Normal holonomy of orbits and Veronese submanifolds
NNORMAL HOLONOMY OF ORBITS AND VERONESESUBMANIFOLDS
CARLOS OLMOS AND RICHAR FERNANDO RIA ˜NO-RIA ˜NO
Abstract.
It was conjectured, twenty years ago, the following result thatwould generalize the so-called rank rigidity theorem for homogeneous Eu-clidean submanifolds: let M n , n ≥
2, be a full and irreducible homogeneoussubmanifold of the sphere S N − ⊂ R N and such that the normal holonomygroup 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 isotropyrepresentation of a semisimple Riemannian symmetric space).If n = 2, then the normal holonomy is always transitive, unless M is ahomogeneous isoparametric hypersurface of the sphere (and so the conjectureis true in this case). We prove the conjecture when n = 3. In this case M must be either isoparametric or a Veronese submanifold. The proof combinesgeometric arguments with (delicate) topological arguments that uses informa-tion from two different fibrations with the same total space (the holonomytube and the caustic fibrations).We also prove the conjecture for n ≥ n ( n + 1). This givesa characterization of Veronese submanifolds in terms of normal holonomy. Wealso extend this last result by replacing the homogeneity assumption by theassumption of minimality (in the sphere).Another result of the paper, used for the case n = 3, is that the numberof irreducible factors of the local normal holonomy group, for any Euclideansubmanifold M n , is less or equal than [ n ] (which is the rank of the orthogonalgroup SO( n )). This bound is sharp and improves the known bound n ( n − Introduction
The holonomy of the normal connection turns out to be a useful tool in Euclideansubmanifold geometry [BCO]. The most important applications of this tool werethe alternative proof of Thorbergsson theorem [Th], given in [O2], and the rankrigidity theorems for submanifolds [O3, CO, DO] (see Section 2.1). Moreover, theextension 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 normalholonomy 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 : September 11, 2018.
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 . a r X i v : . [ m a t h . DG ] J un CARLOS OLMOS AND RICHAR FERNANDO RIA˜NO-RIA˜NO
The starting point for this theory was the normal holonomy theorem [O1] whichasserts 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, thenormal holonomy is a Riemannian non-exceptional holonomy). This implies thatthe so-called principal holonomy tubes have flat normal bundle (holonomy tubesare the image, under the normal exponential map, of the holonomy subbundles ofthe normal bundle). Such tubes, despite to the classical spherical tubes, behavesnicely with respect to products of submanifolds.But the normal holonomy, which is invariant under conformal transformations ofthe ambient space, gives much weaker information in submanifold geometry thanthe Riemannian holonomy in Riemannian geometry. For instance, the reducibilityof the normal holonomy representation does not imply that the manifold splits.So, interesting applications of the normal holonomy can be expected only within arestrictive class of submanifolds. For instance:( ) submanifolds with constant principal curvatures,( ) complex submanifolds of the complex projective space( ) 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 thenormal holonomy, as a submanifold of the sphere, is non-transitive must be eithera 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 M of the complex projective space with non-transitive normal ho-lonomy is the complex orbit (in the projectivized tangent space) of the isotropyrepresentation of a Hermitian symmetric space or, equivalently, M is extrinsicallysymmetric . This result is not true without the completeness assumption.For the class (3) we have the rank rigidity theorem for submanifolds [O3, DO]: ifthe normal holonomy of a full and irreducible Euclidean homogeneous submanifold M n = K.v , n ≥ has a fixed non-null vector, then M is contained in a sphere.If the dimension of the fixed set of the normal holonomy has dimension at least ,then M is 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 thatit was conjectured twenty years ago in [O3]: if the normal holonomy of a full andirreducible homogeneous submanifold M n of the sphere, n ≥ , is non-transitivethen M is 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 thisconjecture. On the other hand, to characterize the classical (Riemannian) Veronesesubmanifolds in terms of normal holonomy.If a submanifold M n of the sphere has irreducible and non-transitive normalholonomy, then the first normal space, as a Euclidean submanifold, coincides withthe normal space (see Remark 2.11). This imposes the restriction that the codi-mension is at most n ( n + 1). We will prove the above mentioned conjecture in thecase that the normal holonomy acts irreducibly and the (Euclidean) codimensionis the maximal one n ( 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 ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 3 also delicate topological arguments involving two different fibrations on a partialholonomy tube: the holonomy tube fibration and the caustic fibration .We extend these resuls by replacing the homogeneity by the property that thesubmanifold is minimal in a sphere. But the proof of this result is simpler than thehomogeneous case and a general proof works also for n = 3.We also prove the sharp bound n on the number of irreducible factors of thenormal holonomy, which implies, from the above mentioned result, the conjecturefor 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 tracelesssymmetric matrices. A Veronese (Riemannian) submanifold M n , 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 M a submanifold with constantprincipal curvatures, the first normal space ν ( M ) coincides with the normal space ν ( M ). Moreover, ν ( M ) has maximal dimension. Namely, the codimension of M is n ( n + 1).The restricted normal holonomy of M , as a submanifold of the sphere, is theimage, under the slice representation, of the (connected) isotropy. Then the nor-mal holonomy representation of M is irreducible and it is equivalent to the isotropyrepresentation of Sl( n ) / SO( n ). So, the normal holonomy of M is non-transitive ifand only if n ≥
3. We have the following geometric characterization of Veronesesubmanifolds 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 submanifoldof the sphere, acts irreducibly, not transitively and the codimension is maximal.
Theorem A.
Let M n ⊂ S n − n ( n +1) , n ≥ , be a homogeneous submanifoldof the sphere. Then M is a (full) Veronese submanifold if and only if the restrictednormal holonomy group of M acts irreducibly and not transitively. For dimension 3 the conjecture on normal holonomy is true. Namely,
Theorem B.
Let M ⊂ S N − be a full irreducible homogeneous -dimensionalsubmanifold of the sphere. Assume that the restricted normal holonomy group of M is non-transitive. Then M is an orbit of an s -representation. Moreover, M iseither a principal orbit of the isotropy representation of Sl (3) / SO (3) or a Veronesesubmanifold. The irreducibility and fullness condition on M is always with respect to theEuclidean ambient space.We can replace, in Theorem A, the homogeneity condition by the assumption ofminimality in the sphere. Theorem C.
Let M n , n ≥ , be a complete (immersed) submanifold of thesphere S n − n ( n +1) . Then M n is, up to a cover, a (full) Veronese submanifold ifand only if M is a minimal submanifold and the restricted normal holonomy groupacts irreducibly and not transitively. The assumptions of homogeneity or minimality, in our main results, cannot bedropped, since a conformal (arbitrary) diffeomorphism of the sphere transforms M CARLOS OLMOS AND RICHAR FERNANDO RIA˜NO-RIA˜NO into a submanifold with the same normal holonomy but in general not any moreminimal. Last theorem admits a local version.We will explain the main ideas in the proof of Theorem A, when n ≥ A be the traceless shape operator of M = H.v , i.e. ˜ A ξ = A ξ − n (cid:104) H, ξ (cid:105) Id ,where H is the mean curvature vector. Let us consider the map ˜ A , from the normalspace ¯ ν q ( M ) to sphere into the traceless symmetric endomorphisms Sim ( T q M ).Then ˜ A maps normal spaces to the Φ( q )-orbits into normal spaces to the SO( n )-orbits, by conjugation, in Sim ( T q M ). By using the results in Section 2, whichare related to Simons theorem, we obtain that ˜ A is a homothecy which maps thenormal holonomy group Φ( q ) into SO( n ). This implies that the eigenvalues of ˜ A ξ do not change if ξ is parallel transported along a loop. From the homogeneity, sincethe group H is always inside the ∇ ⊥ -transvections, we obtain that the eigenvaluesof ˜ A ξ ( t ) are constant, if ξ ( t ) is a parallel normal field along a curve. Now wepass to an appropriate, singular, holonomy tube, M ξ , where A ξ has exactly twoeigenvalues one of them of multiplicity 2. Let ˆ ξ be the parallel normal field of M ξ such that M coincides with the parallel focal manifold ( M ξ ) − ˆ ξ to M ξ . One obtainsthat the three eigenvalue functions, ˆ λ , ˆ λ and ˆ λ = −
1, of the shape operator ˆ A ˆ ξ of M ξ have constant multiplicities. The two horizontal eigendistributions of ˆ A ˆ ξ ,let us say E and E , have multiplicities 2 and ( n −
2) respectively. The verticaldistribution is the eigendistribution associates to the constant eigenvalue −
1. Fromthe above mentioned properties of ˜ A and the tube formulas one obtains that ˆ λ and ˆ λ are functionally related (so if one eigenvalue is constant along a curve theother is also constant). From the Dupin condition, since dim( E ) ≥
2, ˆ λ , andso ˆ λ , as previously remarked, are constant along the integral manifolds of E . If n ≥
4, the the same is true for the distribution E . So, the eigenvalues of ˆ A ˆ ξ are constant along horizontal curves. But any two points in a holonomy tube canbe joined by a horizontal curve. Then ˆ A ˆ ξ has constant eigenvalues and so ˆ ξ isan isoparametric non-umbilical parallel normal field. Then, by the isoparametricrank rigidity theorem, the holonomy tube M ξ , 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 E , 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 recallthe basic notions and results that are needed in this article. We also include in thispart some new results that are auxiliary for our purposes. Some of them have asmall 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 ⊂ R N has constant principal curvatures if the shape operator A ξ ( t ) has constant eigenvalues, for any ∇ ⊥ -parallel normal vector field ξ ( t ) alongany arbitrary (piece-wise differentiable) curve c ( t ) in M . If, in addition, the normalbundle of ν ( M ) is flat, then M is called isoparametric. ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 5
A submanifold M with constant principal curvatures (extrinsically) splits as M = R k × M (cid:48) , where M (cid:48) is 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 principalcurvatures (and, in particular, this family of orbits contains the submanifolds withparallel second fundamental form). But it is not true that all homogeneous subman-ifolds with constant principal curvatures are orbit of polar representations (thereexists a homogeneous focal parallel manifold to an inhomogeneous isoparametric hy-persurface of the sphere [FKM]). It turns out, from Dadok’s classification [Da], thatpolar 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 M is 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 representationis orbit-like to an s -representation.One has the following remarkable result Theorem 2.1. (Thorbergsson, [Th, O3]).
A compact full irreducible isoparametricEuclidean submanifold of codimension at least is homogeneous (and so the orbitof an irreducible s -representation) . The rank at p , of a Euclidean submanifold M , rank p ( M ), is the maximal numberof linearly independent parallel normal fields, locally defined around p . The rankof M , rank( M ), is the minimum, over p ∈ M , of rank p ( M ). If M is homogeneousthen rank p ( M ) = rank( M ), independent of p ∈ M . The submanifold M is said tobe 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 M n , n ≥ , be a Euclidean homogeneous submanifold which is full and irreducible. Then, (a) If rank ( M ) ≥ , if and only if M is contained in a sphere. (b) If rank ( M ) ≥ , then M is an orbit of an s -representation. A parallel normal field ξ of M is called isoparametric if the shape operator A ξ hasconstant eigenvalues. If the shape operator A ξ , of a parallel isoparametric normalfield, is umbilical, i.e. a multiple λ of the identity, then M is contained in a sphere,if λ (cid:54) = 0, or M is 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 M n be a full (local)and locally irreducible submanifold of S N − ⊂ R N which admits a non-umbilicalparallel isoparametric normal field. Then M is an inhomogeneous isoparametrichypersurface or M is (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 M n be a connected, simplyconnected and complete Riemannian manifold and let f : M → R N be an irreducibleisometric immersion. If there exists a non-umbilical isoparametric parallel normalsection then f : M → R N has constant principal curvatures (an so, if f ( M ) is not anisoparametric hypersurface of a sphere, then it is an orbit of an s -representation) . CARLOS OLMOS AND RICHAR FERNANDO RIA˜NO-RIA˜NO
Let K acts (by linear isometries) on R N as an s -representation. Let ( G, K ) be theassociated simple (simply connected) symmetric pair with Cartan decomposition g = k ⊕ p , where p (cid:39) R N . Let M = K.v be an orbit.One has that the normal space to M at v is given by [BCO] ν v ( 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 R N , are not zero.Let M be a full orbit of an s -representation and let p ∈ M . Then the map ξ (cid:55)→ A ξ , from ν p ( M ) into the symmetric endomorphisms of T p M , is injective. Inother words, the first normal space of M at p coincides 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 K acts on R N as an s -representation and let M = K.v be a full orbit. Then the normal holonomy group Φ( v ) of M at v coincides withthe image of the representation of the isotropy K v on ν v ( M ) (the so-called slicerepresentation ) . For a Euclidean vector space ( V , (cid:104) , (cid:105) ), let Sim ( V ) denote the vector space of(real) symmetric endomorphisms of V . The inner product on Sim ( V ) is the usualone, (cid:104) A, B (cid:105) = trace(
A.B ).We denote by
Sim ( V ) the vector space of traceless symmetric endomorphisms. Corollary 2.6.
Let K acts (by linear isometries) on R N as an s -representationand let M = K.v , where | v | = 1 . Assume that the normal holonomy group Φ( v ) acts irreducibly on ¯ ν v ( M ) := { v } ⊥ ∩ ν v ( M ) . Then M is a minimal submanifold ofthe sphere S N − ⊂ R N . Moreover, the map ξ (cid:55)→ A ξ is a homothecy, from ¯ ν v ( M ) onto its image in Sim ( T v M ) .Proof. The mean curvature vector H ( v ) must be fixed by the isotropy, representedon 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 normalholonomy group). Then M is a minimal submanifold of the sphere.Let us consider the following inner product ( , ) of ¯ ν v ( M ): ( ξ, η ) = (cid:104) A ξ , A η (cid:105) Then, ( , ) is Φ( v )-invariant. In fact, if φ ∈ Φ( v ), there exists, from Theorem 2.5, g ∈ K v such that g | ¯ ν v ( M ) = φ . Then( φ ( ξ ) , φ ( η )) = ( g.ξ, g.η ) = (cid:104) A g.ξ , A g.η (cid:105) = (cid:104) gA ξ g − , gA η g − (cid:105) = (cid:104) A ξ , A η (cid:105) = ( ξ, η )Since Φ( p ) acts irreducibly, then ( , ) is proportional to (cid:104) , (cid:105) . Then ξ (cid:55)→ A ξ is ahomothecy. (cid:3) Recall that the normal holonomy (group) representation, of a submanifold ofa 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 theso-called adapted normal curvature tensor R ⊥ (see [O1] and [BCO], Section 4.3 ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 7 c). In fact, if M is an arbitrary submanifold of a space of constant curvaturethen R ⊥ is an algebraic curvature tensor on the normal space ν ( M ). Namely, if p ∈ M and R ⊥ is the normal curvature tensor at p , regarded as a linear map forΛ ( T p M ) → Λ ( ν p ( M )), the adapted normal curvature tensor is defined by R ⊥ = R ⊥ ◦ ( R ⊥ ) t where ( ) t is the transpose endomorphism. This implies that R ⊥ has the sameimage as R ⊥ .From the Ricci identity one has the nice formula, if ξ , ξ , ξ , ξ ∈ ν p ( M ), (cid:104)R ⊥ ξ ,ξ ξ , ξ (cid:105) = − trace([ A ξ , A ξ ] ◦ [ A ξ , A ξ ])= (cid:104) [ A ξ , A ξ ] , [ A ξ , A ξ ] (cid:105) = −(cid:104) [[ A ξ , A ξ ] , A ξ ] , A ξ (cid:105) (**)where A is the shape operator of M .Since R ⊥ (Λ ( ν p ( M ))) = R ⊥ (Λ ( T p M )), one has that R ⊥ ξ ,ξ belongs to thenormal holonomy algebra at p (since curvature tensors, take values in the holonomyalgebra).Since the isotropy representation of a semisimple symmetric space coincides withthat of the dual symmetric space, we may always assume that the symmetric spaceis compact. Let then ( G, K ) be a compact simply connected symmetric pair andlet g = k ⊕ p be the Cartan decomposition associated to such a pair. The isotropyrepresentation of K is naturally identified with the Ad-representation of K on p .The Euclidean metric on p is − B , where B is the Killing form of g . We denote bya dot the Ad-action of K on p . Let 0 (cid:54) = v ∈ p and let us consider the orbit M = K.v (cid:39)
K/K v which is a Euclidean submanifold with constant principalcurvatures (and rank at least 2 if and only if it is not most singular).Let us consider the restriction (cid:104) , (cid:105) of − B to k . This is an Ad-K invariant positivedefinite inner product on k . Let us consider the (normally) reductive decomposition k = k v ⊕ m where k v is the Lie algebra of the isotropy group K v and m is the orthogonalcomplement, with respect to (cid:104) , (cid:105) , of k . The restriction of (cid:104) , (cid:105) to m (cid:39) T [ e ] K/K v (cid:39) T v M induced a so-called normal homogeneous metric on M , which is in particularnaturally reductive, that we also denote by (cid:104) , (cid:105) . Such a Riemannian metric on M will be called the canonical normal homogeneous metric. In general this metric isdifferent from the induced metric as a Euclidean submanifold. Namely, Proposition 2.7.
Let K acts on R N as an irreducible s -representation and let M = K.v , v (cid:54) = 0 . If the (canonical) normal homogeneous metric on M coincides withthe induced metric, then M has parallel second fundamental form (or equivalently, M is extrinsically symmetric [Fe]) .Proof. We keep the notation previous to this proposition. Let ∇ c be the canonicalconnection on M associated to the reductive decomposition k = k v ⊕ m . Thenthe second fundamental form α of M is parallel with respect to the connection¯ ∇ c = ∇ c ⊕ ∇ ⊥ , i.e. ¯ ∇ c α = 0 [OSa, BCO]. Let ¯ ∇ = ∇ ⊕ ∇ ⊥ , where ∇ is theLevi-Civita connection on M associated to the induced metric which coincides, byassumption, with the normal homogeneous metric. Then( ¯ ∇ x α )( y, z ) = α ( D x y, z ) + α ( y, D x z ) CARLOS OLMOS AND RICHAR FERNANDO RIA˜NO-RIA˜NO where D = ∇ − ∇ c We have that D x y = − D y x . This is a general fact, for natu-rally reductive spaces, since the canonical geodesics coincide with the Riemanniangeodesics (see, for instance, [OR]).Then ( ¯ ∇ x α )( x, x ) = 2 α ( D x x, x ) = 0But, from the Codazzi identity, ( ¯ ∇ x α )( y, z ) is symmetric in all of its three vari-ables. Then ¯ ∇ α = 0 and so M has parallel second fundamental form. (cid:3) Corollary 2.8.
Let K acts on R N as an s -representation and let M = K.v , v (cid:54) = 0 .Assume that K v acts irreducibly on T v M . Then M has parallel second fundamentalform (or, equivalently, M is extrinsically symmetric [Fe]) .Remark . 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 M n , ¯ M n ⊂ S N − be submanifolds of the sphere with parallelsecond fundamental forms (or, equivalently, extrinsically symmetric spaces). As-sume also that M is a full submanifold of the Euclidean space R N and that thereexists p ∈ M ∩ ¯ M with T p M = T p ¯ M . Assume, furthermore, that the associatedfundamental forms at p , α, ¯ α of M and ¯ M , respectively, as submanifolds of thesphere, are proportional (i.e. ¯ α = λα , λ (cid:54) = 0 ). Then M = ¯ M (and so λ = 1 ) or M = σ ( ¯ M ) , where σ is the orthogonal transformation of R N which is the identityon R p ⊕ T p M and minus the identity on ¯ ν p ( ¯ M ) = ( R p ⊕ T p M ) ⊥ (and so λ = − ).Proof. Observe, in our assumptions, that the second fundamenal forms of M and¯ M , as Euclidean submaniofolds, are not proportional, unless they coincide (sincethe shapes operators of M and ¯ M , coincides in the direction of the position vector p ).Let us write M = K.p where K acts as an irreducible s -representation. One hasthat the restricted holonomy at p , of the bundle T M ⊕ ¯ ν ( M ), is the representation,of the connected isotropy ( K p ) , on T p M ⊕ ¯ ν p ( M ). This is a well-known fact thatfollows form the following property: if X belongs to the Cartan subalgebra asso-ciated to the symmetric pair ( K, K p ), then d l Exp( tX ) gives the Levi-Civita paralleltransport, when restricted to T p M , along the geodesic γ ( t ) = Exp( tX ) .p , and atthe same time, when restricted to ¯ ν p ( M ), the normal parallel transport along γ ( t ).Since curvature endomorphisms take values in the holonomy algebra, one hasthat ( R x,y , R ⊥ x,y ) ∈ t p , where t p = Lie( K p ) = Lie(( K p ) ) ⊂ so ( T p M ) ⊕ so (¯ ν p ( M ))and R , R ⊥ are the tangent and normal curvature tensors of M at p , respectively.Let R S be the curvature tensor of the sphere S N − at p , restricted to T p M .Then, from the Gauss equation, R x,y = T x,y + R Sx,y where (cid:104) T x,y z, w (cid:105) = (cid:104) α ( x, w ) , α ( y, z ) (cid:105) − (cid:104) α ( x, z ) , α ( y, w ) (cid:105) For ¯ M = ¯ K.p we have similar objects ¯ R, ¯ R ⊥ , ¯ t p , ¯ T . From the assumptions onehas that ¯ T = λ T . So, ¯ R x,y = λ T x,y + R Sx,y (a)From the assumptions, and Ricci equation, one has that¯ R ⊥ x,y = λ R ⊥ x,y (b) ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 9
Now observe that, for any X ∈ t p ⊂ so ( T p M ) ⊕ so (¯ ν p ( M )), X.α = 0 = X. ( λ.α ) = X. ¯ α (c)and the same is true for any ¯ X ∈ ¯ t p (the actions of X and ¯ X are derivations).As we observed, ( R x,y , R ⊥ x,y ) ∈ t p , and ( ¯ R x,y , ¯ R ⊥ x,y ) ∈ ¯ t p . Then, form (a), (b) and(c) one obtains, if λ (cid:54) = ± R Sx,y , .α = 0 = ( R Sx,y , . ¯ α . Since the linear span of { R Sx,y : x, y ∈ T p M } is so ( T p M ), one has that α ( g.x, g.y ) = α ( x, y )for all g ∈ SO( T p M ). Then, from the Gauss equation (cid:104) A ξ x, y (cid:105) = (cid:104) α ( x, y ) , ξ (cid:105) ,one obtains that all the shape operators of M at p commute with any element ofSO( T p M ). Then M is umbilical at p and hence, since it is homogeneous, at anypoint. Then M is an extrinsic sphere. Since M is full we conclude that M = S N − .Then, since n = N − M = ¯ M .Observe that the fullness condition is essential. In fact, if M and ¯ M are umbilicalsubmanifolds of the sphere of different radios, the second fundamental forms at p are proportional.If λ = 1, then M and ¯ M have both the same second fundamental form at p .Since both submanifolds have parallel second fundamental forms, it is well-knownand standard to prove that M = ¯ M .If λ = −
1, then we replace ¯ M by σ ( ¯ M ) and the second fundamental forms of M and ¯ M must coincide. Therefore, M = σ ( ¯ M ). (cid:3) Remark . Let us enounce Theorem 4.1 in [O6]: let M n be a locally full sub-manifold either of the Euclidean space or the sphere, such that the local normalholonomy group at p acts without fixed non- zero vectors. Assume, furthermore,that no factor of the normal holonomy is transitive on the sphere. Then there arepoints in M , arbitrary close to p , where the first normal space coincides with thenormal space. In particular, codim ( M ) ≤ n ( n + 1) . This bound on the codimension is correct. But the better and sharp estimateis codim( M ) ≤ n ( n + 1) −
1. In fact, from the proof one has that if the shapeoperator, at a generic q ∈ M , A ξ is a multiple of the identity (it needs not to bezero, as in that proof), then ξ is in the nullity of the adapted normal curvaturetensor R ⊥ . But this last tensor is not degenerate. This implies that the injectivemap A : ν q ( M ) → Sim ( T q M ) cannot be onto. Then dim( ν q ( M )) = codim( M ) ≤ dim( Sim ( T q M )) − n ( n + 1) − M , in the above assumptions, is is a submanifold of the sphere, then thecodimension of M , as a Euclidean submanifold, is bounded by n ( n + 1).2.2. Holonomy systems.
We recall here some facts about holonomy systems that are useful in submanifoldgeometry.A holonomy system is a triple [ V , R, H ], where V is a Euclidean vector space, H isa connected compact Lie subgroup of SO( V ) and R (cid:54) = 0 is an algebraic Riemanniancurvature tensor on V that takes values R x,y ∈ h = Lie( H ). The holonomy systemis called: - irreducible , if H acts irreducible on V .- transitive , if H acts 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 V must becompact, as it is well-known (since the center of H must be one-dimensional).A holonomy system [ V , R, H ] is the product (eventually, after enlarging H ) ofirreducible 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, R is, up to a scalar multiple, unique.Remark . If [ V , R, H ] is an irreducible symmetric holonomy system, then h coincides with the linear span of R x,y , x, y ∈ V . In this case, since (cid:104) R x,y v, ξ (cid:105) = (cid:104) R v,ξ x, y (cid:105) , one has that the normal space at v to the orbit H.v is given by ν v ( H.v ) = { ξ ∈ V : R v,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 × h coincides with the bracket of h . b) [ X, v ] = − [ v, X ] = X.v , if X ∈ h ⊂ so ( V ) and v ∈ V . c) [ v, w ] = R v,w , if v, w ∈ V .This implies the following: if [ V , R, H ] is an irreducible and symmetric holonomysystem then H acts on V as an irreducible s -representation. Observe that, in this case, the scalar curvature sc ( R ) of R is 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 R x,y = 0 if and only if R T ( x ) ,T ( y ) = 0 . Then T ( R ) = R .Proof. Let R (cid:48) = T ( R ). If ξ ∈ ν v ( K.v ) = { ξ ∈ V : R v,ξ = 0 } , then, from the assump-tions, R (cid:48) v,ξ = T.R T ( v ) ,T ( ξ ) .T − = 0. So, 0 = (cid:104) R (cid:48) v,ξ x, y (cid:105) = (cid:104) R (cid:48) x,y v, ξ (cid:105) , for all x, y ∈ V .Then the Killing field R (cid:48) x,y ∈ so ( V ) of V is tangent to any orbit K.v . This impliesthat R (cid:48) x,y ∈ ˜ h = Lie( ˜ K ), where ˜ K = { g ∈ SO( V ) : g preserves any K -orbit } . Ob-serve that ˜ K is a (compact) Lie subgroup of SO( V ) which is non-transitive (on theunit sphere of V ). Since H ⊂ ˜ K we have that [ V , R, ˜ K ] is also an irreducible andnon-transitive holonomy system. From the Simons holonomy theorem we have that[ V , R, K ] and [ V , R, ˜ K ] are both symmetric. Then h and ˜ h are (linearly) spannedby R x,y , x, y ∈ V . Then h = ˜ h and therefore, K = ˜ K .Since R (cid:48) takes values in ˜ h = h , then [ V , R (cid:48) , K ] is also an irreducible and non-transitive holonomy system. Then, from the uniqueness part of Simons theorem, R (cid:48) = λR , for some scalar λ (cid:54) = 0. Since T is an isometry, it induces an isometry onthe space of tensors. Then λ = ±
1. But 0 (cid:54) = sc ( R ) = sc ( R (cid:48) ). Then λ = 1 andhence R (cid:48) = R . (cid:3) Remark . Let M n = K.v , where K acts (by linear isometries) on R n + n ( n +1) asan s -representation ( | v | = 1). Assume that the restricted normal holonomy group ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 11 Φ( v ) acts irreducibly on ¯ ν v ( M ) = { v } ⊥ ∩ ν v ( M ). In this case M is a minimalsubmanifold of the sphere S n − n ( n +1) (see Corollary 2.6).Let A be the shape operator of M and let Sim ( T p M ) be the space of tracelesssymmetric endomorphisms of T p M . Then the map A : ¯ ν v ( M ) (cid:55)→ Sim ( T v M ) isa linear isomorphism. In fact, it is injective, since the first normal space of M coincides with the normal space, and dim(¯ ν v ( M )) = dim( Sim ( T v M )). Moreover,by the second part of Corollary 2.6, A is a homothecy from ¯ ν v ( M ) onto Sim ( T v M ),let us say, of constant β > Sim ( T p M ) , R, SO( T p M )] and [¯ ν v ( M ) , R ⊥ , Φ( v )] , where R ⊥ is the adapted normal curvature tensor of M at v and R is the curvaturetensor of Sl( n ) / SO( n ) (which is explicitly given by (***) of Section 1.3).Observe that [¯ ν v ( M ) , R ⊥ , Φ( v )] is symmetric since, by Theorem 2.5, the re-stricted normal holonomy group is given byΦ( v ) = { k | ν v ( M ) : k ∈ ( K v ) } and R ⊥ is left fixed by K v .Both algebraic curvature tensors are related by the formula (**) of Section 1.1.This implies that the homothecy A maps R ⊥ into R . Then the isometry β − A maps R ⊥ into β R .Since in a symmetric irreducible holonomy system the Lie algebra of the groupis (linearly) generated by the curvature endomorphisms, we conclude that A mapsΦ( v ) onto SO( T p M ) (cid:39) SO( n ). In particular, the two holonomy systems are equiv-alent and Φ( v ) (cid:39) 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 ofsuch a space is sl ( n + 1) = so ( n + 1) ⊕ Sim ( n + 1)where Sim ( n + 1) denotes the traceless symmetric (real) ( n + 1) × ( n + 1)-matrices.The Ad-representation of SO( n + 1) on Sim ( n + 1) coincides with the action, byconjugation, of SO( n + 1) on Sim ( n + 1).The curvature tensor of X at [ e ] is given (up to a positive multiple) by R A,B C = − [[ A, B ] , C ]and (cid:104) R A,B
C, D (cid:105) = −(cid:104) [[ A, B ] , C ] , D (cid:105) = (cid:104) [ A, B ] , [ C, D ] (cid:105) (***)where A, B, C, D ∈ Sim ( n + 1) (cid:39) T [ e ] X .Let S ∈ Sim ( n + 1) with exactly two eigenvalues, one of multiplicity 1 (whoseassociated eigenspace we denote by E ) and the other of multiplicity n (whoseassociated eigenspace we denote by E ).The orbit V n = SO( n +1) .S = { kSk − : k ∈ SO( n +1) } is called a Veronese-type orbit (see Appendix).The following assertions are easy to verify or well-known.
Facts 2.16. (i)
The Veronese-type orbit V n = SO( n + 1) .S is a full and irreduciblesubmanifold of Sim ( n + 1) which has dimension n and codimension n ( n + 1).Moreover, V n is a minimal submanifold of the sphere of radius (cid:107) S (cid:107) . (ii) An orbit of SO( n + 1) in Sim ( n + 1) has minimal dimension if and only ifit is of Veronese-type; see Lemma 8.1. (iii) The normal holonomy group at S , of the Veronese-type orbit V n , coin-cides with image of the slice representation of the isotropy group (SO( n + 1)) S =S(O( E ) × O( E )) (cid:39) S(O(1) × O( n )). So, from (*), the restricted normal holonomyrepresentation, on ¯ ν S ( V n ) = { S } ⊥ ∩ ν S ( V n ), is equivalent to the isotropy repre-sentation of the symmetric space Sl( n ) / SO( n ) of rank n −
1. Then, this normalholonomy representation is irreducible. Moreover, it is non-transitive (on the unitsphere of ¯ ν S ( V n )) if and only if n ≥ (iv) A Veronese-type orbit V n = SO( n + 1) .S = SO( n + 1) / (SO( n + 1)) S isintrinsically a real projective space R P n . Moreover, (SO( n + 1) , (SO( n + 1)) S )is a symmetric pair and so (SO( n + 1)) S acts irreducibly on T S V n . Then, fromCorollary 2.8, V n has parallel second fundamental form (as it is well known). (cid:3) A submanifold M ⊂ R N is called a Veronese submanifold if it is extrinsicallyisometric to a Veronese-type orbit.
Proposition 2.17.
Let M n = K.v ⊂ R n + n ( n +1) , where K acts on R n + n ( n +1) as an s -representation ( n ≥ ). Assume that the restricted normal holonomy group Φ( v ) of M at v , restricted to ¯ ν v ( M ) = { v } ⊥ ∩ ν v ( M ) , acts irreducibly (eventually,in a transitive way). Then, (i) The normal holonomy representation of Φ( v ) on ¯ ν v ( M ) is equivalent to theisotropy representation of the symmetric space Sl( n ) / SO( n ) . (ii) M n is a Veronese submanifold.Proof. Part (i) is a consequence of Remark 2.15.Since K acts as an s -representation, then the image under the slice representa-tion, of the (connected) isotropy group ( K v ) , coincides with the restricted normalholonomy group Φ( v ). But, from part (i), dim(Φ( v )) = dim(SO( n )) Then theisotropy group K v has dimension at least dim( SO ( n )) = dim( SO ( T v M )).Observe that the isotropy representation of K v on T v M is faithful. Otherwise, M would be contained in the proper subspace which consists of the fixed vector of K v in R N .Then, ( K v ) = SO( T v M ). So, K v acts irreducibly on T v M . Then, from Corol-lary 2.8, M has parallel second fundamental form.Let V n be a Veronese submanifold of R n + n ( n +1) . We may assume that v ∈ V n and that T v M = T v V n = R n ⊂ R n + n ( n +1) . For V n we have, from Corollary 2.6and Remark 2.15, that its shape operator ¯ A : { v } ⊥ ∩ ν v ( V n ) = { v } ⊥ ∩ ν v ( M ) → Sim ( T v V n ) = Sim ( T v M ) is a homothecy which induces an isomorphism fromthe normal holonomy group ¯Φ( v ) of V n onto SO( n ).The same is true, again from Corollary 2.6 and Remark 2.15, for the shape opera-tor A of M . Namely, A : { v } ⊥ ∩ ν v ( M ) → Sim ( T v V n ) = Sim ( T v M ) = Sim ( R n )is a homothecy which induces an isomorphism from the (restricted) normal holo-nomy group Φ( v ) of M onto SO( n ). Then the map A − ◦ ¯ A is a homothecy with ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 13 constant, let us say, β >
0, of the space { v } ⊥ ∩ ν p ( M ). Let h = β − A − ◦ ¯ A . Then h is a linear isometry of { v } ⊥ ∩ ν v ( M ).Let now g be the linear isometry of R n + n ( n +1) defined by the following prop-erties: (i) g ( v ) = v . (ii) g |{ v } ⊥ ∩ ν v ( M ) = h − . (iii) g | T v M = Id.Then V n and g ( M ) have proportional second fundamental forms and satisfy allthe other assumptions of Lemma 2.10. Then, by this lemma, g ( M ), and hence M ,is a Veronese submanifold. (cid:3) Coxeter groups and holonomy systems.
The goal of this section is to prove Proposition 2.21 that will be important forproving our main theorems. In order to prove this proposition we need some basicresults, related to Coxeter groups, that we have not found through the mathematicalliterature. So, and also for the sake of self-completeness, we include the proofs.
Lemma 2.18.
Let C be a Coxeter group acting irreducibly, by linear isometries, onthe Euclidean n -dimensional vector space ( V , (cid:104) , (cid:105) ) . Let H , ..., H r be the family of(different) reflection hyperplanes, associated to the symmetries of C (that generates C ). Let us define the group G = { g ∈ End ( V ) : g permutes H , ..., H r and det ( g ) = ± } . Then G is finite.Proof. Let P r be the (finite) group of bijections of the set { , ..., r } . Let ρ : G → P r be the group morphism defined by ρ ( g )( i ) = j , if g ( H i ) = H j . The group G is finiteif and only if ker( ρ ) is finite. Let us prove that ker( ρ ) is finite. If g ∈ ker( ρ ) thenit induces the trivial permutation on the family H , ..., H r . Then, its transpose g t ,with respect to (cid:104) , (cid:105) , induces the trivial permutation on the set of lines L , ..., L r ,where L i is the line which is perpendicular to H i , i = 1 , ..., r (and hence, anyvector in any line L , ..., L r is an eigenvector of g t . Let us define, for i (cid:54) = j , the2-dimensional subspace V i,j := linear span of ( L i ∪ L j ). This subspace is called generic if there exists k ∈ { , ..., r } , i (cid:54) = k (cid:54) = j such that L k ⊂ V i,j . In otherwords, V i,j is generic if there are at least three different lines of { L , ..., L r } whichare contained in V i,j . We have, if V i,j is generic, that g t : V i,j → V i,j is a scalarmultiple of the identity Id i,j of V i,j . In fact, any vector in L i ∪ L j ∪ L k is aneigenvector of ( g t ) | V i,j . Then, since dim( V i,j ) = 2, ( g t ) | V i,j = λ Id i,j , for some λ ∈ R . Let us define the following equivalence relation ∼ on the set { , ..., r } : i ∼ i (cid:48) if there exist i , ..., i l ∈ { , ..., r } with i = i , i l = i (cid:48) and such that V i s ,i s +1 is generic,for s = 1 , ..., l −
1. Let i ∈ { , ..., r } be fixed. By the previous observations onehas that there must exist λ ∈ R such that for any j ∈ [ i ] (the equivalence class of i ) and for any v j ∈ L j , g t ( v j ) = λv j . In order to prove this lemma, it suffices toshow that there is only one equivalence class on { , ..., r } . In fact, if [ i ] = { , ..., r } ,then g t = λId , since L , ..., L r span V (because of its othogonal complement ispoint-wise fixed by C ). So g = λId . But det( g ) = ±
1. Then λ n = ± λ = ±
1. So, g = ± Id and therefore there are at most two elements in ker( ρ ).Let i ∈ { , ..., r } be fixed. Let us show that [ i ] = { , ..., r } . If j / ∈ [ i ] then L j isperpendicular to any L k , for all k ∈ [ i ]. In fact, assume that this is not true forsome k ∈ [ i ]. Let s j ∈ C be the symmetry across the hyperplane H j . Then s j ( L k )is a line, which belongs to { L , ..., L r } , that is contained in V k,j and it is different from both L k and L j . Then j ∼ k and therefore j ∼ i . A contradiction. Then, if j / ∈ [ i ], L k ⊂ H j , for all k ∈ [ i ]. So, s j acts trivially on V [ i ] , the subspace spannedby (cid:83) k ∈ [ i ] L k . Observe that s j commutes with s k , for all k ∈ [ i ]. Let now V bethe maximal subspace of V such that it is point-wise fixed by all the symmetries s j with j / ∈ [ i ]. Observe that this space is not the null subspace, since V [ i ] ⊂ V . Ifthere exists j / ∈ [ i ], then V must be a proper subspace of V , since s j (cid:54) = Id. On theother hand, if k ∈ [ i ], then s k ( V ) ⊂ V , since s k commutes with all the symmetries s j , j / ∈ [ i ]. Then V is a proper and non-trivial subspace of V which is invariantunder the irreducible Coxeter group C . A contradiction. So, [ i ] = { , ..., r } . (cid:3) Lemma 2.19. . We are under the assumptions and notation of the above lemma.Then G acts by isometries.Proof. By the above lemma, G is finite. By averaging the inner product (cid:104) , (cid:105) overthe elements of G , we obtain a G -invariant inner product ( , ) on V . Since C ⊂ G ,then ( , ) is C -invariant. Since C acts irreducible, (cid:104) , (cid:105) must be proportional to ( , ).Then G acts by isometries on ( V , (cid:104) , (cid:105) ). (cid:3) Corollary 2.20.
Let ( V i , (cid:104) , (cid:105) i ) be a Euclidean vector spaces and let C i be a Cox-eter group acting irreducibly, by linear isometries, on ( V i , (cid:104) , (cid:105) i ) , i = 1 , . Let h : V → V be a linear map such that it induces a bijection from the family ofreflection hyperplanes of C into the family of reflection hyperplanes of C . Then h is a homothetical map.Proof. Let ( , ) = h ∗ ( (cid:104) , (cid:105) ) and let C ∗ = h ∗ ( C ) = h − C h . Observe that thedeterminant of any element of C is ±
1, since it is an isometry of ( V , (cid:104) , (cid:105) ). So,any element in C ∗ has determinant ±
1. From the assumptions, we obtain that thefamily of reflection hyperplanes of the irreducible Coxeter group C ∗ of ( V , ( , ))coincides with the family H , ..., H r of reflection hyperplanes of C . Then anyelement of C ∗ induces a permutation in this family of hyperplanes. Then, byLemma 2.19, C ∗ acts by isometries on ( V , (cid:104) , (cid:105) ). Since C ∗ acts irreducibly, onehas that (cid:104) , (cid:105) is proportional to ( , ). This implies that h is a homothecy (cid:3) Proposition 2.21.
Let ( V , R, K ) and ( V (cid:48) , R (cid:48) , K (cid:48) ) be irreducible, non-transitive(and hence symmetric) holonomy systems. Let h : V → V (cid:48) be a linear isomorphismsuch that, for any K -orbit K.v in V , h ( ν v ( K.v )) = ν h ( v ) ( K (cid:48) .h ( v )) , where ν denotesthe normal space. Then h is a homothecy and h − ∗ ( K (cid:48) ) = K .Proof. Observe that the groups K and K (cid:48) act as irreducible s -representations. Wehave that K.v is a maximal dimensional orbit if and only if K (cid:48) .h ( v ) is so.Recall that, for s -representations, an orbit is maximal dimensional if and only ifit 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 , associatedto this isoparametric submanifold, that acts on the normal space ν v ( K.v ) [Te, PT,BCO]. If H , ..., H r are the reflection hyperplanes of the symmetries of C , then r (cid:91) i =1 H i = { z ∈ ν v ( K.v ) :
K.z is a singular orbit } (a) ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 15 If v (cid:48) = h ( v ) one has the similar objects K (cid:48) .v (cid:48) , ν v (cid:48) ( K (cid:48) v (cid:48) ), C (cid:48) and H (cid:48) , ..., H (cid:48) s and s (cid:91) i =1 H (cid:48) i = { z (cid:48) ∈ ν v (cid:48) ( K (cid:48) .v (cid:48) ) : K (cid:48) .z (cid:48) is a singular orbit } (b)Moreover, from (a) and (b), one has that h maps, bijectively, the family H , ..., H r onto the family H (cid:48) , ..., H (cid:48) s . Then, s = r and so we may assume that h ( H i ) = H (cid:48) i , i = 1 , ..., s .Then, from Corollary 2.20, one has that h : ν w ( K.w ) → ν w (cid:48) ( K (cid:48) .w (cid:48) )is a homothecy, for any principal K -vector w , where w (cid:48) = h ( w ). Denote by λ ( w ) > w ∈ ν w ( K.w ) and w (cid:48) ∈ ν w (cid:48) ( K (cid:48) .w (cid:48) ), that (cid:104) h ( w (cid:48) ) , h ( w (cid:48) ) (cid:105) (cid:48) = λ ( w ) (cid:104) w, w (cid:105) where (cid:104) , (cid:105) and (cid:104) , (cid:105) (cid:48) are the inner products on V and V (cid:48) , respectively.Let v be a fixed K -principal vector and let M = K.v .Let T M = E ⊕ ... ⊕ E r , where E , ..., E r are the (autoparallel) eigendistributionsof T M associated to the commuting family of shape operators A ξ of the isopara-metric submanifold M ⊂ V . Associated to any E i there is a parallel normal field η i , a so-called curvature normal, such that, for any normal field ξ , A ξ | E i = (cid:104) ξ, η i (cid:105) Id E i Let, for q ∈ M , S i ( q ) denote the integral manifold of E i by q . Such integralmanifold is a so-called curvature sphere. If x ∈ S i ( q ) then ν x ( M ) ∩ ν q ( M ) = ( η i ( q )) ⊥ where the orthogonal complement is inside ν q ( M ). Observe that this intersection isnon-trivial, since the codimension of M in V is at least 2. This implies λ ( x ) = λ ( q ).Since the eigendistributions span T M , one has that moving along different curvaturesphere one can reach, from v , any other point of M . Then λ ( x ) = λ ( v ), for all x ∈ M .Observe now that, for any y ∈ V , there exists ¯ x ∈ M such that y ∈ ν ¯ x ( M ).In fact, such an ¯ x can be chosen as a point where the function, from M into R , x → (cid:104) x, y (cid:105) attains a maximum.Then (cid:104) h ( y ) , h ( y ) (cid:105) (cid:48) (cid:104) y, y (cid:105) = (cid:104) h (¯ x ) , h (¯ x ) (cid:105) (cid:48) (cid:104) ¯ x, ¯ x (cid:105) = λ ( x ) = λ ( v ) , for all 0 (cid:54) = y ∈ V . Then h is a homothecy of constant λ := λ ( v ). This proves thefirst assertion.Let g ∈ K (cid:48) and let T = h − ◦ g ◦ h . Since h is a homothecy, T ∈ SO( V ).Then, from the assumptions and Remark 2.13 one has that T satisfies hypothesisof Lemma 2.14. Then, by this lemma, T ( R ) = R . This implies, since the Liealgebra of K is generated by { R x,y } , that T belongs to N ( K ), the normalizer of K in O( V ). Moreover, T must belong to the connected component N ( K ) (becauseof T can be deformed to the identity, since K (cid:48) is connected). But N ( K ) = K ,since K acts as an s -representation (see [BCO] Lemma 6.2.2). Then T ∈ K , thus h − ∗ ( K (cid:48) ) = K . (cid:3) Remark . The above proposition is not true if the holonomy systems are tran-sitive. In fact, let ( V , R, K ) and ( V (cid:48) , R (cid:48) , K (cid:48) ) be the (symmetric) holonomy systemsassociated to the rank 1 symmetric spaces S n = SO(2 n + 1) / SO(2 n ) and C P n =SU( n +1) /S ( U (1) × U( n )), respectively. In this case dim( V ) = dim( V (cid:48) ) = 2 n . Thenany linear isomorphism from V into V (cid:48) , satisfies the assumption of Proposition 2.21,since the normal spaces of non-trivial K or K (cid:48) -orbits are lines.3. non-transitive normal holonomy Let M n = H.v ⊂ S n − n ( n +1) be a homogeneous submanifold of the sphere.Assume that the (restricted) normal holonomy group, as a submanifold of thesphere, acts irreducibly and it is not transitive (on the unit normal sphere).From now on, we will regard M n as a submanifold of the Euclidean space R n + n ( n +1) . Let ν ( M ) be the normal bundle and let Φ( v ) be the restricted nor-mal holonomy group at v (regarding M as a Euclidean submanifold). Observe thatΦ( v ) acts trivially on R .v and that Φ( v ), restricted to ¯ ν v ( M ) := { v } ⊥ ∩ ν v ( M ), isnaturally identified with the (restricted) normal holonomy group of M at v , as asubmanifold of the sphere.Observe that the irreducibility of the normal holonomy group representation on { v } ⊥ ∩ ν v ( M ) implies that rank( M ) = 1. Namely, v is the only vector of ν v ( M )which is fixed by Φ( v ). This implies that M is a full and irreducible submanifoldof the Euclidean space. In fact, if M is not full then any non-zero constant normalvector is a parallel normal field which is not a multiple of the position vector.Then rank( M ) ≥
2. A contradiction. If M is reducible it must be a product ofsubmanifolds contained in spheres. Then rank( M ) ≥
2. Also a contradiction.One has, from Remark 2.11, that the first normal space ν ( M ) coincides withthe normal space νM , regarding M as a Euclidean submanifold. This means, thatthe linear map, from ν v ( M ) into Sim ( T v M ), ξ (cid:55)→ A ξ is injective, where A is theshape operator of M . Since dim( ν v ( M )) = n ( n + 1) = dim( Sim ( T v M )), then A : ν v ( M ) → Sim ( T v M ) is a linear isomorphism .Let R ⊥ ξ ,ξ be the adapted normal curvature tensor (see Section 1). This tensoris given by (cid:104)R ⊥ ξ ,ξ ξ , ξ (cid:105) = − trace([ A ξ , A ξ ] ◦ [ A ξ , A ξ ])= (cid:104) [ A ξ , A ξ ] , [ A ξ , A ξ ] (cid:105) = −(cid:104) [[ A ξ , A ξ ] , A ξ ] , A ξ (cid:105) Observe that the right hand side of the above equality is, with the usual identifica-tions, the Riemannian curvature tensor (cid:104) ˜ R A ξ ,A ξ A ξ , A ξ (cid:105) of the symmetric spaceGl( 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 tracelesssymmetric matrices Sim ( n ).Let us consider the so-called traceless shape operator ˜ A of M . Namely,˜ A ξ := A ξ − n trace( A ξ )Id = A ξ − n (cid:104) ξ, H (cid:105) Idwhere H is the mean curvature vector. ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 17
Observe that (cid:104)R ⊥ ξ ,ξ ξ , ξ (cid:105) = (cid:104) [ ˜ A ξ , ˜ A ξ ] , [ ˜ A ξ , ˜ A ξ ] (cid:105) = (cid:104) ˜ R A ξ ,A ξ A ξ , A ξ (cid:105) = (cid:104) R ˜ A ξ , ˜ A ξ ˜ A ξ , ˜ A ξ (cid:105) (****)where R is the curvature tensor at [ e ] of the symmetric space Sl( T v M ) / SO( T v M )(see formula (***) of Section 1.3).If ¯ ν v ( M ) = { v } ⊥ ∩ ν v ( M ), we have the following two symmetric non-transitiveirreducible holonomy systems: [¯ ν v , R ⊥ , Φ( v )] and [ Sim ( T v M ) , R, SO( T v M )].Recall that for a symmetric irreducible holonomy system [ V , ¯ R, K ], from Remark2.13, the normal space to an orbit
K.v is given by ν v ( K.v ) = { ξ ∈ V : ¯ R v,ξ = 0 } Then, from (****), we have that the map ˜ A is a liner isomorphism that mapsnormal spaces to Φ( v )-orbits into normal spaces to SO( T v M )-orbits. Then, byProposition 2.21, ˜ A is a homothecy and ˜ A : ¯ ν v ( M ) → Sim ( T v M ) transforms Φ( v )into SO( T v M ). Then Φ( v ) is isomorphic to SO( T v M ). Therefore, we have thefollowing result: Lemma 3.1.
Let M n = K.v ⊂ S n − n ( n +1) be a homogeneous submanifold.Assume that the restricted normal holonomy group of M acts irreducibly and itis non-transitive. Then the representation of the normal holonomy group Φ( v ) on ¯ ν v ( M ) is (orthogonally) equivalent to the isotropy representation of the symmetricspace Sl( n ) / SO( n ) (cid:39) Sl( T v M ) / SO( T v M ) . Moreover, the traceless shape operator ˜ A : ¯ ν v ( M ) → Sim ( T v M ) is a homothecy that transforms, equivariantly, Φ( v ) into SO( T v M ) . (In particular, dim(Φ( v )) = n ( n −
1) = dim(SO( n ))). (cid:3) Proposition 3.2.
Let M n = K.v ⊂ S n − n ( n +1) be a homogeneous submanifold.Assume that the restricted normal holonomy group of M acts irreducibly and itis non-transitive. Then, for any ξ ( t ) parallel normal section along a curve, thetraceless shape operator ˜ A ξ ( t ) has constant eigenvalues.Proof. Note that M must be full and irreducible as a Euclidean submanifold (seethe beginning of this section). Let p ∈ M be arbitrary and let K p be the isotropysubgroup of K at p . Let us decomposeLie( K ) = m ⊕ Lie( K p )where m is a complementary subspace of Lie( K p ) Let B r (0) be an open ball, cen-tered at the origin, of radius r of m such that Exp : B r (0) → M is a diffeomorphismonto its image U = Exp( B r (0)), which is a neighbourhood of p (the inner producton Lie( K ) is irrelevant).Let β : [0 , → U be an arbitrary piece-wise differentiable curve with β (0) = p .Since β (1) ∈ U , there exits X ∈ m such that β (1) = Exp( X ) .p . Let γ : [0 , → M be defined by γ ( t ) = Exp( tX ) .p . Let us denote, for k ∈ K , by l k the linear isometry v (cid:55)→ k.v of V . Let τ ⊥ t denote the ∇ ⊥ -parallel transport along γ | [0 ,t ] . Then, fromremarks 6.2.8 and 6.2.9 of [BCO], τ ⊥ t = (d l Exp( tX ) ) | ν p ( M ) ◦ e − t A X (A)where A X belongs to the normal holonomy algebra Lie(Φ( p )) and it is defined by A X = dd t | t =0 τ ⊥− t ◦ (d l Exp( tX ) ) | ν p ( M )8 CARLOS OLMOS AND RICHAR FERNANDO RIA˜NO-RIA˜NO Let τ ⊥ β be the ∇ ⊥ -parallel transport along β and φ = τ ⊥− ◦ τ ⊥ β . Then φ belongs toΦ( p ), the restricted normal holonomy group at p . In fact, φ coincides with the ∇ ⊥ -parallel transport along the null-homotopic, since it is contained in U , loop β ∗ ˜ γ ,obtained from gluing the curve β together with the curve ˜ γ , where ˜ γ ( t ) = γ (1 − t ).We have that τ ⊥ β = τ ◦ φ and so, by (A), τ ⊥ β = ((d l Exp( X ) ) | ν p ( M ) ◦ e −A X ) ◦ φ = ( dl Exp( X ) ) | ν p ( M ) ◦ ¯ φ where ¯ φ = e −A X ◦ φ belongs to Φ( p ). Then, for any ξ ∈ ν p ( M ),˜ A τ ⊥ β ( ξ ) = ˜ A d l Exp( X ) ( ¯ φ ( ξ )) = d l Exp( X ) ◦ ˜ A ¯ φ ( ξ ) ◦ (d l Exp( X ) ) − = Exp( X ) . ˜ A ¯ φ ( ξ ) . (Exp( X )) − Then, from the paragraph just before Lemma 3.1, we have that there exists g ∈ SO( T p ( M )) such that ˜ A ¯ φ ( ξ ) = g. ˜ A ¯ φ ( ξ ) .g − . Then˜ A τ ⊥ β ( ξ ) = (Exp( X ) .g ) . ˜ A ξ . (Exp( X ) .g ) − This shows that the eigenvalues of ˜ A τ ⊥ β ( ξ ) are the same as the eigenvalues of ˜ A ξ .The curve β was assumed to be contained in U . Since p is arbitrary, one obtainsthat the eigenvalue of ˜ A ξ ( t ) are locally constant for any ξ ( t ) parallel normal fieldalong a curve c ( t ). This implies that the eigenvalues of ˜ A ξ ( t ) are constant. (cid:3) 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 M be a submanifold of a space of constantcurvature and let ξ be a parallel normal field such that the eigenvalues of the shapeoperator A ξ have constant multiplicities. Let λ : M → R be an eigenvalue functionof A ξ such that its associated (and integrable from Codazzi identity) eigendistribu-tion E has dimension at least . Then λ is constant along any integral manifold of E (or equivalently, d λ ( E ) = 0 ). Theorem 3.4.
Let M n ⊂ S n − n ( n +1) be a homogeneous submanifold, where n > . Assume that the restricted normal holonomy group acts irreducibly and nottransitively. Then M is a Veronese submanifold.Proof. Note that M must be full and irreducible as a Euclidean submanifold (seethe beginning of this section).We will regard M as a submanifold of the Euclidean space R n + n ( n +1) . Then,as we have observed at the beginning of this section, A : ν p ( M ) → Sim ( T p M ) isan isomorphism ( p ∈ M is arbitrary). Now choose ξ ∈ ν p ( M ) such that A ξ hasexactly two eigenvalues λ ( p ), λ ( p ) with multiplicities m , m ≥ n ≤ m = 2 and m = n −
2. Wemay assume that ξ is small enough such that the holonomy tube [BCO] M ξ is animmersed Euclidean submanifold (see Remark 3.5). We may also assume that ξ isperpendicular to the position (normal) vector p , since A p = − Id .There is a natural projection π : M ξ → M , π ( c (1) + ¯ ξ (1)) = c (1). Moreover,ˆ ξ defines a parallel normal field to M ξ , where ˆ ξ ( q ) = q − π ( q ). In this way M is a parallel focal manifold to M ξ . Namely, M = ( M ξ ) − ˆ ξ . Observe that theholonomy tube M ξ is not a maximal one and so it has not a flat normal bundle ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 19 (this would have been the case, in our situation, where all of the eigenvalues of A ξ 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 thetraceless shape operator ˜ A ¯ ξ ( t ) are constant and hence the same as the eigenvaluesof ˜ A ξ which are ˜ λ = λ ( p ) − n (2 λ ( p ) + ( n − λ ( p )), with multiplicity 2 and˜ λ = λ ( p ) − n (2 λ ( p ) + ( n − λ ( p )), with multiplicity n − H be the mean curvature vector field on M . Then the eigenvalues of theshape operator A ¯ ξ (1) can be written as λ i ( c (1)) = ˜ λ i + 1 n (cid:104) ¯ ξ (1) , H ( c (1)) (cid:105) i = 1 , n −
2, respectively (independent of c (1) ∈ M ).From the tube formula [BCO], one has that the eigenvalues functions ˆ λ andˆ λ of the shape operator ˆ A ˆ ξ of the holonomy tube, restricted to the horizontalsubspace H q of the holonomy tube M ξ , at a point q = c (1) + ¯ ξ (1) are:ˆ λ ( q ) = ˜ λ + n (cid:104) ¯ ξ (1) , H ( c (1)) (cid:105) − ˜ λ − n (cid:104) ¯ ξ (1) , H ( c (1)) (cid:105) and ˆ λ ( q ) = ˜ λ + n (cid:104) ¯ ξ (1) , H ( c (1)) (cid:105) − ˜ λ − n (cid:104) ¯ ξ (1) , H ( c (1)) (cid:105) or, equivalently, ˆ λ ( q ) = ˜ λ + n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) − ˜ λ − n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) and ˆ λ ( q ) = ˜ λ + n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) − ˜ λ − n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) with (constant) multiplicities 2 and n −
2, respectively. Observe that ˆ A ˆ ξ ( q ) , re-stricted to the vertical distribution (tangent to the orbits in M ξ of the normalholonomy group of M at projected points) is minus the identity. So, ˆ A ˆ ξ ( q ) has athird eigenvalue ˆ λ ( q ) = − m = dim( M ξ ) − dim( M ).The real injective function s f (cid:55)→ s s transforms ˆ λ i ( q ) into ˜ λ i + n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) ( i = 1 , λ ( q ) = ˆ λ ( q (cid:48) ) ⇐⇒ ˆ λ ( q ) = ˆ λ ( q (cid:48) ) (I)In fact, any of both equalities implies n (cid:104) ˆ ξ ( q ) , H ( π ( q )) (cid:105) = n (cid:104) ˆ ξ ( q (cid:48) ) , H ( π ( q (cid:48) )) (cid:105) . This,by the above equalities, implies (I).Let now E and E be the (horizontal) eigendistributions associated to eigenvaluefunctions ˆ λ and ˆ λ of the shape operator ˆ A ˆ ξ . Observe that dim( E ) = 2 anddim( E ) = n − ≥ 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). If γ ( t ) is a curve that lies in E then, from the Dupin Condition (see Lemma3.3) we have that ˆ λ is constant along γ . So, by (I), ˆ λ is also constant along γ .The same is true if γ lies in E . This implies that 0 = v (ˆ λ ) = v (ˆ λ ) = v (ˆ λ ) forany vector v that lies in H . Then the eigenvalues of the shape operator ˆ A ˆ ξ areconstant along any horizontal curve. Since any two points, in a holonomy tube, canbe joined by a horizontal curve we conclude that the (three) eigenvalues of ˆ A ˆ ξ areconstant on M ξ .Then ˆ ξ is a parallel isoparametric (non-umbilical) normal section. Observe that M ξ is a full irreducible Euclidean submanifold, since M is so. Moreover, M ξ iscomplete with the induced metric (see Remark 3.5). Then, by [BCO],[DO], M ξ must be a submanifold with constant principal curvatures. Since M = ( M ξ ) − ˆ ξ , wehave that M is also a submanifold with constant principal curvatures. Any principalholonomy tube of M has codimension at least 3 in the Euclidean space, since thenormal holonomy of M , as a submanifold of the sphere, is non-transitive. Then,by the theorem of Thorbergsson [Th, O2, BCO], M is an orbit of an (irreducible) s -representation.The fact that M is a Veronese submanifold follows from Proposition 2.17. (cid:3) Remark . Let M n = H.v be a full irreducible homogeneous submanifold of R N which is (properly) contained in the sphere S N − . We are not assuming that M iscompact (in which case the assertions of this remark are trivial).By making use of the homogeneity of M one obtains that there exists ε > ξ ∈ ν ( M ) with 0 < (cid:107) ξ (cid:107) < ε then any of the eigenvalues λ of the shapeoperator A ξ satisfies | λ | < − a , for some 0 < a < M ) = 1, i.e., M is not a submanifold of higher rank(otherwise, M would be an orbit of an s -representation and hence compact).Let ξ ∈ ν v ( M ) with 0 < (cid:107) ξ (cid:107) < ε and let us consider the normal holonomysubbundle by ξ [BCO] of the normal bundle π : ν ( M ) → M .Hol ξ ( M ) = { η ∈ ν ( M ) : η H ∼ ξ } where H is the horizontal distribution of ν ( M ) and η H ∼ ξ if η and ξ can be joinedby a horizontal curve. Equivalently, η H ∼ ξ y η is the ∇ ⊥ -parallel transport of ξ along some curve.One has that the fibres of π : Hol ξ ( M ) → M are compact. In fact, π − ( { π ( η ) } ) =Φ( π ( η )) .η , where Φ denotes the normal holonomy group. Observe that such agroup is compact, since its connected component acts as an s -representation (seethe discussion inside the proof of Theorem 4.1, Case (2), (c)).Let us consider the normal exponential map exp ν : ν ( M ) → R N , given byexp ν ( η ) = π ( η ) + η . Let η ∈ ν p ( M ) and identify, as usual, via d π , T p M (cid:39) H η . Thevertical distribution ν η = T η ν p ( M ) is canonically identified to ν p ( M ). With thisidentification one has the well-known expression for the differential of the normalexponential map:d(exp ν ) |H η = ( I − A η ) , d(exp ν ) | ν η = Id ν p ( M ) (C) ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 21
Then exp ν : Hol ξ ( M ) → R N is an immersion. The image of this map is theso-called holonomy tube M ξ of M by ξ . It is given by M ξ = { 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 ofsimplifying the notation, the immersed submanifold exp ν : Hol ξ ( M ) → R N will bealso denoted by M ξ .One has that the Euclidean submanifold exp ν : Hol ξ ( M ) → R N , with the inducedmetric (cid:104) , (cid:105) , is a complete Riemannian manifold. In fact, let ( , ) be the Sasaki metricon Hol ξ ( M ). In such a metric the horizontal distribution is perpendicular to thevertical one. Moreover, π is a Riemannian submersion and the metric in the verticalspace Φ( p ) .η is that induced from the metric on the normal space ν p ( M ). Since M is complete and the fibres are compact, then ( , ) is complete. Then, from ( C ), a ( , ) ≤ (cid:104) , (cid:105) . This implies that the induced metric is also complete.4. The proof of the conjecture in dimension 3
Theorem 4.1.
Let M = H.p be a -dimensional homogeneous submanifold of thesphere S N − which is full and irreducible (as a submanifold of the Euclidean space R N ). Assume that the normal holonomy group of M is non-transitive. Then M isan orbit of an s -representation.Proof. Assume that M is not isoparametric (in which case it must be an orbitof an s -representation). Then, by Lemma 4.2, the normal holonomy of M , as asubmanifold of the sphere acts irreducibly and N = 9 = 3 + λ has multiplicity 1. So, we have the Dupin condition only for theeigendistribution E but not for the 1-dimensional eigendistribution E .Let ¯ M = M ξ / E be the quotient of the (partial) holonomy tube M ξ by the(maximal) integral manifolds of the 2-dimensional integrable distribution E .Observe that the (partial) holonomy tube M ξ has dimension 5. In fact, fromLemma 4.2, any focal orbit of the restricted normal holonomy group Φ( p ) (cid:39) SO(3)has dimension 2 (and it is isometric to the Veronese V ).By [BCO], Theorem 6.2.4, part (2) one has that H ⊂ SO(9) acts by (extrinsic)isometries on M ξ . Moreover, the projection π : M ξ → M is H -equivariant.If H. ( p + ξ ) = M ξ , then M ξ is a full and irreducible homogeneous Euclideansubmanifold which is of higher rank. Then, in this case, by the rank rigidity theoremfor submanifolds, M ξ is an orbit of an s -representation. Hence M = ( M ξ ) − ˆ ξ is anorbit of an s -representation.So, we may assume that H. ( p + ξ ) (cid:40) M ξ . Let h = Lie( H ). Let us considerthe subspace h . ( p + ξ ) of T p + ξ M ξ . This subspace has dimension at least 3, sinced π ( h . ( p + ξ )) = h .p = T p M . The horizontal subspace H ( p + ξ ) of T p + ξ M ξ has dimen-sion 3. Since T p + ξ M ξ has dimension 5, dim( H ( p + ξ ) ∩ h . ( p + ξ )) ≥ Case (1): E ( x ) + ( H x ∩ h .x ) = H x , for some x ∈ M ξ We may assume that x = p + ξ . Observe that if the above equality holds at( p + ξ ) then it also holds for q in some open neighbourhood U of ( p + ξ ) in M ξ .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 A ˆ ξ at q are: ˆ λ ( q )with multiplicity 2, ˆ λ ( q ) with multiplicity 1 and ˆ λ ( q ) = − ν q ).On one hand, from the Dupin condition, since dim( E ) = 2, and the equivalence( I ) in the proof of the above mentioned theorem, we have that0 = v (ˆ λ ) = v (ˆ λ ) = v (ˆ λ )for any v ∈ E ( q ). Or, briefly, { } = E ( q )(ˆ λ ) = E ( q )(ˆ λ ) = E ( q )(ˆ λ )On the other hand, if X ∈ h ,0 = ( X.q )(ˆ λ ) = ( X.q )(ˆ λ ) = ( X.q ) . (ˆ λ )In fact, this follows from the fact that the parallel normal field ˆ ξ of M ξ is H -invariantand that ˆ A h. ˆ ξ ( q ) = h. ˆ A ˆ ξ ( q ) .h − , for all h ∈ H .Then, from the assumptions of this case, { } = H q (ˆ λ ) = H q (ˆ λ ) = H q (ˆ λ ) (III)for any q ∈ U .Since M is (extrinsically) homogeneous, the local normal holonomy groups haveall 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 e in the local normal holonomy group, see [CO, DO]). This implies,from (III), that the eigenvalues of ˆ A ˆ ξ ( p + ω.ξ ) are the same as the eigenvalues ˆ λ ( p + ξ ),ˆ λ ( p + ξ ), ˆ λ ( p + ξ ) = − A ˆ ξ ( p + ξ ) , for all ω ∈ Ω. From this it is standard to showthat the eigenvalues of ˆ A ˆ ξ ( p + φ.ξ ) are the same of those of ˆ A ˆ ξ ( p + ξ ) , for all φ ∈ Φ( p ).Therefore, the eigenvalues of ˆ A ˆ ξ are constant on p + Φ( p ) .ξ = π − ( { p } ). Since H acts transitively on M , then H.π − ( { p } ) = M ξ . This implies, since ˆ ξ is H -invariant,that the eigenvalues of ˆ A ˆ ξ are constant on M ξ .Observe that the parallel normal field ˆ ξ is not umbilical, since ˆ A ˆ ξ has threedistinct (constant) eigenvalues. Then, from [DO] (see Theorem 5.5.8 of [BCO]), M ξ has constant principal curvatures. So, M = ( M ξ ) − ˆ ξ has constant principalcurvatures. If ˜ M is a principal holonomy tube of M , then ˜ M is isoparametric [HOT].Observe that ˜ M is not a hypersurface of a sphere (since the normal holonomygroup, in the Euclidean space, is not transitive on the orthogonal complement ofthe position vector), then by the theorem of Thorbergsson [Th, O2] ˜ M is an orbitof an s -representation. Then M is an orbit of an s -representation, since it is a focal(parallel) manifold to ˜ M . Case (2): E ( x ) + ( H x ∩ h .x ) (cid:40) H x , for all x ∈ M ξ or equivalently, ( H x ∩ h .x ) ⊂ E ( x ), since dim( E ( x )) = 2 and dim( H x ) = 3. ORMAL 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 ) ≥ M ). The most difficult case is thegeneric one where dim( H ) = 3. For this case we will have to use topologicalarguments.Note that dim( H ) ≤
6. In fact, H acts effectively on M , since M is a fullsubmanifold. Otherwise, if h ∈ H acts trivially on M then it acts trivially on the(affine) span of M which is R . But the dimension of the isometry group of an n -dimensional Riemannian manifold is bounded by ( n + 1) n (the dimension of theisometry group of an n -dimensional space of constant curvature). In our case, since n = 3, dim( H ) ≤ H cannot be abelian. In fact if H is abelian, since the dimensionof the ambient space N = 9 is odd, the (connected) subgroup H ⊂ SO(9) mustfix a vector, let us say v (cid:54) = 0. So, no H -orbit H.q is a full submanifold, since it iscontained 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 H p has dimension 2 and so it is abelian. We regard H p ⊂ SO( T p M ) (cid:39) SO(3), via theisotropy representation. But the rank of SO(3) is 1 and so it has no abelian twodimensional subgroups. A contradiction.( a ) dim( H ) = 6.In this case we must have that ( H p ) = SO(3), since dim( H p ) = 3. Since SO(3)is simple, the slice representation sr of ( H p ) on the normal space ν p ( M ) must beeither trivial or its image has dimension 3. In the first case we obtain that all shapeoperators A µ of M at p are a multiple of the identity, since they commute all with( H p ) . Note that A µ = A h.µ = h.A µ .h − . So M = M is an umbilical submanifoldof S ⊂ R . So, M is 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 (( H p ) ) ⊂ Φ( p ) where Φ( p ) is the restrictednormal holonomy group of M as a Euclidean submanifold. Since dim(Φ( p )) = 3,we conclude that sr (( H p ) ) = Φ( p ). Then, any holonomy tube of M is an H orbit.In particular the principal ones, which have flat normal bundle. But the holonomytubes are full and irreducible Euclidean submanifolds, which have codimension atleast 3 (since Φ( p ) acts on the 6-dimensional normal space ν p ( M ) with cohomo-geneity 3). Then, by the theorem of Thorbergsson [Th, O2], any holonomy tubeis an orbit of an s -representation and so M is an orbit of an s -representation. ByProposition 2.17 one has that M = M is a Veronese submanifold.( b ) dim( H ) = 4.In this case the isotropy H p has dimension 1. If the slice representation sr of( H p ) is trivial, then, as in (a), all shape operators at p commute with ( H p ) (cid:39) S .A contradiction, since the family of shape operators is Sim ( T p M ).Let us then restrict to the case that the slice representation is not trivial. Forthis we have to use a result of [OS] (see [BCO], Theorem 6.2.7). In fact, we need thefollowing weaker version, which was the main step in the proof of Simons holonomytheorem given in [O6]. Namely, Proposition 2.4 of [O6]: for a full and irreducible H -homogeneous Euclidean submanifold M n , n ≥ , the projections, on the normalspace ν p ( M ) , of the (Euclidean) Killing fields given by the elements of h = Lie ( H ) ,belong to the normal holonomy algebra g . Then, in our situation, since dim( h ) = 4 and dim( g ) = 3, there must exist0 (cid:54) = X ∈ h such that it projects trivially on the normal space. Such an X cannot bein the isotropy algebra, since we assume that the slice representation of ( H p ) (cid:39) S is non-trivial. This implies that 0 (cid:54) = X.p ∈ T p M .Let us consider the H -invariant parallel normal field ˆ ξ of M ξ . Recall that( M ξ ) − ˆ ξ = M (and so M is a parallel focal manifold of M ξ ).Since X projects trivially on ν p ( M ), X.q ∈ H q , for all q ∈ ( p + Φ( p ) .ξ ) =(( π ) − ( { p } )) q ⊂ M ξ .Recall that we are in Case (2) . Then,
X.q ∈ E ( q ), for all q ∈ ( p + Φ( p ) .ξ ). Letus consider the curve γ ( t ) = Exp( tX ) .p of M . One has that γ (cid:48) (0) = X.p (cid:54) = 0. Let q ∈ ( p + Φ( p ) .ξ ) and let ψ ( t ) be the normal parallel transport of ( q − p ) ∈ ν p ( M )along γ ( t ). Then ψ ( t ) = ˆ ξ ( γ ( t ) + ψ ( t )), as it is well known, from the construction ofholonomy tubes [HOT, BCO] (observe that M ξ = M q − p ). From the tube formulaof [BCO], Lemma 4.4.7 (the notation in this lemma permutes our objects), A ( q − p ) = ˆ A ( q − p ) |H . (( Id − ˆ A ( q − p ) ) |H ) − one has that E ( q ) is an eigenspace of the shape operator A ( q − p ) of M .On the one hand, since π ( q ) = q − ˆ ξ ( q ),d π ( E ( q )) = ( Id + ˆ A ˆ ξ )( E ( q )) ⊂ E ( q )On the other hand, since ˆ ξ is H -invariant and ˆ ξ ( q ) = ( q − p ),d π ( X.q ) = dd t | (Exp( tX ) .q − ˆ ξ (Exp( tX ) .q ))= dd t | (Exp( tX ) .q − Exp( tX ) . ( q − p )) = X.p
Therefore,
X.p belongs to an eigenspace of any shape operator A q − p of M , suchthat q ∈ ( p + Φ( p ) .ξ ) (recall that we have assumed, without loss of generality, that ξ is perpendicular to the position vector p ).Observe that Φ( p ) .ξ spans { p } ⊥ , since Φ( p ) acts irreducibly on { p } ⊥ . So X.p isan eigenvector of any shape operator A η , where (cid:104) η, p (cid:105) = 0.Since A p = − Id , we conclude that X.p is an eigenvector of all shape operators of M at p . This is a contradiction, since the family of shape operators at p coincideswith Sim ( T p M ).( c ) dim( H ) = 3.Since we have excluded the case where H is abelian, then H must be simple,with universal cover the (compact) group Spin(3) (cid:39) S . This case is the genericone where the isotropy is finite. Note that M must be compact.Also note that the (full) normal holonomy group ˜Φ( p ) of M is compact. In fact,( ˜Φ( p )) coincides 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( ν p ( M )).Observe that ( N (Φ( p ))) = 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 M ξ is compact.Let us construct the so-called caustic fibration . The eigenvalues functions of ˆ A ˆ ξ are bounded on M ξ . Since M is contained in a sphere, M ξ is contained in a (differ-ent) sphere. If η is the position vector field of M ξ , then η is an umbilical parallel ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 25 normal field. In fact, ˆ A η = − Id. By adding, eventually, to the parallel normal fieldˆ ξ a (big) constant multiple of − η we obtain a new parallel and H -invariant normalfield, such that its associated shape operator has the same eigendistributions as ˆ A ˆ ξ and all of the three eigenvalues functions are everywhere positive and so nowherevanishing. Just for the sake of simplifying the notation, we also denote this per-turbed normal field by ˆ ξ . The eigenvalues of ˆ A ˆ ξ are also denoted by ˆ λ , ˆ λ , ˆ λ ,which differ from the original ones by a (same) constant c .The caustic map ρ , from M ξ into R , q ρ (cid:55)→ q + (ˆ λ ( q )) − ˆ ξ ( q ) has constant rank.In fact, ker (d ρ ) = E has constant dimension 2, since from the Dupin condition,ˆ λ is constant along any integral manifold Q ( q ) of E . Observe that ˆ λ is alsoconstant along Q ( q ), due to equivalence (I) in the proof of Theorem 3.4 (and thesame is true, of course, for the third eigenvalue ˆ λ ≡ − c ).Let ¯ M = M ξ / E be the quotient of M ξ by the family E of (maximal) integralmanifolds of E . From Lemma 4.3 we have that ¯ M is a compact 3-manifold im-mersed in R , via the projection ¯ ρ , of the caustic map ρ , to the compact quotientmanifold ¯ M . Moreover, ¯ π : M ξ → ¯ M is a fibration, where ¯ π : M ξ → ¯ M is theprojection. The distribution E is H -invariant, since ˆ ξ is so. So, the action of H on M ξ projects down to an action on ¯ M . So, ¯ π is H -equivariant.Observe that ρ is H -equivariant, since ˆ ξ is H -invariant. Then, since ¯ π is H -equivariant, the immersion ¯ ρ : ¯ M → R is H -equivariant.We have the following two H -equivariant fibrations on M ξ :0 → Φ( p ) .ξ → M ξ ˜ π → ˜ M → (holonomy tube fibration) → Q → M ξ ¯ π → ¯ M → (caustic fibration) where Q is any integral manifold of E and ˜ M is the quotient manifold M ξ overthe connected component of the fibres of π : M ξ → M , which are orbits of therestricted normal holonomy groups Φ( p ), p ∈ M . We have that ˜ M is a finite coverof M . Recall that we are under the assumptions of Case (2)
We will derive a topological contradiction. This is by using that the holonomytube M ξ is the total space of above two different fibrations.On the one hand the holonomy tube has a finite fundamental group π ( M ξ ). Thisfollows from the long exact sequence of homotopies, associated to the holonomytube fibration. In fact, the fibres are (real) projective 2-spaces (which have a finitefundamental group). Moreover, the base space ˜ M has also a finite fundamentalgroup, since it is an orbit, with finite isotropy, of the group Spin(3) (cid:39) S . Sincethe fibres of the caustic fibration are connected and the total space M ξ has finitefundamental group, then the caustic (base) manifold ¯ M has a finite fundamentalgroup.On the other hand, from Lemma 4.4 we have that the fundamental group of thecaustic manifold ¯ M is not finite (this is by showing that H acts with cohomogeneity1 and without singular orbits on ¯ M ).A contradiction. So we can never be under the assumptions of Case (2) if H (cid:39) Spin(3).
This finishes the proof that M is an orbit of an s -representation. (cid:3) 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 asthe action of SO (3) , by conjugation, on the traceless × -symmetric matrices.Furthermore, the traceless shape operator ˜ A of M at p is SO (3) -equivariant.Proof. Let us regard M as a submanifold of the Euclidean space R N . If M isnot of higher rank one has, from Proposition 6.1, that the (restricted) normalholonomy group Φ( p ) acts irreducibly on ¯ ν ( p ) (the orthogonal complement of theposition vector p ). Since Φ( p ) is non-transitive (on the unit sphere of ¯ ν p ( M )),the first normal space, as a submanifold of the Euclidean space, coincides withthe normal space (see Remark 2.11). Then, the codimension k = N − k ≤ / SO(3). So the codimension of M , in the sphere, is 5 and hence N = 9. The equivariance follows form Lemma 3.1. (cid:3) Lemma 4.3. (Caustic fibration lemma).
Let ˆ M be a compact immersed submani-fold of R N which is contained in the sphere S N − . Let ˆ ξ be a parallel normal fieldto ˆ M such that the eigenvalues of the shape operator A ˆ ξ have constant multiplic-ities on ˆ M . Let ˆ λ : ˆ M → R be an eigenvalue function of A ˆ ξ whose associated(integrable) eigendistribution E has (constant) dimension at least . Let E be thefamily of (maximal) integral manifolds of E . Assume that the eigenvalue function ˆ λ never vanishes (this can always be assumed by adding to ˆ ξ an appropriate constantmultiple of the umbilical position vector) . Then(i) Any integral manifold Q ∈ E is compact.(ii) The quotient space ¯ M = ˆ M / E is a (compact) manifold and the projection π : ˆ M → ¯ M is a fibration (in particular, a submersion).(iii) The caustic map ρ : ˆ M → R N , ρ ( q ) = q + (ˆ λ ( q )) − ˆ ξ ( q ) , projects down toan immersion ¯ ρ : ¯ M → R N (i.e. ρ = ¯ ρ ◦ π ).Proof. From the Dupin condition, see Lemma 3.3, one has that ˆ λ is constant alongany integral manifold Q of E .Consider the caustic map ρ ( q ) = q + ( λ ( q )) − ˆ ξ ( q ) (see the proof of Theorem 4.1,Case (2),(c)). Then ker (d ρ ) = E and so d ρ has constant rank. From the local formof a map with constant rank and the compactness of ˆ M one has that there existsa finite open cover V , ..., V d of ˆ M such that, for any i = 1 , ..., d and q, q (cid:48) ∈ V i , thefollowing equivalence holds: ρ ( q ) = ρ ( q (cid:48) ) ⇐⇒ q and q (cid:48) belong both to a same integral manifold of E. This implies that any (maximal) integral manifold Q of E must be a closedsubset of ˆ M and hence compact. Moreover, the above equivalence implies that thefoliation E is a regular foliation in the sense of Palais [P].In order to prove that the quotient is a manifold we need to prove that thisquotient is Hausdorff. But this can be done as follows: let E ⊥ be the distribution ORMAL 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 (cid:104) , (cid:105) on ˆ M by changing the inducedmetric ( , ) on the distribution E ⊥ in such a way that ρ is locally a Riemanniansubmersion onto its image. Namely, . (cid:104) E, E ⊥ (cid:105) = 0. . (cid:104) , (cid:105) coincides with ( , ) when restricted to E . d | q ρ is a linear isometry from ( E ⊥ ) q onto its image.Such a metric is a bundle-like metric in the sense of Reinhart [Re] Since ˆ M iscompact, (cid:104) , (cid:105) is a complete Riemannian metric. Then, [Re],Corollary 3, pp. 129,the quotient space ¯ M is Hausdorff and π is a fibration (cf. [DO], Proposition 2.4,pp. 83)Then one has that the map ρ projects down to an immersion ¯ ρ : ¯ M → R N and ρ = ¯ ρ ◦ π . (cid:3) Lemma 4.4.
We keep the assumptions of Theorem 4.1. Moreover, we are inthe assumptions and notation of Case (2)(c), inside the proof of this theorem (inparticular, H (cid:39) Spin(3), up to a cover) .(i)
All orbits of the action of H on ¯ M have dimension 2. (ii) The universal cover ˜ M of ¯ M splits off a line and hence the fundamental groupof ¯ M in not finite (since ¯ M is compact). Proof.
The action of H on M ξ projects down to ¯ M , since ˆ ξ is H -invariant and soany eigendistribution of ˆ A ˆ ξ is H -invariant. Let q ∈ M ξ . Then the 3-dimensionalsubspace h .q ⊂ T q M ξ intersects the 3-dimensional horizontal subspace H q in anon-trivial subspace, since dim( M ξ ) = 5). Since we are in Case (2) , { } (cid:54) = ( h .q ∩ H q ) ⊂ E ( q )Let H ¯ q be the isotropy group of H at the point ¯ q = ¯ π ( q ) ∈ ¯ M . Let h ¯ q = Lie( H ¯ q ).Then one has that h ¯ q = { X ∈ h : X.q (cid:48) ∈ E ( q (cid:48) ) } independent of q (cid:48) ∈ S ( q ) = (¯ π ) − (¯ π ( { q } )), since a Killing field that is tangent toan integral manifold S ( q ) of E , at some point, must be always tangent to it (sincethe action projects down to the quotient).If dim( h ¯ q ) = 3. Then h ¯ q = h . Then H leaves invariant the 2-dimensional integralmanifold S ( q ) of E by q . Then the isotropy H q has positive dimension. But H q ⊂ H π ( q ) , where H π ( q ) is the isotropy group of H at the point π ( q ) ∈ M = H.p .A contradiction, since dim( H ) = 3.Observe that dim( h ¯ q ) (cid:54) = 2. In fact, if this dimension is 2, then h ¯ q is an ideal ofthe 3-dimensional (compact type) Lie algebra h . A contradiction, since h is simple.We have used that a Lie subalgebra of codimension 1 of a Lie algebra which admitsa bi-invariant metric must be an ideal. (Also, this 2-dimensional Lie subalgebrashould be abelian, in contradiction with rank ( h ) = 1).Then dim( h ¯ q ) = 1 for all q ∈ M ξ . This implies that all H -orbits in ¯ M havedimension 2. Since H acts with cohomogeneity 1 on ¯ M then, the universal cover of¯ M cannot be compact. Otherwise, as it is well-known, there would exist a singularorbit (after lifting the action to the universal cover). 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 (cid:104) , (cid:105) induced by the immersion ¯ ρ .Since H acts with cohomogeneity 1 on ¯ M , H acts locally polarly. In particular,the one dimensional distribution D on ¯ M , perpendicular to the H -orbits, is anautoparallel distribution. If ¯ q ∈ ¯ M then we put on the orbit H. ¯ q the normalhomogeneous 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-invariantmetric on h .We define (cid:104) , (cid:105) (cid:48) by: a) (cid:104) , (cid:105) (cid:48)|D = (cid:104) , (cid:105) |D b) (cid:104)U , D(cid:105) (cid:48) = 0, where U is the distribution given by the tangent spaces of the H -orbits on ¯ M . c) (cid:104) , (cid:105) (cid:48)|U ¯ q coincides with the normal homogeneous metric of H. ¯ q , for any ¯ q on ¯ M Since ¯ M is compact, the metric (cid:104) , (cid:105) (cid:48) is complete. Let (cid:104) , (cid:105) (cid:48) also denote the liftof the Riemannian metric (cid:104) , (cid:105) (cid:48) to the universal cover ˜ M of ¯ M . Then ( ˜ M , (cid:104) , (cid:105) (cid:48) )is a complete Riemannian manifold. Let us denote by ˜ U and ˜ D the lifts to ˜ M of the distributions U and D , respectively. Let us also lift the H -action on ¯ M to ˜ M Then, since ˜ M is simply connected, the one dimension distribution ˜ D isparallelizable. Namely, there exists a nowhere vanishing vector field ˜ X of ˜ M suchthat R . ˜ X = ˜ D . Let ˜ Z = (cid:107) ˜ X (cid:107) ˜ X , where the norm is with the metric (cid:104) , (cid:105) (cid:48) . Then, theflow φ t , associated to ˜ Z , is by isometries. So ˜ Z is a Killing field. Then (cid:104)∇ . ˜ Z, . (cid:105) (cid:48) is skew-symmetric. So, in particular, (cid:104)∇ v ˜ Z, v (cid:105) (cid:48) = 0, for any vector v that lies in˜ U . But, if ˜ A ˜ Z is the shape operator of the orbit H.x , x ∈ ˜ M , then (cid:104) ˜ A ˜ Z v, v (cid:105) (cid:48) = (cid:104)∇ v ˜ Z, v (cid:105) (cid:48) = 0. Then ˜ U = ˜ D ⊥ is an autoparallel distribution. The distribution ˜ D isalso autoparallel, since the Killing fields induced by H are always perpendicular toit. But two complementary perpendicular autoparallel distribution must be parallel.Then, by the de Rham decomposition theorem, ˜ M is a Riemannian product. Sinceone of the parallel distributions is one dimensional then ˜ M = R × M (cid:48) . (cid:3) Remark . In this paper, for dealing with homogeneous submanifolds of dimension3, we need to know which are the compact Lie groups G of dimension at most 4.For the sake of self-completeness we will briefly show, without using classificationresults, 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: acodimension subgroup, of a Lie group with a bi-invariant metric, must be a normalsubgroup.(i) dim( G ) ≤ G must be abelian. (ii) dim( G ) = 3.If rank( G ) ≥ G must be abelian. If rank( G ) = 1,then G is, up to a cover, Spin(3). This well-known result follows from a topologicalargument that proves that a rank 1 simply connected compact group is isomorphicto Spin(3) (a proof can be found in Remark 2.6 of [OR]). ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 29 (iii) dim( G ) = 4.If G is neither simple nor abelian, then, from the previous cases, we have that afinite cover of G splits as S × Spin(3).If G is simple then rank( G ) ≤
2. Otherwise G would have a codimension 1 (abelian)subgroup (which must be normal).If G is simple, then rank( G ) >
1. Otherwise, G = Spin(3) which has dimension 3.Let G be simple and rank( G ) = 2. Then the Ad -representation of G on g = Lie( G )must have a focal (non-trivial) orbit G.v . Such an orbit must have codimension3. The 3-dimensional normal space ν v ( G.v ) is Lie triple system, since it coincideswith the commutator of v . Then ν v ( G.v ) is an ideal of g . A contradiction. Remark . Let X = G/K be an irreducible simply connected symmetric space ofthe non-compact type and rank at least 2, where G is the connected component ofthe full isometry group of X . Assume that the dimension of X is at most 5. Then, X (cid:39) 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 K on T p X hasa non-trivial focal orbit M = K.v ( p = [ e ]). Such an orbit M must have dimension2. In fact, M cannot have dimension 3. Otherwise, a principal K -orbit must havedimension 4 and so K would act transitively on the sphere. Observe also that thedimension of M cannot be 1. In fact, since K acts irreducibly on T p X , then K actseffectively on any non trivial orbit. If dim( M ) = 1, then dim( K ) = 1. Then, sincedim( X ) > K does not act irreducibly on T p X . A contradiction.Observe that the isotropy K v of the focal orbit M = K.v at v must havepositive dimension (and so dim( K v ) = 1). Moreover, since M is not a principalorbit, the image under the slice representation of K v is not trivial. So, by Corollary2.5, the restricted normal holonomy group Φ( v ) of M at v is not trivial. ThenΦ( v ) must act irreducibly on the 2-dimensional space ¯ ν v ( M ) = { v } ⊥ ∩ ν v ( M ).Observe that the codimension of M is 3 = M is a Veronese submanifold, i.e. orthogonally equivalent to a Veronese-type orbit V of SO(3) on Sim (3) (the action is by conjugation). So, may assume that Sim (3) = T p X and that M = V . Then both K and SO(3) are Lie subgroups of˜ K = { g ∈ SO(
Sim (3)) : g.M = M } . Observe that ˜ K is not transitive on the unitsphere of Sim (3) since the codimension of M is 3. Let R (cid:48) and R be the curvaturetensors at p = [ e ] of X and SL(3) / SO(3). Then we have the following irreduciblenon-transitive holonomy systems: [
Sim (3) , R, ˜ K ] and [ Sim (3) , R (cid:48) , ˜ K ].Then by the holonomy theorem of Simons 2.12, R is unique up to scalar multipleand ˜ K = K = SO(3), since its Lie algebra is spaned by R . This implies that thesymmetric space X is homothetical to SL(3) / SO(3).
Remark . Let M = K.v ⊂ R N be a 3-dimensional full and irreducible homo-geneous (Euclidean) submanifold. Assume that rank( M ) ≥
2. In this case, by therank rigidity theorem, M is an orbit of an s -representation. So, we may assume,that K -acts as an s -representation.Let ξ be a K -invariant parallel normal field to M which is not umbilical. Ifthe shape operator A ξ has two different (constant) eigenvalues then its associatedeigendistributions, let us say E and E are autoparallel distributions that areinvariant under the shape operators of M (recall that A ξ commutes with any other shape operator due to Ricci equality). Then, by the so-called Moore’s lemma [BCO],Lemma 2.7.1, M is product of submanifolds. A contradiction.If A ξ has three eigenvalues, then the multiplicities of any of them are 1. Since A ξ commutes with any other shape operator, all shape operators of M must commute.Then, by the Ricci identity, M has flat normal bundle. Then M is isoparametric,since it is an orbit of an s -representation.Therefore, a full irreducible and homogeneous Euclidean 3-dimensional subman-ifold M , of higher rank, must be isoparametric with exactly three curvature nor-mals. This implies that the irreducible Coxeter group associated to M [Te, PT] hasexactly three reflection hyperplanes. This is only possible if the dimension of thenormal space is 2. Otherwise, the curvature normals must be mutually perpendic-ular and hence M would be a product of circles.This implies that N = 5 and that M is an isoparametric hypersurface of thesphere S . Moreover, from Remark 4.6, M is a principal orbit of the isotropyrepresentation of Sl(3) / SO(3). (cid:3)
Proof of Theorem A. If M n is a (full) Veronese submanifold, n ≥
3, thenthe normal holonomy, as a submanifold of the sphere, acts irreducibly and non-transitively (see Facts 2.16, (iii)).For the converse observe that M must be a full and irreducible Euclidean sub-manifold, since the normal holonomy group (as a submanifold of the sphere) actsirreducibly (see the beginning of Section 3). Then, from Theorem 3.4, Theorem 4.1and Proposition 2.17, M is a Veronese submanifold. (cid:3) Proof of Theorem B.
From Theorem 4.1 M is an orbit of an s -representation.Assume that rank( M ) = 1. Then, by Lemma 4.2, the (restricted) normal holonomygroup of M , as a submanifold of the sphere, acts irreducibly and N = 9 = 3+ M is a Veronese submanifold.If M is of higher rank, then, by Remark 4.7, M is a principal orbit of the isotropyrepresentation of Sl(3) / SO(3). (cid:3) 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 nowthe situation is much more simple, for n = 3. Proof of Theorem C.
Observe that M must be full and irreducible as a Euclideansubmanifold (since the normal holonomy group, as a submanifold of the sphere,acts irreducibly; see Section 3). Note, by the minimality, that the traceless shapeopertator coincides with the shape operator (of vectors which are perpendicular tothe position vector).We keep the notation in the proof of Theorem 3.4.Let p ∈ M be such that the adapted normal curvature tensor R ⊥ ( p ) (cid:54) = 0,or equivalently, R ⊥ ( p ) (cid:54) = 0. Let us consider the irreducible and non-transitiveholonomy systems [¯ ν p ( M ) , R ⊥ ( p ) , Φ( p )] and [ Sim ( T p M ) , R, SO( T p M )].We have, from formula (****) of Section 3 and Proposition 2.21, that the shapeoperator at p , A p : ¯ ν p ( M ) → Sim ( T p M ) is a homothecy and A p Φ( p )( A p ) − = ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 31
SO( T p M ). This implies, if φ ∈ Φ( p ), that the eigenvalues of A pη coincide with theeigenvalues of A pφ ( η ) .Let U be a contractible neighbourhood of p in M such that R ⊥ never vanisheson U .Let now p (cid:48) ∈ U be arbitrary and let γ : [0 , → U be a piece-wise differentiablecurve from p to p (cid:48) . Let τ t be the ∇ ⊥ -parallel transport along γ [0 ,t ] .We have that τ t Φ( p )( τ t ) − = Φ( γ ( t )).Let us choose ξ ∈ ¯ ν p ( M ) such that A pξ ∈ Sim ( T p M ) has exactly two eigenvalues λ = of multiplicity 2 and λ = − n − of multiplicity ( n − A γ ( t ) : ¯ ν γ ( t ) → Sim ( T γ ( t ) ) maps Φ( γ ( t )) intoSO( T γ ( t ) ). Then, the homothecy g t := A γ ( t ) ◦ τ t ◦ ( A p ) − : Sim ( T p M ) → Sim ( T γ ( t ) M )maps the group SO( T p M ) into SO( T γ ( t ) M ). Then g t maps the isotropy subgroupSO( T p M ) A pξ (cid:39) S (O(2) × O( n − T γ ( t ) M ) B ( t ) ,where B ( t ) = A γ ( t ) τ t ( ξ ) . This implies, as it is not difficult to see, that B ( t ) has twoeigenvalues, let us say λ t of multiplicity 2 and λ t of multiplicity n −
2. Since B ( t ) ∈ Sim ( T γ ( t ) ), λ t = − n − λ t .Then the two eigenvalues of B ( t ) are constant up to the multiplication by a ( t ) = λ t (cid:54) = 0. Note, if γ is a loop by p , that τ ∈ Φ( p ). Then, as we have previouslyobserved, the eigenvalues of A pξ are the same as those of B (1). Then a ( t ) dependsonly on γ ( t ). So there is a non-vanishing f : U → R such that a ( t ) = f ( γ ( t )). It isstandard to show that f must be C ∞ . Note that f ( p ) = .Let us consider (eventually, by making U smaller) the holonomy tube U ξ . Weuse the notation in the proof to Theorem 3.4. We will modify the arguments inthis proof.We have the parallel normal field ˆ ξ of U ξ . The eigenvalues of the shape operatorˆ A ˆ ξ at q ∈ U ξ are given by ˆ λ ( q ) = f ( π ( q ))1 − f ( π ( q ))associated to the (horizontal) eigendistribution E of dimension 2ˆ λ ( q ) = − f ( π ( q )) n − f ( π ( q )) n − associated to the (horizontal) eigendistribution E of dimension n − A ˆ ξ , is ˆ λ = −
1, associated to the vertical distribution ν ,tangent to the normal holonomy orbits.By the Dupin condition, d(ˆ λ )( E ) = 0 which implies thatd( f ◦ π )( E ) = 0 (J)If n > E , since it has dimension at least2. 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), d π ( E ( q )) = E ( q ), as linear subspaces. Moreover, E ( q ) is an eigenspaceof A q − π ( q ) = A ˆ ξ ( q ) , where A is the shape operator of M (we drop the supra-index π ( q ) of A ). Let now q ∈ U ξ with π ( q ) = p and let V be the subspace of T p M which is generated by E ( q (cid:48) ), with q (cid:48) ∈ Φ( p ) .q = ( π − ( { p } )) q . If V = T p M , then, from formula (J), d f ( T p M ) = { } . If V is properly contained in T p M , thenlet 0 (cid:54) = v ∈ V ⊥ . We will derive, in this case, a contradiction. In fact, sinceany shape operator A q (cid:48) − p has only two eigenvalues and v is perpendicular to theeigenspace E ( q (cid:48) ) of A q (cid:48) − p , then v is an eigenvector of this shape operator, for any q (cid:48) ∈ Φ( p ) .q . Observe that the linear span of Φ( p ) .q is ¯ ν p ( M ), since q (cid:48) (cid:54) = 0 and Φ( p )acts irreducibly on this normal space. Then v is a common eigenvector for all shapeoperators A η , η ∈ ν p ( M ). But A : ¯ ν p ( M ) → Sim ( T p M ) is an isomorphism. Thisis a contradiction. Then d f ( T p M ) = { } and the same is valid for all p (cid:48) ∈ U . Then f = f ( p ) = is constant on U .Then the eigenvalues ˆ λ , ˆ λ , ˆ λ are constant on U ξ . Then ˆ ξ is a (non-umbilical)parallel normal isoparametric field of U ξ . Then, by [CO] (see [BCO], Theorem5.5.2), U ξ and hence U has constant principal curvatures. But this is true providedone shows that U ξ is full and locally irreducible around some point q ∈ π − ( { p } ).Let us show that the local normal holonomy group of M at p coincides with the re-stricted normal holonomy group. In fact, the holonomy system [¯ ν p ( 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 p coincides with Φ( p ).Then the local rank of M , as submanifold of the Euclidean space, is 1. This impliesthat M is full and locally irreducible around p . Hence U ξ is full and irreduciblearound any point q ∈ π − ( { p } ). Then U is a submanifold with constant principalcurvatures.Since the normal holonomy of M is not transitive on the unit sphere, of thenormal 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], U is locally an orbit of an s -representation. Then (cid:107)R ⊥ (cid:107) is constanton U . From this one obtains that (cid:107)R ⊥ (cid:107) is constant on Ω, where Ω is a connectedcomponent of the open subset { p ∈ M : R ⊥ ( p ) (cid:54) = 0 } . But if p (cid:48) ∈ M is a limitpoint of Ω then, R ⊥ ( p (cid:48) ) (cid:54) = 0. This implies that Ω can be enlarged unless p (cid:48) ∈ Ω.This shows that the open subset Ω is also closed in M . Then M has constantprincipal curvatures. Hence, the image of M (under the isometric immersion), isan embedded submanifold with constant principal curvatures. Moreover, it is anorbit of an s -representation. From Proposition 2.17, the image of M is a Veronesesubmanifold.The converse is true by Facts 2.16, (i) and (iii). (cid:3) Remark . 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 ∈ M be such that R ⊥ ( p ) (cid:54) = 0. Then, since the shape operator A maps Φ( p )-orbits into SO( T p M )-orbits of Sim ( T p M ), one obtains that the second fundamental form is λ -isotropic.That is, the length of α ( X, X ) is λ ( p ) independent of X in the unit sphere of T p M , where α is the second fundamental form. The function λ must be a constantmultiple of f . Then, by Proposition 4.1. of [IO], λ , and hence f , must be constant( n ≥ ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 33 The number of factors of the normal holonomy
In this section we will prove a sharp linear bound, depending on the dimension n of the submanifold, of the number of irreducible factors of the local normal holo-nomy representations. This improves, substantially, the quadratic bound n ( n − Proposition 6.1.
Let M n be a submanifold of the Euclidean space R N . Assumethat at any point of M the local normal holonomy group and the restricted normalholonomy group coincide (or, equivalently, the dimensions of the local normal holo-nomy groups are constant on M ). Let p ∈ M and let r be the number of irreducible(non-abelian) subspaces of the representation of the restricted normal holonomygroup Φ( p ) on ν p ( M ) . Then r ≤ n . Moreover, this bound is sharp for all n ∈ N (also in the class of irreducible submanifolds).Proof. Let us decompose ν p ( M ) = ν p ( M ) ⊕ ν p ( M ) ⊕ ... ⊕ ν rp ( M ), where Φ( p ) actstrivially on ν p ( M ) and irreducibly on ν ip ( M ), for i = 1 , ..., r . From the assumptionswe obtain that ν ip ( M ) extends to a ∇ ⊥ -parallel subbundle ν i of the normal bundle ν ( M ), i = 0 , ..., r (eventually, by making M smaller around p ). Note that we havethe decomposition ν ( M ) = ν ( M ) ⊕ ν ( M ) ⊕ ... ⊕ ν r ( M ). Moreover, we obtainfrom the assumptions, for any q ∈ M , that the local normal holonomy group Φ( q )acts trivially on ν q ( M ) and irreducibly on ν iq ( M ), for any i = 1 , ..., r .Let R ⊥ ξ,ξ (cid:48) be the adapted normal curvature tensor (see Section 1.1). From theexpression of R ⊥ in terms of shape operators A , one has that R ⊥ ξ,ξ (cid:48) = 0 if and onlyif [ A ξ , A ξ (cid:48) ] = 0.Observe, if i (cid:54) = j , that R ⊥ ξ i ,ξ (cid:48) j = 0 if ξ i , ξ (cid:48) j are normal sections that lie in ν i ( M )and ν j ( M ), respectively.There must exist q ∈ M , arbitrary close to p , such that R ⊥ ν iq ,ν iq (cid:54) = { } , for all i = 1 , ..., r . In fact, there exists q ∈ M , arbitrary close to p such that R ⊥ ν q ,ν q (cid:54) = { } (otherwise, ν ( M ) would be flat). The above inequality must be true in aneighbourhood V of q . Now choose q ∈ V such that R ⊥ ν q ,ν q (cid:54) = { } . Continuingwith this procedure we find q := q r with the desired properties.Let us show that for any i = 1 , ...., r there exist ξ i , ξ (cid:48) i en ν iq ( M ) such that [ A ξ i , A ξ (cid:48) i ]does not belong to the algebra of endomorphisms generated by { A η i } , where η i ∈ ν q ( M ) has no component in ν iq ( M ). In fact, if this is not true, then, for any ξ i , ξ (cid:48) i in ν iq ( M ), [ A ξ i , A ξ (cid:48) i ] commutes with A ξ i (since the shape operators of elements ofthe subspaces ν jq ( M ) commute with con A ξ i , if j (cid:54) = i ). Then (cid:104) [[ A ξ i , A ξ (cid:48) i ] , A ξ i ] , A ξ (cid:48) i (cid:105) = 0 = −(cid:104) [ A ξ i , A ξ (cid:48) i ] , [ A ξ i , A ξ (cid:48) i ] (cid:105) and hence [ A ξ i , A ξ (cid:48) i ] = 0. A contradiction, since R ⊥ ν iq ,ν iq (cid:54) = { } . This proves ourassertion.Observe that [ A ξ , A ξ (cid:48) ] , ..., [ A ξ r , A ξ (cid:48) r ] are linearly independent and commutingskew-symmetric endomorphisms of T q M . Then r ≤ rank (SO( T p M )) = [ n ] (theinteger part of n ). This proves the inequality.Let us see that it is sharp. For M ⊂ S k − , ¯ M ⊂ S k − be a surface and a 3-dimensional submanifold and such that the normal holonomies have one irreduciblefactor (for example, the Veronese V and V ). Let n > n = 2 d is n iseven or n = 2 d + 3 if n is odd. Let M n be the product of d times M or M n be the product of d times M by ¯ M . Such submanifolds are contained in the product of Euclidean ambientspaces. Moreover, the number of irreducible factors of the normal holonomy group(representation) of M n is exactly the upper bound [ n ]. Moreover, since M n iscontained in a sphere, we can apply to M n a conformal transformation of thesphere (the normal holonomy group is a conformal invariant) in such a way that M n is an irreducible (Riemannian) submanifold of the Euclidean space. (cid:3) Further comments
Remark . There is a beautiful result of Little and Phol [LP] which characterizesVeronese submanifolds M n , modulo projective diffeomorphisms, by the two-pieceproperty and the fact that the codimension is the maximal one n ( n + 1) (forsubmanifolds with the two-piece property). Note that a tight submanifold has thetwo-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 . A natural question that arises, since the normal holonomy group is aconformal invariant, is the following: is a compact submanifold M n ⊂ S n − n ( n +1) ,with irreducible and non-transitive (restricted) normal holonomy, equivalent, mod-ulo conformal transformations of the sphere, to a Veronese submanifold?.Remark . The symmetric space X = SU(4) / SO(4), dual to Sl(4) / SO(4), isisometric to the Grassmannian SO(6) / SO(3) × SO(3). In this last model, T [ e ] X = R × and the isotropy representation is given by ( g, h ) .T = gT h − , ( g, h ) ∈ SO(3) × SO(3). The Veronese submanifold V is given bySO(3) × SO(3) .Id = SO(3) × { Id } .Id = SO(3) ⊂ R × Thus V is also an orbit of the smaller group SO(3) (cid:39) SO(3) × { Id } . The otherorbits SO(3) .A , where A is invertible and near Id , must be full and irreduciblesubmanifolds of R × , since V is so. Note that the action of SO(3) on R × isreducible. In fact, it is the sum of three times the standard representation of SO(3)on R . The orbit, SO(3) .A is not minimal in the sphere, for A generic. So, thenormal 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 r A − of R × , r A − ( T ) = T A − , transformsSO(3) .A into V . In particular, since V is 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]). ORMAL HOLONOMY OF ORBITS AND VERONESE SUBMANIFOLDS 35 Appendix
The Veronese embedding.
We recall here some basic definitions and facts about the well-known Veronesesubmanifolds.Let S n , n ≥
2, be the unit sphere of the Euclidean space R n +1 and let R n +1 ⊗ s R n +1 be space of symmetric 2-tensors of R n +1 . Let h : R n +1 ⊗ s R n +1 → Sim ( n + 1) theusual isomorphism onto the symmetric matrices of R n +1 . Namely, let e , ..., e n +1 be the canonical basis of R n +1 . Then, h ( e i ⊗ e j + e j ⊗ e i ) is the matrix whosecoefficients a k,l are all zero except: a i,j = a j,i = 1 , if i (cid:54) = j ; a i,i = 2 , if i = j The Veronese map Q : S n → Sim ( n + 1) is defined byQ( v ) = h ( v ⊗ v )Observe that (Q( v )) i,j = v i v j , where v = ( v , ..., v n +1 ). Let (cid:104) , (cid:105) be the innerproduct on Sim ( n + 1) given by (cid:104) A, B (cid:105) = trace ( AB ). Then Q is an isometricimmersion. Observe that trace (Q( v )) = 1, for all v ∈ S n . So, the image of Q iscontained in the affine hyperplane of Sim ( n + 1), given by the linear equation (cid:104) · , Id (cid:105) = 12Let ˜ ρ : S n → Sim ( n +1) be defined by ˜ ρ ( v ) = Q ( v ) − n +1 Id , where Sim ( n +1)are the symmetric traceless matrices. The map ˜ ρ is called the Veronese Riemannianimmersion of the sphere S n into Sim ( n + 1). One has that ˜ ρ , (as well as Q) isO( n + 1)-equivariant. Namely, if g ∈ O( n + 1), then˜ ρ ( g.v ) = g. ˜ ρ ( v ) .g − In fact, if we regard v ∈ R n +1 as a column vector, then˜ ρ ( v ) = v.v t − n + 1 Id From the above formula it follows easily the O( n + 1)-equivariance of ˜ ρ . Itis also not difficult to verify, as it is well known, that ˜ ρ ( v ) = ˜ ρ ( w ) if and only if w = ± v . Therefore, ˜ ρ projects down to an isometric O( n +1)-equivariant embedding ρ : R P n → Sim ( 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) ⊕ Sim ( 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 Sim ( n + 1). Then,the image of the Veronese embedding, is the orbit M = SO( n + 1) .S where S ∈ Sim ( n +1) is the diagonal matrix with exactly two eigenvalues. Namely,1 − n +1 and − n +1 . The first one, with multiplicity 1, is associated to the eigenspace R e and the second one, with multiplicity n , is associated to the eigenspace ( R e ) ⊥ .Let S (cid:48) ∈ Sim ( n +1) with exactly two eigenvalues λ of multiplicity 1 and λ withmultiplicity n . Assume that (cid:107) S (cid:48) (cid:107) = (cid:107) S (cid:107) (i.e. S and S (cid:48) have the same length). It iseasy to verify that either λ = 1 − n +1 , λ = − n +1 or λ = − n +1 , λ = n +1 . In the first case one has that S (cid:48) ∈ S n + 1) .S = ρ ( R P n ). In the second case, − S (cid:48) ∈ S n + 1) .S .Observe that S (cid:48) and − S (cid:48) cannot be both in the image of the Veronese embedding,since the respective eigenvalues of multiplicity 1 are different. In general, if ¯ S ∈ Sim ( n + 1) has two different eigenvalues, one of multiplicity 1 and the other ofmultiplicity n , then ¯ S = λS , for some 0 (cid:54) = λ ∈ R . The orbit SO( n + 1) . ¯ S iscalled a Veronese-type orbit (see Section 1.1). Observe that there are exactly twoVeronese-type orbits in any given sphere, centered at 0, of
Sim ( n + 1). Moreover,any of these two Veronese-type orbits is isometric to the other, via the isometry − Id Sim ( n +1) of Sim ( n + 1).We have the following well-known fact. Lemma 8.1.
Let SO ( r ) acts by conjugation on Sim ( r ) , the traceless symmetric r × r -matrices, and let M = SO ( r ) .A be an orbit, A (cid:54) = 0 . Then r − ≤ dim ( M ) .Moreover, the equality holds if and only if M is an orbit of Veronese-type.Proof. Let us assume that M has minimal dimension. We will first prove that A hasexactly two eigenvalues. If not, let λ , ..., λ d be the different eigenvalues of A withassociated eigenspaces E , ..., E d ( d ≥ r ) A =S(SO( E ) × ... × SO( E d )) has less dimension than S(SO( E ) × SO( E ⊕ ... ⊕ E d )),which is the isotropy group of some (cid:54) = A (cid:48) ∈ Sim ( r ) with two different eigen-values whose associated eigenspaces are E and E ⊕ ... ⊕ E d . Then dim( M ) > dim(SO( r ) .A (cid:48) ). A contradiction. Therefore, d = 2. (Observe that d = 1 impliesthat A = 0, since it is traceless).Let now k = dim( E ) and so r − k = dim( E ).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/ r/ , r ]. So, the minimum of q , restricted to the finite set { , ..., r − } is attained at both, x = 1 and x = r −
1. Then k = 1 or k = r −
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