On homogeneous composed Clifford foliations
aa r X i v : . [ m a t h . DG ] M a r ON HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS
CLAUDIO GORODSKI AND MARCO RADESCHI
Abstract.
We complete the classification, initiated by the second named au-thor, of homogeneous singular Riemannian foliations of spheres that are liftsof foliations produced from Clifford systems.
A singular Riemannian foliation of a Riemannian manifold M is, roughly speak-ing, a partition F of M into connected complete submanifolds, not necessarily ofthe same dimension, that locally stay at a constant distance one from another.Singular Riemannian foliations of round spheres ( S n , F ) are of special importancesince, other than producing submanifolds with interesting geometrical properties,they provide local models around a point of general singular Riemannian foliations.The special case of singular Riemannian foliations in spheres whose leaves ofmaximal dimension have codimension one is better known as the case of isopara-metric foliations , and its study dates back to ´E. Cartan, who showed the existenceof a number of non-trivial examples. However, his examples were all homogeneous ,i.e., given as orbits of isometric group actions on S n . The first inhomogeneous ex-amples were found much later by Ozeki and Takeuchi [OT75]. A while later, Ferus,Karcher and M¨unzner [FKM81] developed an algebraic framework based on Cliffordalgebras (or, equivalently, Clifford systems, see subsection 1.2) to construct a largefamily of examples of isoparametric foliations (so called of FKM-type ), includingmany inhomogeneous examples, and completely classified the homogeneous ones.Whereas the theory and classification of isoparametric foliations of spheres areby now rather well understood, the situation of singular Riemannian foliations inhigher codimensions is still largely terra incognita . In [Rad14], inspired by theideas in [FKM81], two new classes of foliations were introduced. Namely, the classof
Clifford foliations , and the class of composed foliations which properly containsthe first one. A Clifford foliation ( S n , F C ) is constructed from a Clifford system C , and a composed foliation ( S n , F ◦ F C ) is constructed from C and a singularRiemannian foliation ( S m , F ) of a lower dimensional sphere. The natural questionof determining which ones are homogeneous was also solved in [Rad14], with theexception of composed foliations based on Clifford systems of type C , and C , (seesubsection 1.2). The goal of the present work is to deal with these two remaining,more involved cases. Main Theorem.
Let ( S n , F ◦ F C ) be a homogeneous composed foliation, witheither C = C , or C = C , . • If C = C , , then n = 15 , and there are exactly six examples of homogeneousfoliations ( S , F ◦ F C , ) , listed in Tables 1 and 2. • If C = C , , then n = 31 , and the only homogeneous foliation ( S , F ◦F C , ) is the isoparametric one induced by the action of Spin(10) on S viathe spin representation. In this case, m = 9 and the corresponding foliation ( S , F ) consists of one leaf. There is a general idea that it should be possible to recover many geometric prop-erties of the singular Riemannian foliations from the geometry of the underlying
Date : July 13, 2018. leaf (or quotient) space, compare e.g. [Lyt10, LT10, Wie14, GL14a, GL14b, GL15,AR15]. In this regard, it was shown in [Rad14] that Clifford foliations are character-ized as those singular Riemannian foliations of spheres whose leaf spaces isometricto either a sphere or a hemi-sphere of constant curvature 4. More generally, it wasbelieved that any foliation whose leaf space has constant curvature 4 should be acomposed foliation. Our result shows that this belief is now dismissed. Comparingour Main Theorem with [Str94, Table II] and [GL14a, Table 1], we observe thatthere are exactly two homogeneous foliations on S whose quotient space has con-stant sectional curvature 4 and which are not composed, namely, those given bythe orbits of Spin(9) and Spin(9) · SO(2) actions on S with quotient a quarter of around sphere S and an eighth of a round sphere S , respectively. Togetherwith results in [Rad14, GL14a] this implies: Corollary 1.
The foliations given by the
Spin(9) and
Spin(9) · SO(2) actions on S are the only homogeneous foliations of spheres whose leaf space has constantcurvature and which are not composed. The case of composed foliations ( S , F ◦ F C , ) is also very interesting, as theycoincide with those foliations that contain the fibers of the octonionic Hopf fibration S → S . Based on the fact that the Cayley projective plane O P is the mappingcone of S → S , it was shown in [Rad14] that there corresponds to any singularRiemannian foliation ( S , F ) a singular Riemannian foliation ( O P , ˜ F ) which ishomogeneous if and only if F ◦ F C , is homogeneous. It thus follows from ourMain Theorem that there is a large amount of inhomogeneous foliations of O P : Corollary 2.
The foliation ( O P , ˜ F ) is inhomogeneous for any foliation ( S , F ) except for those six (homogeneous) examples listed in Tables 1 and 2. About this paper: after a section on preliminaries, we first consider the case offoliations with closed leaves and treat the cases C , and C , in separate sections,as they have very different features. The short, last section is devoted to foliationswith non-closed leaves.It is our pleasure to thank Alexander Lytchak for several very informative dis-cussions as well as for his hospitality during our stay at the University of Cologne.1. Preliminaries
In this section, we quickly review some definitions and results from [Rad14].1.1.
Singular Riemannian foliations.
Definition . Let M be a Riemannian manifold, and F a partition of M intocomplete, connected, injectively immersed submanifolds, called leaves . The pair( M, F ) is called: • A singular foliation if there is a family of smooth vector fields { X i } thatspans the tangent space of the leaves at each point. • A transnormal system if any geodesic starting perpendicular to a leaf staysperpendicular to all the leaves it meets. Such geodesics are called horizontalgeodesics . • A singular Riemannian foliation if it is both a singular foliation and atransnormal system.Given a singular foliation ( M, F ), the space of leaves , denoted by M/ F , is theset of leaves of F endowed with the topology induced by the canonical projection π : M → M/ F that sends a point p ∈ M to the leaf L p ∈ F containing it. If inaddition the leaves of F are closed, then M/ F inherits the structure of a Hausdorffmetric space, by declaring the distance d ( π ( p ) , π ( q )) to be equal to the distance N HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS 3 d ( L p , L q ) in M between the corresponding leaves. Moreover, M/ F is stratifiedby smooth Riemannian manifolds, and the projection π is global submetry, and aRiemannian submersion along each stratum.1.2. Clifford systems and Clifford foliations.
A Clifford system, denoted by C ,is an ( m + 1)-tuple C = ( P , . . . , P m ) of symmetric transformations of a Euclideanvector space V such that P i = Id for all i , P i P j = − P j P i for all i = j .Two Clifford systems ( P , . . . , P m ), ( Q , . . . , Q m ) are called geometrically equiva-lent if there exists an element A ∈ O( V ) such that ( AP A − , . . . , AP m A − ) and( Q , . . . , Q m ) span the same subspace in Sym ( V ). Geometric equivalence classesof Clifford systems are completely classified, and: • A Clifford system { P , . . . , P m } on V exists if and only if dim V = 2 kδ ( m ),where k is a positive integer and δ ( m ) is given by: m nδ ( m ) 1 2 4 4 8 8 8 8 16 δ ( n )Given integers m , k , we denote by C m,k any Clifford system consisting of m + 1 symmetric matrices on a vector space of dimension 2 kδ ( m ). • If m C m,k , for any fixed k . • If m ≡ (cid:4) k (cid:5) + 1 equivalence classes of Cliffordsystems of type C m,k . They are distinguished by the invariant | tr( P P · · · P m ) | .Given a Clifford system C = ( P , . . . , P m ) on R l , l = kδ ( m ), we can define amap π C : S l − ⊂ R l −→ R m +1 x ( h P x, x i , . . . , h P m x, x i ) . The
Clifford foliation ( S l − , F C ) associated to C is given by the preimages of themap π C . This foliation is a singular Riemannian foliation, it only depends on thegeometric equivalence class of C , and its quotient is isometric to either a roundsphere S m if l = m , or a round hemisphere S m +1+ if l ≥ m + 1.1.3. Composed foliations.
Fix a Clifford system C = C m,k = ( P , . . . , P m )with associated Clifford foliation ( S n , F C ), and fix a singular Riemannian folia-tion ( S m , F ). Alternatively, we can view F as: a foliation of the boundary ofthe leaf space of F C , namely ∂ ( S n / F C ) = ∂ ( S m +1+ ), in case l ≥ m + 1; and afoliation of S m in case l = m . Such a foliation can be extended by homotheties toa foliation ( S m +1+ , F h ). The composed foliation ( S n , F ◦ F C ) is then defined bytaking the π C -preimages of the leaves of F h .Given any Clifford system C = C m,k and any singular Riemannian foliation( S m , F ), the composed foliation ( S n , F ◦ F C ) is a singular Riemannian foliation.1.4. Homogeneous composed foliations.
Recall that a singular Riemannian fo-liation ( M, F ) is called homogeneous if its leaves are orbits of an isometric Lie groupaction G → Isom( M ). In [Rad14] appears a complete classification of homogeneousClifford foliation and a partial classification of composed foliations: Theorem 1.2 ([Rad14]) . Let C = C m,k = ( P , . . . , P m ) be a Clifford system on R l and let ( S m , F ) be a singular Riemannian foliation. Then: CLAUDIO GORODSKI AND MARCO RADESCHI (1) The Clifford foliation ( S l − , F C ) is homogeneous if and only if m = 1 , or m = 4 and P P · · · P = ± Id , in which cases it is respectively spannedby the orbits of the diagonal action of SO( k ) on R k × R k ( m = 1 ), SU( k ) on C k × C k ( m = 2 ) or Sp( k ) on H k × H k ( m =4).(2) If C = C , , C , then ( S l − , F ◦F C ) is homogeneous if and only if both F and F C are homogeneous. If C = C , and ( S l − , F ◦F C ) is homogeneous,then F is homogeneous. By the classification of Clifford systems, both C , and C , consist of a uniquegeometric equivalence class of Clifford systems. Moreover, for C = C , the corre-sponding Clifford foliation ( S , F C ) is given by the fibers of the octonionic Hopffibration S → S , while for C = C , the Clifford foliation ( S , F C ) is given bythe fibers of π C : S → S .2. The case C = C , In this section we will show that there are no new examples of homogeneouscomposed foliations originating from the Clifford system C = C , . More precisely,we will see that a composed foliation ( S , F ◦F C ) is homogeneous if and only if F is the codimension one foliation of S consisting of concentric 9-spheres; recall thatin that case, the composed foliation is the isoparametric foliation ˜ F C of FKM-typegiven by the orbits of the spin representation Spin(10) → SO(32) [FKM81]. Recallalso that the maximal connected Lie subgroup of SO(32) whose orbits coincide withthe leaves of ˜ F C is Spin(10) · U(1) = Spin(10) × Z U(1) [Dad85, EH99].In this section we will only consider closed Lie subgroups of SO(32), which cor-respond to proper isometric actions on S , and postpone the case of non-closedLie subgroups to section 4. So suppose the leaves of F ◦ F C are orbits of a closedconnected Lie subgroup G of SO(32). Since F ◦ F C is contained in ˜ F C , i.e. theleaves of F ◦ F C are contained in those of ˜ F C , G preserves each leaf of ˜ F C . By theabove maximality property, G ⊂ Spin(10) · U(1).
Lemma 2.1.
The foliation ( S , F ) induced by G ⊂ Spin(10) · U(1) is of the form F ◦ F C if and only if F C is contained in F .Proof. The only if part is clear. Suppose now that the orbits of G contain the leavesof F C . Any element in Spin(10) · U(1) preserves the submanifold M + ⊂ S definedas the preimage of the north pole of S / F C = S , and therefore so does G . Since G acts by isometries, the projection of any G -orbit to the quotient S is eitherentirely contained in the interior of S or entirely contained in the boundary. Itfollows that for every leaf L of F , the restriction ( L, F C | L ) is a regular foliation,and its quotient L/ F C ⊂ S is a submanifold. The partition { L/ F C } L ∈F is easilyseen to form a singular Riemannian foliation F h on S with the north pole as a0-dimensional leaf and, by the Homothetic Transformation Lemma (see e.g. [Rad12,Lemma 1.1]), this foliation is determined by its restriction F on the boundary S .By definition of composed foliation, F is of the form F ◦ F C . (cid:3) It follows from Lemma 2.1 that we need only consider maximal connected closedsubgroups of Spin(10) · U(1).The orbital geometry of the spin representation Spin(10) → SO(32) (or its exten-sion to Spin(10) · U(1)) is well understood. The orbit space S / Spin(10) is isometricto an interval of length π/
4, where the endpoints parametrize singular orbits M + , M − of dimensions 21 and 24 (cf. [HPT88, p. 436]; see also [Bry, pp. 8-9] for a moreelementary discussion). The orbit M + is particularly interesting, as it is also a leaf of F ◦F C for any homogeneous foliation F of S , namely, the π C -fiber over the ori-gin of S . As a homogeneous space, M + ∼ = Spin(10) / SU(5) ∼ = Spin(10) · U(1) / U(5)
N HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS 5 (this also follows from the fact that M + is the orbit of a highest weight vector ofthe spin representation). Since G is transitive on M + , we must have dim G ≥ · U(1) are, up to conjugacy,Spin(10) , U(5) · U(1) , Spin(10 − k ) · Spin( k ) · U(1)for k = 1 , . . . ,
5, and ρ ( H ) · U(1)where H is simple and ρ is irreducible of real type and degree 10 (cf. [Dyn00];see also [KP03, Prop. 8]). We have already remarked that Spin(10) is an orbitequivalent subgroup of Spin(10) · U(1); we shall not need to discuss its subgroups,because they are subgroups of the other maximal subgroups of Spin(10) · U(1). Inthe sequel, we first analyse which of the other maximal subgroups of Spin(10) · U(1)can act transitively on M + .The group U(5) · U(1) cannot act transitively on M + since its semisimple partSU(5) is coincides with an isotropy subgroup of Spin(10) on M + .The simply-connected compact connected simple Lie groups H of rank at most 5and dimension between 20 and 44 are Spin(7), Spin(8), Spin(9), Sp(3), Sp(4), SU(5)and SU(6); none admits irreducible representations of real type and degree 10.In order to determine if the groups Spin(10 − k ) · Spin( k ) · U(1) can act transitivelyon M + , one can compute the intersection of the Lie algebra so (10 − k ) ⊕ so ( k ) withthe so (10)-isotropy subalgebra su (5). It does not matter that the subalgebras aredefined only up to conjugacy (corresponding to the fact that one can choose adifferent basepoint in M + ). We view su (5) inside so (10) as consisting of matricesof the form (cid:18) A B − B A (cid:19) where A , B are real 5 × A is skew-symmetric, B is symmetric of tracezero. A standard choice of embedding of so (10 − k ) ⊕ so ( k ) into so (10) is given bymatrices of the form (cid:18) C D (cid:19) where C , D are skew-symmetric (10 − k ) × (10 − k ), resp. k × k , matrix blocks.Then their intersection is isomorphic to su (5 − k ) ⊕ so ( k ). Therefore the dimensionof the Spin(10 − k ) · Spin( k )-orbit through the basepoint is 21 − k ( k − for k ≤ k = 5. We deduce that Spin(9) and Spin(9) · U(1) act transitively on M + ;besides those, only Spin(8) · SO(2) · U(1) has a chance of acting transitively on thatmanifold. In order to discard the latter group, we choose a different embeddingof so (8) ⊕ so (2) into so (10), namely, that in which the ( i, j )-entry is zero if i ∈{ , , , , , , , } and j ∈ { , } or j ∈ { , , , , , , , } and i ∈ { , } . Now( so (8) ⊕ so (2)) ∩ su (5) ∼ = s ( u (4) ⊕ u (1)) and the corresponding Spin(8) · SO(2)-orbithas dimension 29 −
16 = 13, showing that Spin(8) · SO(2) · U(1) is not transitiveon M + .Finally, we need to show that the Spin(9) · U(1)-orbits cannot coincide with theleaves of F ◦ F C for any F . Suppose the contrary for some F . Since π C : S → S is equivariant with respect to the double covering Spin(10) → SO(10), wesee that SO(9) preserves the leaves of F . We already know that F is homoge-neous (Theorem 1.2(2)), and SO(9) is a maximal connected subgroup of SO(10).Therefore F must be given by the orbits of SO(9). It follows that the leaf spaceof F ◦ F C is S / SO(9), which is isometric to S . On the other hand, thequotient space S / Spin(9) · U(1) is one-eighth of a round sphere S [Str94, Ta-ble II, Type III ]. We reach a contradition and deduce that ( S , F ◦ F C ) cannotbe homogeneous under Spin(9) · U(1).
CLAUDIO GORODSKI AND MARCO RADESCHI
Remark . Let ( S , F ) denote the homogeneous foliation given by the orbitsof SO(9), and let ( S , F ◦ F C ) the corresponding composed foliation. By theresult above, F ◦ F C is not homogeneous and, in particular, it is different from thehomogeneous foliation induced by the orbits of Spin(9) · U(1). Nevertheless, bothfoliations have cohomogeneity 2, and both have quotients of constant curvature 4.Moreover, they both contain the homogeneous foliation induced by the orbits ofSpin(9). Since S / Spin(9) = S , the orbits of Spin(9) have codimension 1 inthe leaves of F ◦ F C , which makes F ◦ F C very close to a homogeneous foliation.3. The case C = C , In this section, we determine the list of homogeneous composed foliations origi-nating from the Clifford system C = C , . Namely, we determine the orbit equiva-lence classes of the isometric group actions that yield such foliations. In this sectionwe only consider closed subgroups of SO(16) and defer the analysis of non-closedLie subgroups to section 4. The foliation F C is given by the fibers of the inhomo-geneous octonionic Hopf fibration S → S . Fix a singular Riemannian foliation( S , F ), and suppose that F ◦ F C is homogeneous, given by the orbits of a closedconnected subgroup G of SO(16). Recall that if X denotes the leaf space X = S / F then the orbit space S /G is isometric to X . In particular, the sectional curvatureof (the regular part of) S /G is everywhere ≥ G cannot act polarly,unless it acts with cohomogeneity 1.3.1. Criteria to recognize composed foliations.
Before we start the classifi-cation in detail, we want to present some results that will be helpful to identifyfoliations that can be written as F ◦ F C , where C = C , . We start with thestraightforward remark that a foliation F can be written in the form F ◦ F C if andonly if every fiber of the Hopf fibration S → S is contained in a leaf of F (compareLemma 2.1). In particular, if F is a homogeneous composed foliation induced bythe action of a group G ⊂ SO(16), then any other group G with G ⊂ G ⊂ SO(16)will also generate a homogeneous composed foliation.As a special case of the above situation, which will be useful later on, supposethat ( S , F ◦ F C ) is homogeneous given by the orbits of G ⊂ SO(16), and supposethat ( S , F ) is homogeneous given by the orbits of H ⊂ SO(9). Then for any group H ⊂ SO(9) containing H , there is a canonical enlargement G ⊂ SO(16) of G whoseorbits yield a composed folation, as follows. Since the Hopf fibration S → S isequivariant with respect to the covering map Spin(9) → SO(9), we can lift H to agroup ˜ H ⊂ Spin(9) ⊂ SO(16). Now G is defined as the closure of the subgroup inSO(16) generated by G and ˜ H . By the discussion above, the orbits of G define ahomogeneous composed foliation on S .Next we prove a criterion to distinguish some foliations that cannot be writtenin the form F ◦ F C . Proposition 3.1.
Let ( S , F ) denote the homogeneous, codimension 1 foliationgiven by the orbits of Sp(2) · Sp(2) , under the representation ν ˆ ⊗ H ν ∗ . Then anyfoliation ( S , F ) which is contained in F (i.e., every leaf of F is contained in aleaf of F ) cannot be written in the form F ◦ F C , .Proof. If F could be written as F ◦ F C , , by the remarks above, so could F .Therefore it is enough to prove the proposition for F and, to do so, it is enoughto provide a leaf of F that cannot be foliated by totally geodesic 7-spheres. Wethus consider the singular orbit M + containing the point Id ∈ Hom R ( H , H ) ∼ = H ⊗ H H ∗ , which is diffeomorphic to Sp(2). N HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS 7
Suppose now that M + = Sp(2) is foliated by totally geodesic S . Then the leavesare all simply connected, which implies that there is no leaf holonomy, and thusthe quotient M + / F is a manifold B and M + → B is a Riemannian submersionwith totally geodesic fibers. Then it is also a fibration, and from the long exactsequence in homotopy, B is simply connected (and 3-dimensional). Therefore itmust be B = S , and we have a fibration S → Sp(2) → S . Again from the longexact sequence in homotopy, we have π (Sp(2)) −→ π ( S ) −→ π ( S ) = 0However, on the one hand π (Sp(2)) = 0 (for example, cf. [MT64]), and on theother π ( S ) = 0, which gives a contradiction. (cid:3) As an application of Proposition 3.1 above, consider the Clifford foliation F ¯ C gen-erated by ¯ C = ( P , . . . , P ) with P P P P P = ± Id . This foliation is homogeneousand given by the orbits of the diagonal action of Sp(2) on H ⊕ H (Theorem 1.2)and thus, by Proposition 3.1 above, it cannot be written as F ◦ F C , . In fact, thisis the only Clifford foliation of S with this property: Proposition 3.2.
For any Clifford system C ′ on R with C ′ = ¯ C , the foliation F C ′ can be written in the form F C ′ = F ◦ F C , , for some foliation ( S , F ) .Proof. Let C , = ( P , . . . , P ) and, for every i = 1 , . . . ,
7, let C i denote the sub-Clifford system ( P , . . . , P i ). Since F C i is given by the preimages of the map π C i : S → R i +1 ,π C i ( x ) = (cid:0) h P x, x i , . . . , h P i x, x i (cid:1) , it is clear that π C i factors as π i ◦ π C , , where π C , : S → S is the Hopf fibration,and π i : S ⊂ R → D i +1 ⊂ R i +1 is the projection onto the first i + 1 components.In particular, F C i can be written as F ◦ F C , , where ( S , F ) is given by thefibers of π i . Notice that F in this case is homogeneous and given by the orbits ofSO(8 − i ), embedded in SO(9) as the lower diagonal block.Moreover, any Clifford system C ′ = C m,k on R must satisfy the equation kδ ( m ) = 8, and the only possibilities are( m, k ) = (8 , , (7 , , (6 , , (5 , , (4 , , (3 , , (2 , , (1 , . For any m C m,k can be identified with the sub-Clifford system C m ⊂ C , . For m ≡ ⌊ k ⌋ + 1 geometrically distinct Cliffordsystems of type C m,k . Therefore, there is a unique C , , and two distinct classes oftype C , . One of them is C ⊂ C , , which is composed by the discussion above,and the other is ¯ C . Since this exhausts all possible Clifford systems on R , itfollows that all of them are composed, with the exception of ¯ C . (cid:3) Gathering all the information together, we obtain the following
Corollary 3.3.
A composed foliation ( S , F ◦ F C m,k ) can also be written as F ′ ◦F C , for some ( S , F ′ ) , if and only if C m,k = ¯ C .Proof. If C m,k = ¯ C then, by Proposition 3.2, F C m,k can be written as F ′ ◦ F C , and, by the initial remark, the same holds for F ◦ F C m,k since it contains F C m,k .On the other hand, any composed foliation F ◦ F ¯ C is contained in the foliation F ◦ F ¯ C where ( S , F ) is the trivial foliation with one leaf. Since F ◦ F ¯ C coincideswith the foliation F of Proposition 3.1, F ◦ F C cannot be written as F ′ ◦ F C , forany ( S , F ′ ). (cid:3) CLAUDIO GORODSKI AND MARCO RADESCHI
We can now proceed with the classification of composed foliations of S homo-geneous under a closed Lie group G . The diameter of X = S / F is either equalto π , or it is at most π/
2. We will consider these two cases separately.3.2.
Case I: diam X = π . Suppose first that the diameter of X is π . Then thereis a copy of S ⊂ S consisting of 0-dimensional leaves, and ( S , F ) decomposes asa spherical join ( S , F ) = S ⋆ ( S , F ) , for some foliation F . In particular, X is isometric to a spherical join S ⋆ Y , where Y = S / F . In this case, X has diameter π/ G acts reducibly) and itcontains two points x + , x − at distance π/
2. Moreover, any unit speed geodesic in X starting from x − meets x + at the same time t = π/
2. Therefore the preimages S ± of x ± are orthogonal round spheres of curvature 1, i.e., they are the unit spheresof subspaces V ± of R such that R = V + ⊕ V − . Since we are assuming that the G -orbits contain the fibers of the Hopf fibration, it must be dim S ± ≥
7. Thereforeequality must hold, and dim V + = dim V − = 8. Moreover, G acts transitively on S ± . Given p ∈ S + , the isotropy G p acts on the unit sphere in the normal space ν p S + , which is isometric to S − via the map v exp p π v . Moreover, the foliation( S − , G p ) coincides with the infinitesimal foliation of F at π C ( p ) ∈ S , which inturn coincides with ( S , F ). In particular, F is homogeneous and given by theaction of G p on R = R ⊕ V − given by ǫ ⊕ λ | G p , where ǫ : G → R is the trivialrepresentation, and λ : G → SO(8) denotes the representation of G on V − (or V + ). Remark . Since the infinitesimal foliation of F ◦F C at any point of S − coincideswith the infinitesimal foliation at a point in S + (because they both coincide with( S , F )), the slice representations at S + and S − must be orbit equivalent.If the G action on S ± is not effective, then the kernels K ± of G → SO( V ± ) arenormal subgroups of G with K + ∩ K − = { e } . Since G is compact, it admits anormal subgroup L such that G = K + · L · K − , where K + · L acts effectively on S − and L · K − acts effectively on S + . Let k + , k − , l denote the Lie algebras of K + , K − , L respectively. From the list of all groups acting transitively on the 7-sphere, weget the following possibilities:1. k + = k − = 0. Then G = L up to a finite cover, and the possible such representa-tions are: Type G G → SO(16)I.1 SO(8) ρ ⊕ ρ I.2 SU(4) µ ⊕ µ I.3 U(4) µ ⊕ µ II.1 Spin(8) ∆ +8 ⊕ ∆ − II.2 Spin(7) ∆ ⊕ ∆ II.3 SU(4) · U(1) µ ˆ ⊗ ( µ r ⊕ µ s ) ( r = s )III.1 Sp(2) ν ⊕ ν III.2 Sp(2) · Sp(1) ( ν ˆ ⊗ ν ) ⊕ III.3 Sp(2) · U(1) ν ˆ ⊗ ( µ r ⊕ µ s )The actions of type I induce the Clifford foliations F C , and F C , respectively(actions I.2 and I.3 are orbit equivalent) and, by Proposition 3.2, they indeed canbe written as F ◦ F C , . Therefore, the same is true for the foliations coming fromactions of type II, since each of them contains a foliation of type I. On the otherhand, the foliations of type III are containted in the orbits of the representationof Sp(2) · Sp(2) given by ν ˆ ⊗ H ν ∗ , and therefore are not of the form F ◦ F C , by Proposition 3.1. Therefore, the homogeneus composed foliations in this caseare given by the orbits of the groups listed in Table 1, where we have put together N HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS 9 orbit equivalent actions. As we have seen, the foliation ( S , F ) is also homogeneous,given by the orbits of the isotropy group H of G at a certain point. G for F ◦ F C G → SO(16) H for F H → SO(9) X Spin(8) ∆ +8 ⊕ ∆ − Spin(7) ǫ ⊕ ∆ [0 , π ]SO(8) ρ ⊕ ρ SO(7) ǫ ⊕ ρ S Spin(7) ∆ ⊕ ∆ G ǫ ⊕ φ SU(4) µ ⊕ µ SU(3) ǫ ⊕ µ S U(4) µ ⊕ µ U(3) ǫ ⊕ µ SU(4) · U(1) µ ˆ ⊗ ( µ r ⊕ µ s ) ( r = s ) U(3) · U(1) ǫ ⊕ µ r − s ⊕ µ ⊗ µ − s S Table 1. diam X = π , and k + = k − = 0. Remark . (a) Any pair of equivalent or inequivalent 8-dimensional irreduciblerepresentations of Spin(8) could occur in the table, but some are not listed sincethey differ from the two listed by an outer automorphism of Spin(8). In particularthose representations are not only orbit equivalent to a representation in the list,but their image in SO(16) is the same as the image of a representation in the list.2. l = 0. Then G = K + · K − , and each K ± acts transitively on S . All these casesare orbit equivalent among themselves, and also to the first entry in Table 1, so weget no new examples.3. l = 0 and k + = 0. Since L · K + is a nontrivial product, and it acts effectively andtransitively on S − , it must be L · K + ∈ { Sp(2) · Sp(1) , SU(4) · U(1) , Sp(2) · U(1) } . If L = Sp(2), then the foliation is contained in the foliation of Proposition 3.1,so it is not composed.If L = SU(4) then K + = U(1), and K − can be either U(1) or trivial. Then G isgiven by U(1) · SU(4) · U(1), resp., U(1) · SU(4), and it acts via ( µ ˆ ⊗ µ ) ⊕ ( µ ˆ ⊗ µ ),resp., ( µ ˆ ⊗ µ ) ⊕ µ . Those actions are orbit equivalent to the representation ofSU(4) · U(1) in Table 1 given by µ ˆ ⊗ ( µ r ⊕ µ s ) with r = s (including the case( r, s ) = (1 , L = Sp(1) or U(1), then K + , K − ∈ { Sp(2) , SU(4) } and the action hascohomogeneity 1, and they are all orbit equivalent to the first entry in Table 1.Hence we get no new examples in this case.3.3. Case II: diam X ≤ π/ . In this case the diameter of S /G = X is at most π/ G acts irreducibly. We distinguish between possible cases, accordingto the dimension of X .Suppose first that dim X = 1, i.e., F ◦ F C is an isoparametric family in S . Itfollows from the classification of cohomogenity 1 actions in spheres that the onlypossible actions on S with quotient of diameter ≤ π/ ν ˆ ⊗ ν ∗ for G = Sp(2) · Sp(2), µ ˆ ⊗ µ for G = S(U(2) · U(4)), and ρ ˆ ⊗ ρ for G = SO(2) · SO(8)(or SU(2) · SU(4) and SO(2) · Spin(7), which are orbit equivalent subgroups of G , G , resp.). By Proposition 3.1, the action of G is ruled out, but the othertwo actions give rise to composed foliations; in fact, those actions yield foliationscontaining foliations given in Table 1. Since G and G are contained in Spin(9),in each case they project to a subgroup H of SO(9) which generates a codimensionone isoparametric foliation F in S . We summarize the discussion above in thefollowing table:If 2 ≤ dim X ≤
4, then G acts irreducibly on S with cohomogeneity ≤
4, andthe action is not polar. From the classification of low cohomogeneity representations G for F ◦ F C G → SO(16) H for F H → SO(9) X SU(2) · SU(4) µ ˆ ⊗ C µ SO(3) · SO(6) ρ ˆ ⊕ ρ [0 , π/ · SO(8) ρ ˆ ⊗ R ρ SO(2) · SO(7) ρ ˆ ⊕ ρ [0 , π/ Table 2. diam X = π/ G must act on S with cohomogeneity 2,and there are exactly two possible actions, µ ˆ ⊗ C ν for G = U(2) · Sp(2), and S ( µ ) ˆ ⊗ H ν ∗ for G = SU(2) · Sp(2) [Str94, Table II]. Again these actions are ruledout by Proposition 3.1. In fact it is clear that G is contained in Sp(2) · Sp(2).As for G being contained in that group, note that the Sp(2)-representation C restricts to C ⊕ C ∗ along the embedding U(2) ⊂ Sp(2), so the result follows fromthe following representation theoretic lemma.
Lemma 3.6. If V and W are representations of complex, resp., quaternionic type,then ( V ⊕ V ∗ ) ⊗ H W ∗ is equivalent as a real representation to the realification of V ⊗ C W .Proof. The representations have equivalent complexifications. Indeed the complex-ification of the first representation is ( V ⊕ V ∗ ) ⊗ C W whereas that of the second is( V ⊗ C W ) ⊕ ( V ⊗ C W ) ∗ , where W ∼ = W ∗ over C . (cid:3) If dim X ≥
5, then the foliation ( S , F ) has leaves of dimension ≤ F cannot have dimension 1 o 2 (i.e., dim X = 6,7). In fact, in those cases F would have to be generated by a representation H ⊂ SO(9), where H = S or T . In particular, H would be contained in amaximal torus of SO(9), and every such maximal torus acts on S fixing at leasttwo antipodal points. In particular the diameter of X would be π which contradictsour assumption.We are thus left with the case in which ( S , F ) is homogeneous under a closedconnected subgroup H of SO(9) and dim X = 5. For the same reasons above, F cannot be generated by a T -action. The principal orbits are 3-dimensional witheffective (transitive) actions of H . Therefore a principal isotropy group H princ doesnot contains a normal subgroup of H , H princ is a subgroup of O(3), dim H ≤ H is locally isomorphic to SU(2) × SU(2). We deducethat H is one of SU(2), SU(2) × T , SU(2) × SU(2), up to cover.The only almost effective 9-dimensional representation of SU(2) × SU(2) withoutfixed directions is ρ ˆ ⊗ ρ , which has 6-dimensional principal orbits.Assume H = SU(2) × T and V is a 9-dimensional representation with coho-mogeneity 6 and no fixed directions. The identity component of H princ on V is acircle with non-trivial projection into SU(2). It follows that the only admissible ir-reducible components of V are (SU(2) , R ), (U(2) , C ), ( T , C ). Since 9 is odd, thefirst representation must occur exactly once. We get two possiblities: R ⊕ C ⊕ C and R ⊕ C ⊕ C ⊕ C . The first one has trivial principal isotropy groups, so it isexcluded. The second one can be extended to an action of H = SU(2) × T actingon S with cohomogeneity 3. If F ◦ F C were homogeneous, induced by some group G , then the extension H of H would induce an extension G of G that would act on S with cohomogeneity 3. This action would be non-polar and irreducible, howeverthere is no such group [GL14b, Table 1].The only 9-dimensional representations of H = SU(2) without fixed directionsare λ , µ ⊕ λ and ρ ⊕ ρ ⊕ ρ . N HOMOGENEOUS COMPOSED CLIFFORD FOLIATIONS 11
The representation ρ ⊕ ρ ⊕ ρ can be extended to an action of H = SO(3) via the outer sum ρ ˆ ⊕ ρ ˆ ⊕ ρ , acting on S with cohomogeneity 2. If F ◦ F C werehomogeneous, induced by some group G , then the extension H of H would inducean extension G of G , that would act on S , with quotient isometric to S andthree most singular orbits of dimension 9 (they would be preimages of most singular H orbits, of dimension 2). However, from the classification of non-polar irreducibleisometric actions of cohomogeneity 2 on S there is no such group [Str94], andtherefore F ◦ F C cannot be homogeneous in this case.The representation µ ⊕ λ can be extended to an action of H = SU(2) × SU(2)via the representation µ ˆ ⊕ λ , again acting on S with cohomogeneity 2. If F ◦ F C were homogeneous, induced by some group G , then the extension H of H wouldinduce an extension G of G , with quotient isometric to S /H = ( S /D ), where D denotes a dihedral group. The group G must then be Sp(1) · Sp(2) [Str94], whichis 13-dimensional and thus acts on S with finite principal isotropy. In particular G must act with finite principal isotropy as well, and since the cohomogeneity of H on S is 5, we have dim G = 10. However, a quick check shows that there areno 10-dimensional groups of rank at most 3 acting irreducibly (and non-polarly)on R . In particular in this case F ◦ F C cannot be homogeneous.The representation λ has isolated singular orbits, and therefore the quotient X has no boundary (compare [GL14b, § F ◦F C is homogeneous, given by the action of G on S . Since the quotient X has no boundary, there are no nontrivial reductions of ( G, R ), i.e., there are noother representations ( G ′ , R n ) with dim G ′ < dim G such that S n − /G ′ is isometricto S /G = X , cf. [GL14b, Prop. 5.2]. In particular, G must act with trivialprincipal isotropy, since otherwise we could produce a nontrivial reduction [GL14b,p. 2]. Since the principal isotropy is trivial and dim X = 5, again it must bedim G = 10. The only 10-dimensional group acting irreducibly (and non-polarly)on R is G = SU(2) × U(1) acting by ˆ ⊗ ( µ ) ˆ ⊗ µ ; however a pure tensor v ⊗ v ⊗ v has isotropy subgroup T , so this action has as an orbit of dimension 7. Since λ has no fixed points in S , this shows that the G -orbits cannot yield a foliation ofthe form F ◦ F C . 4. Non-proper actions
We treat the cases of C = C , and C = C , simultaneously. Suppose F ◦ F C is a homogeneous composed foliation of S , resp., S given by the orbits of a non-closed connected Lie subgroup G of SO(32), resp., SO(16). Then the closure of G isa closed connected subgroup whose orbits also comprise a homogeneous composedfoliation, so it is already described in sections 2 or 3. However, most of the groupstherein listed admit no dense non-closed connected Lie subgroups in view of thefollowing: Lemma 4.1.
A compact connected Lie group U with at most a one-dimensionalcenter admits no dense non-closed connected Lie subgroups.Proof. Suppose, to the contrary, that G is a dense connected proper Lie subgroupof U . If G is a normal subgroup of U , then either G is contained in the semisimplepart of U or it contains the center of U . Owing to [Rag72], normal subgroups ofsemisimple Lie groups are closed. It follows that G cannot be normal in U . Let N be the normalizer of G in U . This is a proper subgroup of U , thus cannot be closedby denseness of G . On the other hand, N must be closed in U because it coincideswith the normalizer in U of the Lie algebra of G (here we use connectedness of G ),a contradiction. (cid:3) The closed groups U yielding homogeneous composed foliations described insections 2 or 3 which do not satisfy the assumptions of Lemma 4.1 occur in case C = C , only and have two dimensional tori as centers, and they are of two types:1. U = K + · K − where K ± ∈ { SU(4) · U(1) , Sp(2) · U(1) } . In both cases there are dense connected Lie subgroups G which however yield orbitequivalent subactions.2. U = K + · L · K − where K ± = U(1), L = SU(4), and K + · L acts effectively on C ⊕ L · K − acts effectively on 0 ⊕ C . The non-closed dense connected Liesubgroups of U are of the form G = R × SU(4), where R is an irrational line inthe center T of U . Note that G and U share a common singular orbit through p ∈ C ⊕
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