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Revista de la Unión Matemática Argentina
versión impresa ISSN 0041-6932versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
Riemannian G-manifolds as Euclidean submanifolds
Ruy Tojeiro
Abstract: We survey on some recent developments on the study of Riemannian G-manifolds as Euclidean submanifolds.
2000 Mathematics Subject Classification. 53 A07, 53 C40, 53 C42
Key words and phrases. Riemannian G-manifolds, rotation hypersurfaces, polar actions, rigidity of hypersurfaces, warped products
Let be a complete Riemannian manifold of dimension
acted on by a connected closed subgroup
of its isometry group
. We refer to
, or simply to
for short, as a Riemannian
-manifold. The codimension of a maximal dimensional
-orbit is called the
-cohomogeneity of
. In particular,
is
-homogeneous if it has
-cohomogeneity zero, that is, if
acts transitively on
.
Riemannian -manifolds have been extensively studied from various points of view. In particular, those of low cohomogeneity have been used as a rich source of examples of Riemannian manifolds whose metrics have several important geometric properties, including metrics of positive sectional curvatures, Einstein and Ricci-flat metrics, and metrics with exceptional holonomy groups.
An interesting and natural problem is to study Riemannian -manifolds as submanifolds of Euclidean space
. These include submanifolds that are invariant by the action of a closed subgroup of the isometry group of
. By a result of Moore [Mo], every compact homogeneous Riemannian manifold
can be realized in this way, namely,
admits an isometric embedding into some Euclidean space that is equivariant with respect to its full isometry group. On the other hand, a theorem of Kobayashi [Ko] states that an isometric immersion
,
, of a compact homogeneous Riemannian manifold must be equivariant with respect to the full isometry group of
, thus implying
to be an embedding of
as a round hypersphere of
.
The aim of this paper is to survey on some recent developments on this problem, mostly for the case of Riemannian -manifolds as Euclidean hypersurfaces.
The study of Riemannian -manifolds as Euclidean submanifolds was initiated by the following result of Kobayashi [Ko] for compact homogeneous hypersurfaces.
Theorem 1. Let be an isometric immersion of a compact homogeneous Riemannian manifold. Then
embeds
as a round sphere.
The main step in the proof of Kobayashi's theorem is to show that can be realized as a group of extrinsic isometries of Euclidean space, i.e., that there exists a Lie-group homomorphism
with respect to which
is equivariant, in the sense that
for all
. As in several subsequent results, this is accomplished by establishing the rigidity of
. Recall that an isometric immersion
is rigid if any other isometric immersion
differs from
by a rigid motion of
. Under the assumptions of Kobayashi's theorem, one can start by using the well-known fact that any compact Euclidean hypersurface has a point with strictly positive sectional curvatures (and hence with nonvanishing principal curvatures) and then use the homogeneity of
to conclude that this must hold everywhere on
. For
, the local rigidity theorem of Cartan-Beez-Killing can thus be applied: a Euclidean hypersurface of dimension
is rigid whenever it has at least three nonzero principal curvatures everywhere. Applying this result to the pair of isometric immersions
and
, for each fixed
, one gets a rigid motion
such that
. Once we have this, it is not difficult to prove that the correspondence
defines a Lie-group homomorphism
such that
for all
. Moreover, the subgroup
of
being compact, it has a fixed point by a well-known result of Cartan. This implies that
is contained in (a conjugacy class of) the orthogonal group
. Since the image
lies in a
-orbit, it must be a round hypersphere of
. Finally, that
is an embedding follows by a standard covering map argument.
The extension of Kobayashi's theorem to the noncompact case came soon afterwards with the contribution by Nagano and Takahashi [NT], who proved that if is an isometric immersion of a homogeneous Riemannian manifold then
is isometric to
,
, provided that the rank
of the second fundamental form of
is different from
at some point. Moreover,
splits as
if
at some (and hence at any) point of
, where
is the identity map and
,
, is an umbilical inclusion, and otherwise it is a cylinder over a plane curve. The restriction on
was later removed by Harle [Ha], as a consequence of a more general result on complete Euclidean hypersurfaces with constant scalar curvature. A simpler proof of Harle's theorem making use of the Gauss parameterization of hypersurfaces whose second fundamental forms have constant rank was subsequently given by Dajczer-Gromoll [DG
].
3. Homogeneous submanifolds of codimension two
Non-equivariant isometric immersions ,
, of homogeneous Riemannian manifolds can be easily constructed by taking, for instance, a composition
of an umbilical inclusion
with an isometric immersion
of an open subset
containing
. This indicates that classifying homogeneous Euclidean submanifolds of higher codimension should be much harder than in the hypersurface case. Nevertheless, for compact submanifolds of dimension
and codimension two this was accomplished by Castro-Noronha [CN]:
Theorem 2. Let ,
, be an isometric immersion of a compact homogeneous Riemannian manifold. Then one of the following possibilities holds:
is equivariant with respect to an orthogonal representation
is a homogeneous isoparametric hypersurface of
;
is isometric to
;
is isometrically covered by
.
The authors have actually been able to obtain the same conclusion by assuming, instead of compactness of , that
has at least one point where the index of relative nullity of
, that is, the dimension of the kernel of its second fundamental form, is not greater than
.
The main tool in the proof is a rigidity result for Euclidean submanifolds due to do Carmo-Dajczer [dCD], which implies that a Euclidean -dimensional submanifold of dimension
and codimension two is rigid whenever its index of relative nullity is everywhere less than or equal to
and there are at least three nonzero principal curvatures in every normal direction. Using this result, the authors prove that a compact homogeneous Euclidean submanifold of codimension two is rigid if it has at least one point where the index of relative nullity is not greater than
, unless at every point one can find an orthonormal basis
of the normal space at that point such that the shape operators
and
satisfy
and
for every intrinsic isometry
of the submanifold. Rigidity forces the submanifold to be extrinsically homogeneous, as previously discussed, giving the first possibility in the statement. The proof proceeds by considering separately the cases in which the rank of
is either
or
. The former leads to the two remaining possibilities in the statement, whereas in the most delicate case where
it is shown that the submanifold is an extrinsic product
.
As far as we know, it remains an open problem to determine the cases in which a homogeneous Euclidean submanifold of codimension greater than two can fail to be extrinsically homogeneous. In the next sections we go back to the case of -hypersurfaces, but allow the intrinsic
-action to be no longer transitive.
4. Complete -hypersurfaces of cohomogeneity one
In this section we discuss Riemannian -manifolds of cohomogeneity one isometrically immersed in Euclidean space as hypersurfaces. We start with the simplest examples:
4.1. Hypersurfaces of revolution. Consider a curve that is either contained in the interior of a half-space
or meets its boundary
orthogonally. Now let the subgroup
of
that fixes
act on
. This gives rise to a
-invariant hypersurface
of
, called a hypersurface of revolution with
as profile. The group
acts on
with one-codimensional round spheres as principal orbits. In particular, they are umbilical submanifolds of
, i.e., at any point their principal curvatures coincide. It was shown by Podestá-Spiro [PS] that this property characterizes hypersurfaces of revolution among compact hypersurfaces of cohomogeneity one with dimension
:
Theorem 3. Let ,
, be an isometric immersion of a compact Riemannian manifold. Assume that a closed connected subgroup
of
acts on
with cohomogeneity one. If the
-principal orbits are umbilical in
then
is a hypersurface of revolution.
For any manifold of
-cohomogeneity one, and any open interval
in the interior of the orbit space
(which is diffeomorphic to either
,
,
or
), the inverse image
by the quotient map is
-equivariantly diffeomorphic to
, where
denotes an isotropy subgroup of principal type (see, e.g., [AA]). One of the main ingredients in Podestá-Spiro's proof of Theorem 3 is the observation that umbilicity of the principal orbits implies the metric induced on
by this diffeomorphism to be a warped product metric
.
The assumption of umbilicity of the principal orbits in Podestá-Spiro's theorem has been weakened in subsequent developments. Namely, still in the compact case the same conclusion was obtained by assuming, instead, that and all
-principal orbits have constant sectional curvature [AMN], and that
and all
-principal orbits have positive sectional curvatures [CN].
The noncompact case of Podestá-Spiro's result was studied by Mercuri-Seixas in [MS]. Here, even for cohomogeneity one isometric actions of compact Lie groups it is easy to construct non-rotational examples. For instance, consider the standard action of on
and isometrically immerse
into
as a cylinder over a plane curve. Moreover, asking the hypersurface not to be everywhere flat is not enough, as pointed out in [MS]. For actions of compact groups, the right assumption turns out to be that no connected component of the flat part of the hypersurface be unbounded. The case of complete noncompact cohomogeneity one hypersurfaces under the action of a closed and noncompact Lie group (with umbilical principal orbits) was also considered in [MS].
4.2. Standard examples. More general examples of Euclidean -hypersurfaces of cohomogeneity one, with
compact and connected, are produced as follows. Start with a cohomogeneity two closed connected subgroup
, so that the orbit space
is a two dimensional manifold, possibly with boundary. Now consider the hypersurface
of
given by the inverse image under the canonical projection onto
of a curve that is either contained in the interior of
or meets its boundary orthogonally. We call
a standard example.
The natural question that emerges is whether, under reasonable global assumptions, the standard examples comprise all Euclidean hypersurfaces of cohomogeneity one. The answer is given by the following result of [MPST]:
Theorem 4. Let be an isometric immersion of a complete Riemannian manifold acted on with cohomogeneity one by a compact connected subgroup of
. If either
and
is compact, or if
and the connected components of the flat part of
are bounded, then
is either rigid or a hypersurface of revolution. In particular, it is a standard example.
The proof of Theorem 4 relies on a global rigidity result due to Sacksteder [Sa], which states that a compact Euclidean hypersurface of dimension
is rigid whenever the subset of totally geodesic points of
does not disconnect
. In fact, the arguments in an unpublished proof of Sacksteder's theorem due to Ferus actually show more than the preceding statement. Namely, if
are isometric immersions of a complete Riemannian manifold of dimension
such that there exists no complete leaf of dimension
or
of the relative nullity distribution of
, then it is shown that their second fundamental forms are related by
at every
, where
is a vector bundle isometry between the normal bundles of
and
. This allows one to prove the following general result [MPST], [MT]:
Theorem 5. Let ,
, be a complete hypersurface. Assume that there exists no complete leaf of relative nullity of
of dimension
or
. Then there exists a Lie-group homomorphism
of the identity component of
into
such that
for all
.
Notice that Theorem 5 easily implies the results of Kobayashi and, more generally, of Nagano-Takahashi mentioned in Section . Namely, if
is an isometric immersion of a homogeneous Riemannian manifold such that
at some point, then either
is flat, and hence
is a cylinder over a plane curve by a well-known theorem of Hartman-Nirenberg, or
and Theorem 5 applies: it follows that
is an orbit of a cohomogeneity one subgroup of
, in which case
is an isoparametric hypersurface of
, whence an extrinsic product
,
.
The proof of Theorem 5 is simple enough to be included here in almost full detail: given , let
denote the second fundamental form of
. On one hand, using that
is an isometry we have
![0 n g ∈ Iso (M )](/img/revistas/ruma/v47n1/1a08228x.png)
![n x ∈ M](/img/revistas/ruma/v47n1/1a08229x.png)
![n X, Y ∈ TxM](/img/revistas/ruma/v47n1/1a08230x.png)
![Θx : Iso0(M n) → Sym(TxM n × TxM n → Tx⊥M nf )](/img/revistas/ruma/v47n1/1a08231x.png)
![TxM n × TxM n](/img/revistas/ruma/v47n1/1a08232x.png)
![T ⊥M n x f](/img/revistas/ruma/v47n1/1a08233x.png)
![n X, Y ∈ TxM](/img/revistas/ruma/v47n1/1a08235x.png)
![xf ∘ g(x) = φg(x) ∘ xf(x)](/img/revistas/ruma/v47n1/1a08236x.png)
![xf ∘ g(x) = - φg(x) ∘ xf (x)](/img/revistas/ruma/v47n1/1a08237x.png)
![φg](/img/revistas/ruma/v47n1/1a08238x.png)
![T ⊥M fn](/img/revistas/ruma/v47n1/1a08239x.png)
![T ⊥M fn∘g](/img/revistas/ruma/v47n1/1a08240x.png)
![Θx](/img/revistas/ruma/v47n1/1a08241x.png)
![{xf(x),- xf (x)}](/img/revistas/ruma/v47n1/1a08242x.png)
![Iso0(M n)](/img/revistas/ruma/v47n1/1a08243x.png)
![Θx(id) = xf(x)](/img/revistas/ruma/v47n1/1a08244x.png)
![αf ∘g(x) = φg(x) ∘ αf(x)](/img/revistas/ruma/v47n1/1a08245x.png)
![g ∈ Iso0(M n)](/img/revistas/ruma/v47n1/1a08246x.png)
![x ∈ M n](/img/revistas/ruma/v47n1/1a08247x.png)
![0 n g ∈ Iso (M )](/img/revistas/ruma/v47n1/1a08248x.png)
![n+1 ˜g ∈ Iso(ℝ )](/img/revistas/ruma/v47n1/1a08249x.png)
![f ∘ g = ˜g ∘ f](/img/revistas/ruma/v47n1/1a08250x.png)
![g ↦→ ˜g](/img/revistas/ruma/v47n1/1a08251x.png)
![Φ : Iso0(M n) → Iso(ℝn+1)](/img/revistas/ruma/v47n1/1a08252x.png)
![Iso0(ℝn+1)](/img/revistas/ruma/v47n1/1a08253x.png)
As a consequence of Theorem 5, if denotes the set of totally geodesic points of
as in the statement, then
![]() | (1) |
Before we sketch a proof of this assertion, we first observe that it already implies the conclusion of Theorem 4 in the compact case. Namely, if is a compact connected subgroup of
acting with cohomogeneity one, then (1) and Sacksteder's theorem imply that
has a fixed vector
, hence every orbit of
through a point not in the line spanned by
is a hypersphere of an affine hyperplane orthogonal to
. Thus
is a hypersurface of revolution. If
is not compact, one still has to consider the case in which
carries a complete leaf of relative nullity of
of dimension
. Here a key rôle is played by the results of [CN], applied to the orbits of
as compact homogeneous submanifolds of
with codimension two. In this case
turns out to be an extrinsic product
, hence rigid, with products
as the
-principal orbits.
Now, to prove (1) one first uses the equivariance of in order to show that
is invariant under
. It then follows that the orbit of
through a point
is a connected subset of totally geodesic points, whence must be contained in a hyperplane
that is tangent to
along it by a lemma of [DG
]. Therefore, a unit vector
orthogonal to
spans
for every
. Since
for every
, because
is equivariant with respect to
, the connectedness of
implies that it must fix
.
5. Compact locally polar -hypersurfaces
An isometric action of a compact Lie group on a Riemannian manifold
is said to be locally polar if the distribution of normal spaces to principal orbits on the regular part of
is integrable. Then
is called a locally polar Riemannian
-manifold. For instance, any isometric action of cohomogeneity one is locally polar. It was shown by Heintze-Liu-Olmos [HLO], answering a question posed by Palais-Terng [PT], that for any complete locally polar Riemannian
-manifold there exists a connected complete immersed submanifold
of
that intersects orthogonally all
-orbits. Such a submanifold is called a section, and it is always a totally geodesic submanifold of
. 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. Polar orthogonal representations have been classified by Dadok [D]. As a consequence of his classification, he obtained that every polar representation is orbit equivalent to (i.e., has the same orbits as) the isotropy representation of a semi-simple symmetric space.
It was shown in [BCO] (see Proposition ) that if a closed subgroup of
acts polarly on
and leaves invariant a submanifold
, then its restricted action on
is locally polar. It is not true, however, that every locally polar action on a Euclidean submanifold arises in this way, even assuming it to be compact. For instance, consider a compact submanifold
that is invariant under the action of one of the closed subgroups
that act non-polarly on
with cohomogeneity three. Then the induced action of
on
has cohomogeneity one, whence is locally polar. Nevertheless, for compact hypersurfaces of dimension
it was shown in [MT] that this is indeed the case.
Theorem 6. Let ,
, be an isometric immersion of a compact Riemannian manifold. Assume that a closed connected subgroup
of
acts locally polarly on
with cohomogeneity
. Then there exists an orthogonal representation
such that
acts polarly on
with cohomogeneity
and
for every
.
For the proof of Theorem 6, one starts by recalling that is equivariant with respect to a Lie-group homomorphism
by Theorem 5, which here must actually be an orthogonal representation of
as a closed subgroup of
by the compactness and connectedness of
. Thus, it suffices to prove that
acts polarly on
and set
. For that, the idea is to make use of a result of Palais-Terng [PT], according to which an orthogonal representation in
of a compact Lie group
is polar whenever it has an orbit that is an isoparametric submanifold
and, in addition, either
is full (i.e., not contained in any affine subspace) or, if otherwise,
acts trivially on the orthogonal complement of the linear span of
. Recall that a submanifold
is isoparametric if it has flat normal bundle and the principal curvatures with respect to every parallel normal vector field along any curve in
are constant (with constant multiplicities). The proof then consists of proving that these conditions are satisfied by the action of
, for which the main tool is to use that principal orbits of locally polar isometric actions are characterized by the fact that equivariant normal vector fields are parallel with respect to the normal connection.
The simplest examples of hypersurfaces that are invariant by a polar action on
of a closed subgroup of
are the hypersurfaces of revolution (with possibly higher dimensional profiles), which are produced by the action on a hypersurface
(possibly with boundary) of a half-space of
, disjoint to the boundary
or orthogonal to it, of a closed subgroup
of
that fixes
and acts transitively on the hyperspheres of the orthogonal complement
of
. The following result of [MT] gives several sufficient conditions for a compact Euclidean hypersurface as in Theorem 6 to be a hypersurface of revolution.
Corollary 7. Let ,
, be an isometric immersion of a compact Riemannian manifold. Assume that a closed connected subgroup
of
acts locally polarly on
with cohomogeneity
. Then any of the following additional conditions implies that
is a hypersurface of revolution:
- there exists a totally geodesic (in
)
-principal orbit;
;
- the
-principal orbits are umbilical in
;
- there exists a
-principal orbit with nonzero constant sectional curvatures;
- there exists a
-principal orbit with positive sectional curvatures.
Moreover, in this case is isomorphic to one of the closed subgroups of
that act transitively on
.
For the case of hypersurfaces of cohomogeneity , the same conclusion was also derived in the noncompact case in [MPST] under the assumptions of Theorem 4. Another sufficient condition obtained in [MPST] for a hypersurface of
-cohomogeneity one and dimension
as in Theorem 4 to be a hypersurface of revolution is that some principal
-orbit be homeomorphic to a sphere. This was accomplished by combining Theorem 4 and the fact that an isoparametric hypersurface of the sphere of dimension
can not have a sphere as its universal covering, unless it is a round sphere. The proof of the latter uses several of the known restrictions on the number
of principal curvatures of isoparametric hypersurfaces and on their multiplicities in order to show that the only possibility allowed for an isoparametric hypersurface of dimension
covered by a sphere is that
.
For the proof of Corollary 7, since one already knows from Theorem 6 that there exists an orthogonal representation such that
acts polarly on
and
for every
, it suffices to show that any of the conditions in the statement implies that
![]() | (2) |
Indeed, since acts polarly on
it fixes the subspace orthogonal to the linear span of
, and hence
is a hypersurface of revolution. Moreover, a standard covering map argument shows that
must be an embedding if
. Thus, if
and
is the identity for some
then it follows from
and the injectivity of
that
must fix any point of
. Since
is a principal orbit, this easily implies that
, and hence that
is an isomorphism of
onto
.
Now, to prove (2) one uses the fact that immerses
as an isoparametric submanifold, since
is equivariant with respect to
and principal orbits of polar representations are isoparametric submanifolds [PT]. If some principal
-orbit is totally geodesic, then the first normal spaces of
in
, that is, the subspaces of the normal spaces spanned by the image of its second fundamental form, are one-dimensional. Since for isoparametric submanifolds the first normal spaces always form a parallel subbundle of the normal bundle with respect to the normal connection, a standard reduction of codimension result implies that
is contained as a hypersurface in some affine hyperplane
, and thus is a round hypersphere of
(a circle if
). Conditions
an
both imply
. In fact, if
then the principal orbit of maximal length must be a geodesic. Similarly, if all principal orbits are umbilical in
then the one of maximal volume is both minimal and umbilical, whence totally geodesic. Finally, one can show that an isoparametric submanifold that has either constant and nonzero or positive sectional curvatures must be a round sphere, which yields the statement under the remaining conditions
and
, respectively.
Corollary 7 and the results of [MPST] for cohomogeneity one hypersurfaces mentioned after it generalize and give simpler proofs of Podestá-Spiro's theorem and some of its extensions described in Section . Further extensions have been recently obtained in [Mou], where the case of complete Euclidean hypersurfaces of dimension
acted on locally polarly with umbilical principal orbits of codimension
by a closed connected subgroup
of its isometry group was considered. In case
is compact and
the hypersurface was shown to be of revolution if the connected components of the flat part of the hypersurface are bounded. If
is not compact, the hypersurface is not everywhere flat and, for a given section
, the connected components of the subset of
formed by flat points of the hypersurface are assumed to be bounded, it was shown that the hypersurface is either an extrinsic product
, the principal orbits being the fibers
,
, or an extrinsic product
,
, in which case the principal orbits are the fibers
,
. For
these results reduce to the ones in [MS] referred to at the end of Section
.
In part of Corollary 7, it was shown in [MT] that weakening the assumption to non-negativity of the sectional curvatures of some principal
-orbit implies
to be a multi-rotational hypersurface in the sense of [DN]:
Corollary 8. Under the assumptions of Theorem 6, suppose further that there exists a -principal orbit
with nonnegative sectional curvatures. Then there exist an orthogonal decomposition
into
-invariant subspaces, where
, and connected Lie subgroups
of
such that
acts on
, the action being transitive on
, and the action of
on
given by
![G˜](/img/revistas/ruma/v47n1/1a08461x.png)
![Gp](/img/revistas/ruma/v47n1/1a08462x.png)
![n = 2 i](/img/revistas/ruma/v47n1/1a08463x.png)
![Gi](/img/revistas/ruma/v47n1/1a08464x.png)
![SO(2)](/img/revistas/ruma/v47n1/1a08465x.png)
![i = 1, ...,k](/img/revistas/ruma/v47n1/1a08466x.png)
Corollary 7 was used in [MT] to study a problem that is seemingly unrelated to isometric actions. Let be a hypersurface of revolution as described in the paragraph preceding Corollary 7. Then the open and dense subset of
that is mapped by
onto the complement of the axis
is isometric to the warped product
, where
is the orbit of some fixed point
under the action of
, and the warping function
is a constant multiple of the distance to
. Recall that a warped product
of Riemannian manifolds
and
with warping function
is the product manifold
endowed with the metric
where
,
, denote the canonical projections. The problem is then to determine whether hypersurfaces of revolution are characterized by their intrinsic warped product structure. For compact Euclidean hypersurfaces of dimension
this was answered affirmatively in [MT]:
Theorem 9. Let ,
, be a compact hypersurface. If there exists an isometry onto an open and dense subset
of a warped product
with
connected and complete (in particular if
is isometric to a warped product
with
connected) then
is a hypersurface of revolution.
Theorem 9 can be seen as a global version in the hypersurface case of the local classification in [DT] of isometric immersions in codimension of warped products
,
, into Euclidean space. It follows from the results of [DT] that an isometric immersion
,
, of a warped product with no flat points is either a hypersurface of revolution or an extrinsic product
, where
denotes the cone over a hypersurface
of
.
Acknowledgment. This survey is a slightly extended version of a talk given at the "II Encuentro de Geometria Diferencial" held in La Falda (Argentina) on June 2005. The author would like to take the opportunity to thank Carlos Olmos and the other colleagues from the Mathematics Department of the Universidad Nacional de Cordoba for the hospitality during his stay in La Falda and for the invitation to contribute with this paper.
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Ruy Tojeiro
Universidade Federal de São Carlos
13565-905-São Carlos, Brazil
tojeiro@dm.ufscar.br
Recibido: 7 de setiembre de 2005
Aceptado: 29 de agosto de 2006