbit such that the
position vector of f is nowhere tangent to f(Mn) along Gp. Then any normal vector ξ̃
to f |Gp at gp ∈ Gp can be written as ξ̃ = af(gp) + f∗(gp)ξgp, with ξgp normal to Gp in
Mn. Therefore the shape operator A
f |Gp
is a multiple of the identity tensor, hence f |Gp
is umbilical. Since, by (ii), the dimension of Gp can be assumed to be at least two, the
claim is proved also in this case. As for conditions (iv) and (v), the claim follows from
Proposition 17-(i) and the last assertion in Proposition 17-(ii), respectively.
Proof of Corollary 5. By Proposition 19 and part (ii) of Proposition 17 we have that the
orbit G̃(f(p)) = f(Gp) of G̃ is an extrinsic product Sn1−11 ×. . .×S
of round spheres or
circles. In particular, this implies that the orthogonal decomposition Rn+1 =
where Rni is the linear span of S
i for 1 ≤ i ≤ k, is G̃-stable, that is, R
ni is G̃-invariant
for 0 ≤ i ≤ k (cf. [GOT], Lemma 6.2). By [D], Theorem 4 there exist connected Lie
subgroups G1, . . . , Gk of G̃ such that Gi acts on R
ni and the action of Ḡ = G1× . . .×Gk
on Rn+1 given by
(g1 . . . gk)(v0, v1, . . . , vk) = (v0, g1v1, . . . , gkvk)
is orbit equivalent to the action of G̃. Moreover, writing q = f(p) = (q0, . . . , qk) then
G̃(q) = {q0} × G1(q1) × . . . × Gk(qk), and hence Gi(qi) is a hypersphere of R
ni for
1 ≤ i ≤ k. The last assertion is clear.
7 Proof of Theorem 6.
Let Lk×ρN
n−k be a warped product with Nn−k connected and complete and let ψ: Lk×ρ
Nn−k → U be an isometry onto an open dense subset U ⊂Mn. Since Mn is a compact
Riemannian manifold isometrically immersed in Euclidean space as a hypersurface, there
exists an open subset W ⊂Mn with strictly positive sectional curvatures. The subset U
being open and dense inMn, W ∩U is a nonempty open set. Let L1×N1 be a connected
open subset of Lk×Nn−k that is mapped intoW∩U by ψ. Then the sectional curvatures
of Lk ×ρ N
n−k are strictly positive on L1 ×N1.
For a fixed x ∈ L1, choose a unit vector Xx ∈ TxL1. For each y ∈ N
n−k, let X(x,y)
be the unique unit horizontal vector in T(x,y)(L1 × N
n−k) that projects onto Xx by
(π1)∗(x, y). Then the sectional curvature of L
k ×ρ N
n−k along a plane σ spanned by
X(x,y) and any unit vertical vector Z(x,y) ∈ T(x,y)(L1 ×N
n−k) is given by
K(σ) = −Hess ρ(x)(Xx, Xx)/ρ(x).
Observe that K(σ) depends neither on y nor on the vector Z(x,y). Since K(σ) > 0 if
y ∈ N1, the same holds for any y ∈ N
n−k. In particular, L1×ρN
n−k is free of flat points.
If n − k ≥ 2, it follows from [DT], Theorem 16 that f ◦ ψ immerses L1 ×ρ N
either as a rotation hypersurface or as the extrinsic product of an Euclidean factor Rk−1
with a cone over a hypersurface of Sn−k+1. The latter possibility is ruled out by the fact
that the sectional curvatures of Lk ×ρ N
n−k are strictly positive on L1 ×N1. Thus the
first possibility holds, and in particular f ◦ ψ immerses each leaf {x} × Nn−k, x ∈ L1,
isometrically onto a round (n−k)-dimensional sphere. It follows that Nn−k is isometric
to a round sphere.
In any case, Iso0(Nn−k) acts transitively on Nn−k and each g ∈ Iso0(Nn−k) induces
an isometry ḡ of Lk ×ρ N
n−k by defining
ḡ(x, y) = (x, g(y)), for all (x, y) ∈ Lk ×Nn−k.
The map g ∈ Iso0(Nn−k) 7→ ḡ ∈ Iso(Lk ×ρ N
n−k) being clearly continuous, its image
Ḡ is a closed connected subgroup of Iso(Lk ×ρ N
n−k). For each ḡ ∈ Ḡ, the induced
isometry ψ ◦ ḡ ◦ ψ−1 on U extends uniquely to an isometry of Mn. The orbits of the
induced action of Ḡ on U are the images by ψ of the leaves {x} ×Nn−k, x ∈ Lk, hence
are umbilical in Mn. Moreover, the normal spaces to the (principal) orbits of Ḡ on U
are the images by ψ∗ of the horizontal subspaces of L
k ×ρ N
n−k. Therefore, they define
an integrable distribution on U , whence on the whole regular part of Mn. Thus, the
action of Ḡ on Mn is locally polar with umbilical principal orbits. We conclude from
Corollary 4-(iii) that f is a rotation hypersurface.
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Universidade Federal Fluminense Universidade Federal de São Carlos
24020-140 – Niteroi – Brazil 13565-905 – São Carlos – Brazil
E-mail: ion.moutinho@ig.com.br E-mail: tojeiro@dm.ufscar.br
	Introduction
	Proof of Theorem ??
	Proof of Theorem ??
	Isoparametric submanifolds
	Proof of Theorem ??
	Proofs of Corollaries ?? and ??.
	Proof of Theorem ??.
