versión On-line ISSN 1669-9637
Rev. Unión Mat. Argent. v.47 n.1 Bahía Blanca ene./jun. 2006
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.
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].
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]:
- is equivariant with respect to an orthogonal representation
- 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.
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
As a consequence of Theorem 5, if denotes the set of totally geodesic points of as in the statement, then
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 .
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
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
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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Universidade Federal de São Carlos
13565-905-São Carlos, Brazil
Recibido: 7 de setiembre de 2005
Aceptado: 29 de agosto de 2006