Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces

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1 manuscripta math. 97, (1998) c Springer-Verlag 1998 Sergio Console Carlos Olmos Clifford systems, algebraically constant second fundamental form and isoparametric hypersurfaces Received: 22 April 1998 Abstract. In this paper we prove that a submanifold with parallel mean curvature of a space of constant curvature, whose second fundamental form has the same algebraic type as the one of a symmetric submanifold, is locally symmetric. As an application, using properties of Clifford systems, we give a short and alternative proof of a result of Cartan asserting that a compact isoparametric hypersurface of the sphere with three distinct principal curvatures is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes. Introduction Isoparametric hypersurfaces of spaces of constant curvature are one of the first examples of submanifolds of space forms characterized by simple geometrical invariants (in this case: the constancy of their principal curvatures). The notion of isoparametric hypersurface was extended to that of isoparametric submanifold and, more generally, of submanifold with constant principal curvatures [PT, HOT]. From Dadok s Theorem, it turns out that homogeneous isoparametric submanifolds are principal orbits of s-representations (that is to say, isotropy representations of semisimple symmetric spaces) [PT]. In a previous article [CO], the authors have shown that a locally irreducible submanifold M of the sphere admitting a non trivial parallel isoparametric normal section is a submanifold with constant principal curvatures (ξ is called an isoparametric section if the corresponding shape operator A ξ has constant eigenvalues). Moreover, if M is not an isoparametric hypersurface of the sphere, it is an orbit of an s-representation. In the same paper, the authors studied submanifolds of spaces of constant curvature with algebraically constant second fundamental form as an extrinsic version of curvature homogeneous manifolds [TV]. The second fundamental form α of a submanifold M of a space of constant curvature S. Console: Dipartimento di Matematica, Università di Torino, via Carlo Alberto, 10, I Turin, Italy. console@dm.unito.it C. Olmos: Fa.M.A.F., Universidad Nacional de Córdoba, Ciudad universitaria, 5000 Córdoba, Argentina. olmos@mate.uncor.edu Mathematics Subject Classification (1991): 53C40

2 336 S. Console, C. Olmos Xκ N is said to be algebraically constant if, for any p, q M there exists an isometry F of the ambient space Xκ N such that F(p) = q,f pt p M = T q M, and Fp α q = α p (i.e. the algebraic type of the second fundamental form is independent of the point). Observe that both homogeneous submanifolds and isoparametric submanifolds have algebraically constant second fundamental form. In particular, using the above result, the authors obtained the following corollary [CO]: Corollary. Let M be a locally irreducible full submanifold of S N with non zero parallel mean curvature (i.e., M is not minimal in S N ). Assume furthermore that the second fundamental form is algebraically constant. Then M is a submanifold with constant principal curvatures. Moreover, if M is not an (isoparametric) hypersurface, it is an orbit of an s-representation. This implies, in particular, that a homogeneous submanifold of the sphere with non vanishing parallel mean curvature is an orbit of an s-representation [Ol]. Since any compact Lie subgroup of SO(n) has a minimal orbit in the sphere[hl], the assumption that the mean curvature is not zero cannot be deleted. The goal of the present article is twofold. First of all, we extend the above corollary to the case of minimal immersions, making of course further assumptions. Theorem. Let M n Xκ N be a submanifold with parallel (possibly vanishing) mean curvature and suppose that its second fundamental form is algebraically constant and it is the same as that of an extrinsic symmetric submanifold M n Xκ N. Then M has parallel second fundamental form (so, up to a rigid motion of the ambient space, M is equivalent to an open part of M). Observe that, in case Xκ N = SN, if the extrinsic symmetric submanifold is irreducible, then the minimality of M is automatic, since irreducible symmetric submanifolds are minimal [Fe]. Note also that the above generalizes a uniquess theorem for immersions with parallel second fundamental form due to Reckziegel [Re]: if two immersions with parallel second fundamental form at some point have the same tangent space and second fundamental form, then they (locally) coincide. Then, we use this Theorem to give a short and alternative proof of the following well known result of Cartan [Ca], which can be phrased as follows Cartan s Theorem. Let M be a compact isoparametric hypersurface of the sphere S 3m+1 with three distinct principal curvatures. Then M is a tube around the Veronese embedding of the real, complex, quaternionic or Cayley projective planes (in particular M is homogeneous and so an orbit of a rank two irreducible s-representation).

3 Clifford systems 337 Our proof relates Clifford systems and isoparametric hypersurfaces with three distinct principal curvatures. It turns out that the homogeneity of these hypersurfaces can be read off from properties of these systems. We remark that in the case of four distinct principal curvatures there is also a connection, though in different way, with Clifford systems [FKM]. We would like to point out that Karcher [Ka] and Knarr and Kramer [KK] have also given, with very different approaches, alternative proofs of Cartan s Theorem. While the approach of Karcher simplifies arguments of Cartan and Münzner [Mü], using polynomials and division algebras, the one of Knarr and Kramer uses topology and incidence geometry. 1. Proof of the Theorem Since M has the same second fundamental form as a (given) symmetric submanifold M, the second fundamental form of M has constant length and Mis a semi-symmetric submanifold (cf. [Me]). The latter means that R xy α = 0, (1.1) where R xy acts (as a derivation) in the tangent space as the Riemannian curvature tensor and in the normal space as the normal curvature tensor. (Observe that, due to the Gauss and Ricci equations, both the Riemannian and the normal curvature tensors are determined by the shape operator.) By the Codazzi equations, the Ricci formula 2 xy α 2 yx α = R xy α, and (1.1), it follows that xy 2 α is symmetric in all its four entries. By [CdCK, formula (3.12)], we have 1 2 α 2 = α 2 + α, α, where is the Laplace Beltrami operator defined by trace 2, and the norm and inner products are the usual ones of tensors. So we have, since α has constant length, α 2 + α, α =0. Thus it sufficies to show that α = 0. But, if (e m ) is a local orthonormal frame of M and x,y are tangent vectors to M, using the symmetry of 2 α, we have ( α)(x, y) = ( e 2 m e m α)(x, y) = ( xy 2 α)(e m,e m )= xy 2 H = 0, m m since the mean curvature H is parallel.

4 338 S. Console, C. Olmos 2. Isoparametric hypersurfaces with three distinct principal curvatures Let M 3m be a complete isoparametric hypersurface of the sphere S 3m+1 such that the shape operator has three distinct (constant) eigenvalues. We will always regard M as a codimension two submanifold of Euclidean space. Let n 1,n 2 and n 3 be the three distinct curvature normals, E 1,E 2 and E 3 the associated eigendistributions and S 1 (q), S 2 (q) and S 3 (q) the corresponding curvature spheres (i.e., the leaves at some q M of E 1,E 2 and E 3 ). Associated to M there is a Weyl group generated by the reflections with respect to the three focal lines, acting on the normal space [PT]. Since the number of reflections is odd, this Weyl group acts transitively on the reflection lines. This implies that the multiplicities of the three eigenvalues are all the same, i.e., the dimensions of the curvature spheres S 1,S 2,S 3 equal m. Let M i = M ni / n i 2, i = 1, 2, 3 be the focal parallel manifolds. We recall (cf. [HOT]) that the shape operators  and A of M and M i respectively are related by the following tube formula A η =  η H [(id  ni / n i 2 (q)) H ] 1, (2.1) where η ν q (M) ν q (M i ), q = q + n i / n i 2 (q) and H is the horizontal distribution in the submersion p i : M M i (i.e., the sum of the two eigendistributions different from E i ). Recall that ν q (M i ) = p p 1 i ( q) ν p(m), see [PT]. For the sake of simplicity we will set i = 1 in the sequel. Let q M be fixed and V be the affine subspace generated by S 1 (q). Each ξ V can be regarded as a normal vector to M 1 at q = q + n 1 / n 1 2 (q). From the tube formula (2.1), one has that the shape operator of M 1 at q has two constant eigenvalues both of multiplicities m, for any ξ S 1 (q). The linear map onv defined by trace A ξ is clearly constant on the sphere S 1 (q), so it vanishes. Remark. It is well known that the focal manifolds are minimal in the sphere. This can be seen as follows. The focal manifolds have parallel mean curvature, because they have constant principal curvatures. If the mean curvature would not be trivial, since it is an isoparametric section, we could find a more focal manifold. Observe also that, by the minimality of the focal manifold M 1, one gets that V is perpendicular to the position vector at q, since the trace of the shape operator (in the Euclidean space) is null on V. Let now A be the linear map from V to the space of traceless symmetric endomorphisms of T q M 1 defined by ξ A ξ.

5 Clifford systems 339 Note that, for any ξ S 1 (q), A ξ is constant, since the eigenvalues of A ξ are constant on S 1 (q) (here, is endowed with the usual inner product given by A, B =trace(a B)). Thus A is a homothetic map (with constant equal to λ r 1 2m, where λ>0 and λ are the eigenvalues of the shape operator on S 1 (q) and r 1 = 1/ n 1 is the radius of S 1 (q)). Observe that A has to be one-to-one and that, since all shape operators are conjugated and have two eigenvalues with the same multiplicities m, the image of A is contained into the canonically embedded Grassmannian G = SO(2m)A ξ0 SO(2m) 1 of m-planes in R 2m, where ξ 0 is a fixed element of S 1 (q). Moreover, A(S 1 (q)) is a totally geodesic m-sphere in a sphere of radius λ 2m in. So, A(S 1 (q)) is also a totally geodesic submanifold of the canonically embedded Grassmannian G (which is contained in the above sphere). Next we analyse the algebraic structure of the family of shape operators {A ξ } ξ S1 (q), proving that one can construct a Clifford system. Clifford systems. Let ξ be in the m-sphere S 1 (q) V.The shape operator at ξ has two distinct eigenvalues, λ and λ, with the same multiplicities m. Let ξ 0,...,ξ m S 1 (q) be pairwise orthogonal to each other and consider the symmetric endomorphisms of the tangent space of M 1 at q. P 0 = 1 λ A ξ 0,...,P m = 1 λ A ξ m. We will show that {P 0,...,P m }is a Clifford system, i.e. (1) P i P i = id, (2) P i P j +P j P i = 0, i j. The first is clear. So, we have also, since all eigenvalues are constant on S 1 (q), that ( ) P i + 2 P j = id. Hence 1 ( P 2 2 i + P i P j + P j P i + Pj 2 ) = id, which implies (2). Now, given an irreducible Clifford system Q 0,...,Q n on R 2δ, δ is not arbitrary. It depends on n according to following rule (see [LM, pp. 33] or also [FKM]) n: n+8... δ(n): δ(n)... Since in our case we have δ(m) = m, if the Clifford system P 0,...,P m is irreducible, m can only be 1, 2, 4 or 8. If P 0,...,P m is reducible, then

6 340 S. Console, C. Olmos P i = (Pi 1,P2 1 i ),i = 0,...,m, where Pi, acts on R2δ 1 and Pi 2 acts on R 2δ 2 (δ = δ 1 + δ 2 ) and the system {Pi 1 } is irreducible. Then, from the table, we would have m>δ 1 =δ(m) m, which is absurd. So, we have shown that the multiplicities of the eigenvalues can only be 1,2,4or8. Observe that for all these multiplicities there exists a homogeneous example of isoparametric hypersurface of the sphere with three distinct principal curvatures. Namely, these examples are tubes around the standard embeddings (also called the Veronese embeddings) of the real, complex, quaternionic and Cayley projective planes which are achieved as singular orbits of the isotropy representations of the following symmetric spaces: SU(3)/SO(3), SU(3) = SU(3) SU(3)/ (SU(3)), SU(6)/Sp(3), E 6 /F 4. Recall that the focal manifolds of these hypersurfaces are extrinsic symmetric. Proof of Cartan s Theorem. Let M be an isoparametric hypersurface of S 3m+1 with three distinct principal curvatures and multiplicities m = 1, 2, 4 or 8 and q M. Let M be a homogeneous example with the same multiplicities as M and let x M. Since both M and M have the same Weyl group (and using that the multiplicities are the same) we may assume, perhaps by passing to a parallel manifold, that q = x, the tangent and normal spaces coincide, S i (q) = S i (x), i = 1, 2, 3 and so, the corresponding affine spaces V and V coincide. Thus, if q is the centre of S i (q), q belongs to both focal manifolds M i and M i and both tangent spaces at q coincide. Let ξ 0 S 1 (q). Then, the shape operators A ξ0 and Ā ξ0 of M 1 and M 1 have the same eigenvalues. This is because all this information can be read off from the curvature normals and the tube formula (2.1). Then by what we have previously remarked, A(S 1 (q)) and Ā( S 1 (q)) are both totally geodesic spheres in the canonically embedded Grassmannian G. By [He, Theorem 11.1, pp. 334] any two totally geodesic spheres in a compact irreducible symmetric space are conjugated. So, by the equivariance of the embedding, there exists an orthogonal transformation h of the tangent space of the focal manifold such that A(S 1 (q)) = hā( S 1 (q))h 1, which yields A V = hā V h 1. Observe that the conjugation by h induces an isometry between the linear spaces A V and Ā V. Since both A and Ā : V are homothetic maps with the same constant, there must exist a linear isometry g of V such that Ā ξ = ha gξ h 1 for any ξ V. Note that, in our case, the algebraic form of the second fundamental forms of the focal manifolds are completely determined by the set of shape operators at vectors inv. Thus we get that the second fundamental forms of the focal

7 Clifford systems 341 manifolds of M and M have the same algebraic type. Using the Theorem, we are done, since the focal manifolds of M are extrinsic symmetric. Remark. The considered immersion into the extrinsic Grassmannian G shows the well known fact that two Clifford systems, with the same number of elements, acting on the same vector space, are geometrically equivalent (cf. [FKM, pp. 483] and [LM]). Observe moreover that the property of being the three multiplicities equal can also be read off from the fact that in a Clifford system the number of positive eigenvalues is equal to the number of the negative ones. Acknowledgements. We wish to thank Jürgen Berndt for the helpful discussions and Linus Kramer for the many valuable comments. S. Console was supported by Universidad Nacional de Córdoba, and partially supported by MURST. C. Olmos was supported by Universidad Nacional de Córdoba, CONICET and CNR of Italy; he was also partially supported by CONICOR. S. Console and C. Olmos would like to thank the FaMAF of the Universidad Nacional de Córdoba and the Department of Mathematics of the University of Turin, respectively, for the hospitality received during their visits to Córdoba and Turin. References [Ca] Cartan, É.: Sur des familles remarquables d hypersurfaces isoparamétriques dans les éspaces spheriques. Math. Z. 45, (1939) [CdCK] Chern, S.M., do Carmo, S., Kobayashi, S.: Minimal Submanifolds of a Sphere with second fundamental form of constant Length. In: Functional Analysis and Related Fields, Springer, Berlin, 1970, pp [CO] Console, S., Olmos, C.: Submanifolds of higher rank. Quart. J. Math. Oxford (2) 48, (1997) [Fe] Ferus, D.: Symmetric submanifolds of Euclidean space. Math. Ann. 247, (1980) [FKM] Ferus, D., Karcher, H. and Münzner, H.F.: Clifford Algebren und neue isoparametriche Hyperflächen. Math. Z. 177, (1981) [HOT] Heintze, E., Olmos, C. and Thorbergsson, G.: Submanifolds with constant principal curvatures and normal holonomy groups. Int. J. Math. 2, (1991) [He] Helgason, S.: Differential Geometry, Lie groups and symmetric spaces. Academic Press, New York, 1978 [HL] Hsiang W.-Y. and Lawson, B.H.: Minimal submanifolds of low cohomogeneity. J. Differ. Geom. 5, 1-38 (1971) [Ka] Karcher, H.: A geometric classification of positively curved symmetric spaces and the isoparametric construction of the Cayley plane. Astérisque , (1988) [KK] Knarr, N., Kramer, L.: Projective Planes and Isoparametric Hypersurfaces. Geometriae Dedicata 58, (1995) [LM] Lawson, B.H. and Michelson, M.: Spin Geometry. Princeton University Press, Princeton, N.J., 1990 [Me] Mercuri, F.: Parallel and semi-parallel immersions into space forms. Riv. Mat. Univ. Parma 17, (1991) [Mu] Münzner, H.F.: Isoparametrische Hyperflächen in Sphären, I, II. Math. Ann. 251, (1980); 256, (1981) [Ol] Olmos, C.: Homogeneous submanifolds of higher rank and parallel mean curvature. J. Differ. Geom. 39, (1994)

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