Homogeneous and inhomogeneous isoparametric hypersurfaces in rank one symmetric spaces

HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES IN RANK ONE SYMMETRIC SPACES JOS´ E CARLOS D´ IAZ-RAMOS, MIGUEL DOM´ INGUEZ-V´ AZQUEZ, AND ALBERTO RODR´ IGUEZ-V´ AZQUEZ Abstract. We conclude the classification of cohomogeneity one actions on symmetric spaces of rank one by classifying cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. As a by-product of our proof, we produce uncountably many examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces. 1. Introduction Riemannian geometry, in a very broad sense, can be understood as the study of those properties of a smooth manifold that are invariant under isometries. Among Riemannian manifolds with large isometry groups, Riemannian symmetric spaces stand out as a class of their own, not only in Riemannian Geometry, but also in Lie group theory or Global Analysis. In this class, Euclidean spaces and symmetric spaces of rank one are the most popular in Riemannian geometry. It has been an interesting problem to study isometric actions on manifolds with large isometry groups, and several types of them have been investigated over the years. One of the most important families of isometric actions is that of cohomogeneity one, that is, proper isometric actions whose orbit space is one-dimensional, or in other words, whose principal orbits are hypersurfaces. Cohomogeneity one actions have recently been of great interest for the construction of geometric structures, such as Einstein metrics, Ricci solitons, special holonomy, or minimal hypersurfaces, among others. However, it is also a natural and important problem to find all cohomogeneity one actions on a given Riemannian manifold, usually just up to orbit equivalence. This is a classical problem in submanifold geometry that traces back to the time of ´ E. Cartan, and which turns out to be equivalent to the classification of homogeneous hypersurfaces up to isometric congruence. By both historical and mathematical reasons, it has frequently been linked to the investigation of the so-called isoparametric hypersurfaces. 2020 Mathematics Subject Classification. Primary 53C35, Secondary 57S20, 53C40. Key words and phrases. Isoparametric hypersurface, cohomogeneity one action, homogeneous submanifold, constant principal curvatures, symmetric space, quaternionic hyperbolic space, K¨ahler angle. The authors have been supported by the projects PID2019-105138GB-C21 (AEI/FEDER, Spain) and ED431C 2019/10, ED431F 2020/04 (Xunta de Galicia, Spain). The second and third authors acknowledge support of the Ram´on y Cajal program (Agencia Estatal de Investigaci´on, Spain) and the FPU program (Ministry of Education, Spain), respectively. 1 2 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ A hypersurface is called isoparametric if its nearby equidistant hypersurfaces have constant mean curvature. Thus, every homogeneous hypersurface is isoparametric. In the 30s, Cartan himself studied the converse implication. Since all the examples known to him (which included all isoparametric hypersurfaces in Euclidean and real hyperbolic spaces, and all isoparametric hypersurfaces with up to three principal curvatures in spheres) were homogeneous, he posed the question: is it true that an isoparametric hypersurface is homogeneous? A surprising negative answer to this question arrived almost forty years later, with the construction of the first examples of inhomogeneous isoparametric hypersurfaces in spheres by Ozeki and Takeuchi [32], soon generalized by Ferus, Karcher and M¨unzner [22]. These examples led to an added difficulty in the classification problem of isoparametric hypersurfaces in spheres, which has given rise to outstanding results over the last few years [12, 24, 30, 34, 13, 14]. Other inhomogeneous isoparametric hypersurfaces have been found in symmetric spaces such as complex and quaternionic projective spaces [20, 21] and in certain symmetric spaces of non-compact type, such as complex hyperbolic spaces [17], or more generally, the symmetric spaces with Dynkin diagram of (BCr)-type [15, 19]. However, none of these examples, unlike the ones in spheres, have constant principal curvatures, with only one remarkable exception: one inhomogeneous family of isoparametric hypersurfaces with constant principal curvatures in the Cayley hyperbolic plane [15]. The classification of cohomogeneity one actions up to orbit equivalence in Euclidean spaces follows from the classification of isoparametric hypersurfaces in Rnobtained by Segre [33]. In symmetric spaces of compact type and rank one, the corresponding classification follows from several works. In spheres it was obtained by Hsiang and Lawson [23], in complex projective spaces by Takagi [36], and in quaternionic projective spaces and the Cayley plane by Iwata [25, 26]. There is also a classification of cohomogeneity one actions on irreducible symmetric spaces of compact type due to Kollross [28]. The problem is more difficult in the non-compact case. The main reason is that, unlike in the compact setting, there are two main types (namely, reductive and parabolic) of maximal subgroups of the isometry group of a symmetric space of non-compact type, and parabolic subgroups contain many subgroups that act transitively on the space. Thus, the investigation of orbits of subgroups of a parabolic subgroup frequently leads to complicated linear algebra or combinatorial problems (in a certain sense similar, for example, to the ones arising in the outstanding classification problem of totally geodesic submanifolds [27]), for which very few ideas have been developed (cf. [4, 6, 16]). The first classification result of cohomogeneity one actions on a symmetric space of non-compact type was given by Cartan [11] for real hyperbolic spaces, while he was studying isoparametric hypersurfaces in spaces of constant curvature. However, the classification in complex hyperbolic spaces and the Cayley hyperbolic plane, due to Berndt and Tamaru [5], only arrived seventy years later. There are several structural results for symmetric spaces of non-compact type [3, 6], but a full classification is still not available, not even in quaternionic hyperbolic spaces. This is precisely the point where we start our study. The main aim of this article is to classify cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence. Our method relies partially on the ideas developed in [5], where it is proved that this classification can be reduced to a certain problem that we solve in this paper. HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 3 The first main result of this article can be stated in terms of quaternionic algebra. We denote by Hthe real division algebra of the quaternions, endowed with its standard complex structures i,jand k. Let Hnbe a right quaternionic vector space of dimension n. The compact symplectic group Sp(n) is the group of quaternionic matrices (acting on the left on Hn) that preserve the standard quaternionic bilinear form Pn i=1 ¯viwi, where v, w ∈Hn, and bar denotes conjugation. This bilinear form naturally induces an inner product in Hn that makes it isometric with R4n. By Jwe will denote the quaternionic structure of Hn, that is, the subspace of real endomorphisms of Hngenerated by the right multiplications by i, j and k, which can therefore be seen as the Lie algebra of Sp(1). We also consider the Lie group Sp(1)Sp(n) = Sp(1) ×Sp(n)/Z2, which acts on Hnas (q, A)·v=Avq−1. This is an important group in Differential Geometry, as it arises in Berger’s holonomy list, that is, the list of Lie groups which can be realized as the holonomy of irreducible, simply connected and non-locally symmetric Riemannian manifolds. Thus, a Riemannian manifold is called quaternionic K¨ahler if it has dimension 4n, is not Ricciflat, and its holonomy is isomorphic to a subgroup of Sp(1)Sp(n), n≥2. The simplest examples of symmetric, quaternionic K¨ahler spaces are the quaternionic projective spaces, and their non-compact duals, the quaternionic hyperbolic spaces. In any case, understanding algebraic properties linked to holonomy groups is a first fundamental step towards the study of more geometric questions, such as those related to curvature (e.g. the celebrated LeBrun-Salamon conjecture [29]) or submanifolds (e.g. the theory of calibrations [10]). Similarly, the problem of submanifold geometry that we address in this paper relies on a linear algebraic problem that we describe below. We say that a real subspace Vof Hnis protohomogeneous if there exists a connected Lie subgroup of Sp(1)Sp(n) that acts transitively on the unit sphere of V. A protohomogeneous subspace of Hnhas constant quaternionic K¨ahler angle, a concept that is central in our study and that we recall now. Let πVdenote the orthogonal projection onto a vector subspace V, and define PJ=πV◦J, where J∈J. We say that Vhas constant quaternionic K¨ahler angle (ϕ1, ϕ2, ϕ3), with ϕ1≤ϕ2≤ϕ3, if for any v∈Vthe symmetric bilinear form Lv:J×J→R, Lv(J, J0) = hPJv, PJ0vi, has eigenvalues cos2(ϕi)hv, vi,i∈ {1,2,3}. We point out here the fact that the bilinear forms Lv,v∈V, described above do not necessarily diagonalize simultaneously (although we can prove a priori that they do so for protohomogeneous subspaces of dimension greater or equal than 5, see Corollary 4.2, and by classification results for dimension different from 3). The first main result of this article is to classify, up to congruence by elements in Sp(1)Sp(n), protohomogeneous subspaces of Hn. We state here the moduli space of such subspaces of dimension kin Hnby presenting their possible quaternionic K¨ahler angles. In Theorem A, and in what follows, tdenotes disjoint union. 4 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ Theorem A. The moduli space Mk,n of non-zero protohomogeneous subspaces of dimension kin Hn, up to congruence in Sp(1)Sp(n), is described in the following table: Mk,n k≤n n < k ≤4n 3 4n 3< k ≤2n k > 2n k≡0 (mod 4) (R+ 4\R− 4)t(R− 4×Z2)S{(0, ϕ, ϕ)}ϕ∈[0,π 2]{(0,0,0)} k≡2 (mod 4) {(ϕ, π 2,π 2)}ϕ∈[0,π 2]{(0,π 2,π 2)} {(0,π 2,π 2)} ∅ k6= 3 odd {(π 2,π 2,π 2)} ∅ ∅ ∅ k= 3 (R+ 3\R− 3)t(R− 3×Z2)∅ {(ϕ, ϕ, π 2)}ϕ∈{0,π 3}{(0,0,π 2)} where Λ = {(ϕ1, ϕ2, ϕ3)∈[0, π/2]3:ϕ1≤ϕ2≤ϕ3}, and R+ 3={(ϕ, ϕ, π/2) ∈Λ : ϕ∈[0, π/2]}, R− 3={(ϕ, ϕ, π/2) ∈Λ : ϕ∈[π/3, π/2)}, R+ 4={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2)−cos(ϕ3)≤1}, R− 4={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2) + cos(ϕ3)≤1, ϕ36=π/2}, S={(ϕ1, ϕ2, ϕ3)∈Λ : cos(ϕ1) + cos(ϕ2) + εcos(ϕ3)=1,for ε= 1 or ε=−1}. This classification includes typical examples such as totally real subspaces (precisely those with quaternionic K¨ahler angle (π/2, π/2, π/2)), totally complex subspaces (with quaternionic K¨ahler angle (0, π/2, π/2)), quaternionic subspaces (with quaternionic K¨ahler angle (0,0,0)), subspaces of constant K¨ahler angle ϕ∈(0, π/2) inside a totally complex vector subspace (with quaternionic K¨ahler angle (ϕ, π/2, π/2)), complexifications of subspaces of constant K¨ahler angle ϕ∈(0, π/2) in a totally complex subspace (with quaternionic K¨ahler angle (0, ϕ, ϕ)), and (Im H)v,v∈Hn,v6= 0 (with quaternionic K¨ahler angle (0,0, π/2)). However, there are some other non-classical examples. Some of them were introduced in [15], but there are some others, which are basically presented and classified in Section 5. A basis of these subspaces can be calculated explicitly, but for R± 3and R± 4 its expression is rather long. See Proposition 5.3 for R± 3and Propositions 5.10 and 6.1 for R± 4to get further details. Furthermore, there are non-congruent subspaces of Hnwith the same K¨ahler angles. These correspond precisely to the intersections R+ 3∩R− 3=R− 3and R+ 4∩R− 4=R− 4. We point out here three main tools that have been essential to obtain this classification. First we use the classical generalization of the hairy ball theorem regarding the possible rank of continuous distributions on spheres [35] in order to reduce the classification problem of real subspaces of Hnwith constant quaternionic K¨ahler angle to subspaces of dimensions 3 and multiples of 4 (Section 3). Secondly, we provide a Lie theoretic argument relying on results by Borel [9] and Montgomery and Samelson [31] on groups acting effectively and transitively on spheres, to prove that, for subspaces of dimension greater or equal than 5, the maps Lvthat are used to define quaternionic K¨ahler angle diagonalize simultaneously (Corollary 4.2). In third place, using the previous results, we can show that a protohomogeneous subspace of dimension 4lis the sum of protohomogeneous subspaces HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 5 of dimension 4 with the same quaternionic K¨ahler angle (Section 4.2). All this reduces the classification of protohomogeneous subspaces to dimensions 3 and 4. At this stage, we actually obtain the more general classification of real subspaces of dimensions 3 and 4 with constant quaternionic K¨ahler angle. This is a (hard) problem of linear algebra that is solved in Section 5. The first consequence of Theorem A is the classification of cohomogeneity one actions on quaternionic hyperbolic spaces HHn+1 up to orbit equivalence. In fact, Berndt and Tamaru explained in [5] how to obtain this classification. Consider the symmetric pair (G, K) = (Sp(1, n + 1),Sp(1) ×Sp(n+ 1)) representing the symmetric space HHn+1. We denote by g=k⊕pthe corresponding Cartan decomposition, and let abe a maximal abelian subspace of p, which is one-dimensional because HHn+1 is of rank one. Let g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe the restricted root space decomposition of gwith respect to a. Then, gαis isomorphic to a quaternionic vector space Hnendowed with the standard quaternionic bilinear form, and K0∼ =Sp(1) ×Sp(n), the connected Lie subgroup of Gwhose Lie algebra is k0=g0∩k=Nk(a), normalizes gαand acts on gαin the canonical way. The classification of cohomogeneity one actions on HHn+1 can be obtained if we determine the protohomogeneous subspaces Vof gα∼ =Hn. If Vis such a protohomogeneous subspace, we define the Lie subalgebra sV=a⊕(gαV)⊕g2αof g, and denote by SV the connected Lie subgroup of Gwith Lie algebra sV(throughout this article denotes the orthogonal complement of a vector subspace). Then N0 K0(SV)SV=N0 K0(V)SVacts on HHn+1 with cohomogeneity one, where N0 K0(·) denotes the connected component of the identity of the normalizer in K0. Knowing all such subspaces Vup to congruence by an element of Sp(1)Sp(n) determines all cohomogeneity one actions on HHn+1 up to orbit equivalence. Roughly twenty years after Berndt and Br¨uck [2] announced the first examples of cohomogeneity one actions using this procedure, we obtain the full classification of cohomogeneity one actions on quaternionic hyperbolic spaces up to orbit equivalence as a consequence of Theorem A. Together with the results by Berndt and Tamaru [5], this finishes the classification of cohomogeneity one actions on non-compact symmetric spaces of rank one: Theorem B. The moduli space of cohomogeneity one actions on HHn+1 up to orbit equivalence is given by the disjoint union {N, K, SU(1, n + 1)}t 4n G k=1 Mk,n. The actions referenced here are: (1) N: the action that produces a horosphere foliation. (2) K: the action that produces a family of geodesic spheres centered at a point. (3) SU(1, n+1): the action that produces a family of tubes around a totally geodesic CHn+1. (4) Mk,n: the cohomogeneity one actions of the connected Lie subgroups of Sp(1, n + 1) with Lie algebras Nk0(V)⊕a⊕(gαV)⊕g2α, where Vis a protohomogeneous subspace of dimension kof gα∼ =Hn. 6 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ We note that, in this classification, the action of Sp(1, `)×Sp(n+ 1 −`)⊂Sp(1, n + 1) which gives tubes around a totally geodesic lower dimensional quaternionic hyperbolic space HH`,`∈ {1, . . . , n}, in HHn+1 are included in item (4), where in this case Vis a quaternionic subspace of gα∼ =Hn(hence, of quaternionic K¨ahler angle (0,0,0)) of real dimension k= 4(n−`+ 1). Moreover, if we take Va line in gα(i.e. k= 1), then N0 K0(V) is trivial and we recover the action that gives rise to the so-called solvable foliation [4]. In our study of protohomogeneous subspaces of Hnwe have also encountered noncongruent pairs of subspaces with the same constant quaternionic K¨ahler angles. Moreover, we prove in Section 6 that an H-orthogonal direct sum of subspaces of dimension 4 with the same constant quaternionic K¨ahler angle is protohomogeneous if and only if any two factors are congruent under an element of Sp(n). However, even if that direct sum is not protohomogeneous, it has constant quaternionic K¨ahler angle in some cases. Thus, if we take Va non-protohomogeneous subspace with constant quaternionic K¨ahler angle as above, and denote by SVthe subgroup of Gwhose Lie algebra is sV=a⊕(gαV)⊕g2α, then: (1) since Vhas constant quaternionic K¨ahler angle, tubes around SV·oare isoparametric and have constant principal curvatures by [15, Theorem 4.5], and (2) these tubes are not homogeneous by [5, Theorem 4.1]. Hence, we have the following remarkable consequence: Theorem C. There exist uncountably many inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures in HHn+1 with n≥7, up to congruence. We recall that the only examples of inhomogeneous isoparametric families of hypersurfaces with constant principal curvatures known so far in any irreducible Riemannian symmetric space are the celebrated examples in spheres by Ferus, Karcher and M¨unzner [22] and a single example found in the Cayley hyperbolic plane [15]. Thus, this is the first time an uncountable collection of such examples is produced in some symmetric space. This article is organized as follows. We recall some basic facts about symmetric spaces in § 2.1, and of cohomogeneity one actions in § 2.2. The fundamental concept of quaternionic K¨ahler angle is recalled in Subsection 2.3 together with some important notation that will be used throughout this article. In Section 3 we use a generalization of the hairy ball theorem to rule out several possibilities for quaternionic K¨ahler angles. Then, in Subsection 4.1 we prove a simultaneous diagonalization result for subspaces of constant quaternionic K¨ahler angle. This is used in § 4.2 to prove a factorization theorem for protohomogeneous subspaces of dimension multiple of 4. Altogether, this reduces our study to dimensions 3 ( § 5.1) and 4 ( § 5.2). The existence of inhomogeneous isoparametric hypersurfaces with constant principal curvatures in quaternionic hyperbolic spaces (Theorem C) is established in Section 6. We finally prove Theorems A and B in Section 7. 2. Preliminaries We start this section by recalling the main known results concerning cohomogeneity one actions on symmetric spaces of non-compact type and rank one. Cohomogeneity one actions with a non-totally geodesic singular orbit are built using the concept of quaternionic K¨ahler angle, which we recall in this section. Also, we will present some properties and HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 7 sumarize all the examples of subspaces with constant quaternionic K¨ahler angle known up to the present. The main references for these notions and results are [2], [5], and [15]. 2.1. Symmetric spaces of non-compact type and rank one. Hurwitz’s theorem asserts that any normed real division algebra Fis isomorphic to R, C,Hor O. The hyperbolic spaces over these algebras constitute the symmetric spaces of non-compact type and rank one. In other words, if Mis a symmetric space of non-compact type and rank one, then Mis either a real hyperbolic space RHn+1,n≥1, a complex hyperbolic space CHn+1,n≥1, a quaternionic hyperbolic space HHn+1,n≥1, or the Cayley hyperbolic plane OH2. As a symmetric space, any of these manifolds Mcan be identified with a quotient G/K of Lie groups, where Gis the connected component of the identity of the isometry group of M, up to a finite covering, and Kis the isotropy subgroup of Gcorresponding to a certain point o∈Mthat we fix from now on. Then one can take G=SO0(1, n + 1), SU(1, n + 1), Sp(1, n + 1), F−20 4and K=SO(n+ 1), S(U(1) ×U(n+ 1)), Sp(1) ×Sp(n+ 1), Spin(9), depending on whether F=R,C,H,O, respectively. We denote by gand kthe Lie algebras of Gand K, respectively, by Bthe Killing form of g, and by θthe Cartan involution of gwith respect to k. Let g=k⊕pbe the Cartan decomposition of ginduced by θ. We have that hX, Y i=−B(X, θY ) is an inner product that restricted to pinduces a Riemannian metric on G/K that makes G/K isometric to M, up to homothety. Let abe a maximal abelian subspace of p, which is one-dimensional as Mhas rank one, and let g=g−2α⊕g−α⊕g0⊕gα⊕g2αbe the corresponding restricted root space decomposition of g. Here, the root space g0splits as g0=k0⊕a, where k0is the Lie algebra of K0=NK(a), the normalizer of ain K, which also normalizes gαand g2α. Moreover, g=k⊕a⊕n, where n=gα⊕g2α, is an Iwasawa decomposition of g. When F=R, we have g−2α=g2α= 0 and nis abelian. Otherwise, nis only two-step nilpotent. In fact, nis isomorphic to the (2n+ 1)-dimensional Heisenberg algebra when F=Cand to a certain generalized Heisenberg algebra if F∈ {H,O}(see [7]). Moreover, g2α, the center of n, is equal to the derived algebra of n, and has dimension 1, 3 or 7 for F=C,Hor O, respectively. In addition to this, we can identify gαwith Rn,Cn,Hn,Ofor F=R,C,H, O, respectively. Indeed, gαis a Clifford module over Cl(m), where m= dim g2α, which is the sum of equivalent Clifford modules if m= 3, and is irreducible if m= 7. The subalgebra a⊕nof gis solvable and nis its derived subalgebra. We denote by Aand by Nthe connected closed Lie subgroups of Gwith Lie algebras aand n, respectively. Then, G=KAN is an Iwasawa decomposition of Gand AN is diffeomorphic to M. Furthermore, if we pull back the metric on Mto AN we get a left-invariant Riemannian metric on AN. Thus, Mis isometric to the solvable Lie group AN endowed with a left-invariant metric. As such, it is an example of a Damek-Ricci space (see [7]). 2.2. Cohomogeneity one actions on hyperbolic spaces. We can distinguish three different classes of cohomogeneity one actions on symmetric spaces of non-compact type and rank one, up to orbit equivalence. It was shown in [2] that any such action has at most one singular orbit. 8 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ Actions with no singular orbit. Berndt and Tamaru [4] classified actions without singular orbits. They proved that there are exactly two such actions up to orbit equivalence. (i) The action of Non FHn+1 has cohomogeneity one. The orbits of this action are mutually congruent horospheres which form a regular Riemannian foliation on FHn+1, called the horosphere foliation. (ii) Let Sbe the connected Lie subgroup of AN with Lie algebra s=a⊕w⊕z, where wis a vector subspace of gαof codimension one. Different choices of wlead to conjugate actions. The action of Son FHn+1 has cohomogeneity one and its orbits form a regular Riemannian foliation on FHn+1, called the solvable foliation. Actions with a totally geodesic singular orbit. Berndt and Br¨uck [2] classified cohomogeneity one actions on FHn+1 with a totally geodesic singular orbit F. The remaining orbits, which are principal, are tubes around the totally geodesic submanifold F, where: (i) F=R:F∈ {point,RH1,...,RHn−1}; (ii) F=C:F∈ {point,CH1,...,CHn,RHn+1}; (iii) F=H:F∈ {point,HH1,...,HHn,CHn+1}; (iv) F=O:F∈ {point,OH1,HH2}. Actions with a non-totally geodesic singular orbit. Berndt and Tamaru [5] gave a construction method of all cohomogeneity one actions with a non-totally geodesic singular orbit in hyperbolic spaces. Such actions only appear if F6=R. We recall that K0acts on gαby the adjoint representation, and hence, if Vis a real subspace of gα,N0 K0(V) will denote the connected component of the identity of the normalizer of Vin K0. Theorem 2.1. Let g=k⊕a⊕nbe an Iwasawa decomposition of the Lie algebra of the isometry group of the hyperbolic space M=FHn+1,F∈ {C,H,O}. (i) Let Vbe a non-zero vector subspace of gαsuch that N0 K0(V)acts transitively on the unit sphere of V. Denote by gαVthe orthogonal complement of Vin gα. Then the connected subgroup of Gwith Lie algebra Nk0(V)⊕a⊕(gαV)⊕g2α acts on Mwith cohomogeneity one, and the orbit through ois singular, provided that dim V≥2. Furthermore, every cohomogeneity one action on Mwith a non-totally geodesic singular orbit can be obtained in this way up to orbit equivalence. (ii) Let Vand V0be vector subspaces of gαas in (i), and assume that the corresponding cohomogeneity one actions have non-totally geodesic singular orbits. Then, these actions are orbit equivalent if and only if there exists k∈K0such that Ad(k)V=V0. 2.3. Quaternionic K¨ahler angle. The metric and the quaternionic K¨ahler structure on HHn+1 induce a positive definite inner product h·,·i on gαand a quaternionic structure Jon gα, respectively, such that gα is isomorphic to Hnas a (right) quaternionic Euclidean space. Here, by a quaternionic structure Jwe understand a 3-dimensional vector subspace of EndR(Hn), the space of real HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 9 endomorphisms of Hn∼ =R4n, admitting a basis {J1, J2, J3}of orthogonal transformations of Hn∼ =R4nsuch that J2 i=−Id and JiJi+1 =Ji+2 =−Ji+1Ji, for each i∈ {1,2,3} (indices modulo 3). Such a basis is called a canonical basis of the quaternionic structure J. Sometimes it is helpful to regard Jas endowed with a positive definite inner product that makes it isometric to the Euclidean 3-space R3, and such that the elements of Jthat are orthogonal complex structures of Hnconstitute the unit sphere S2⊂Jwith respect to such inner product. Throughout this article, if vis a vector in Hnand Vis a real subspace of Hn (i.e. a vector subspace of the real vector space R4nwith the underlying real vector space structure of Hn), we will denote by Hv=Rv⊕Jvand by HV=V+JVthe quaternionic spans of v∈Hnand of V⊂Hn, respectively; sometimes we will also write (Im H)vto refer to Jv. Theorem 2.1 shows the crucial role played by real subspaces Vof gα∼ =Hnand their behavior with respect to K0in the classification problem of cohomogeneity one actions on HHn+1. Note that the effectivization of K0on gα∼ =Hnis the Lie group Sp(1)Sp(n) = (Sp(1) ×Sp(n))/{±(1,Id)}, which acts in the standard way: (q, A)·v=Avq−1, where q∈Sp(1) and A∈Sp(n). Thus, in this subsection we gather some important terminology and useful facts to study real subspaces of a quaternionic Euclidean space, up to congruence by elements of Sp(1)Sp(n). Firstly, motivated by Theorem 2.1, we will say that a real subspace V⊂Hnis protohomogeneous if there is a connected subgroup of Sp(1)Sp(n) that acts transitively on the unit sphere of V. Equivalently, Vis protohomogeneous if the connected Lie group N0 Sp(1)Sp(n)(V) acts transitively on the unit sphere of V. Note that protohomogeneous subspaces Vof gα∼ =Hnare precisely those inducing cohomogeneity one actions on HHn+1 via the construction in Theorem 2.1(i). We will also say that two real subspaces Vand Wof Hnare equivalent if there exists an element T∈Sp(1)Sp(n) such that TV =W. Observe that, by Theorem 2.1(ii), Vand Ware equivalent protohomogeneous subspaces of gα∼ =Hnif and only if they induce orbit equivalent cohomogeneity one actions on HHn+1. Let us now recall a useful description of the action of Sp(1)Sp(n) on Hn. Let {X1, . . . , Xn} and {Y1, . . . , Yn}be two H-orthonormal bases of Hn, and let {J1, J2, J3}and {J0 1, J0 2, J0 3} be two canonical bases of the quaternionic structure of Hn. Then, there exists a unique T∈Sp(1)Sp(n) such that T(Xi) = Yiand TJj=J0 jTfor all i∈ {1, . . . , n}and all j∈ {1,2,3}. Conversely, any R-linear endomorphism of Hnwhich maps H-orthonormal bases of Hnto H-orthonormal bases of Hnand intertwines canonical bases of the quaternionic structure of Hnin the above described fashion lies in Sp(1)Sp(n). Let Vbe a real vector subspace of the quaternionic Euclidean space Hn. The K¨ahler angle of a non-zero vector v∈Vwith respect to a non-zero J∈Jand Vis defined to be the angle between Jv and V. Equivalently, it is the value ϕ∈[0, π/2] such that hPJv, PJvi= cos2(ϕ)hv, vi, where PJ:= πVJand we denote by πVthe orthogonal projection onto V. The following lemma was essentially proved by Berndt and Br¨uck [2, Lemma 3]. We state it in a somewhat different form following [15, Theorem 3.1], where it was proved in the more general context of subspaces of Clifford modules. 16 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ only if ϕ36=π/2. Furthermore, we can easily check that ¯ Pi¯ Pj=−¯ Pj¯ Pi, for i6=j. Indeed, if v, w ∈V, then Lemma 2.2 yields 0 = h¯ Pi(v+w),¯ Pj(v+w)i=h¯ Piv, ¯ Pjwi+h¯ Pjv, ¯ Piwi=−h¯ Pj¯ Piv, wi−h¯ Pi¯ Pjv, wi. Hence, Vhas a module structure over the Clifford algebra Cl(3) if ϕ36=π/2, or over Cl(2) if ϕ3=π/2. It is well known that there are exactly two inequivalent irreducible Clifford modules over Cl(3), both of dimension 4 (we will denote them by V0and V1), whereas there is exactly one irreducible Cl(2)-module up to equivalence, again of dimension 4 (we will denote it by V0). Moreover, Clifford modules are semisimple. This implies that, if ϕ36=π/2, we can decompose Vinto a direct sum of irreducible Cl(3)-modules as follows V= l0 MV0!⊕ l1 MV1!, where l0+l1=l, whereas if ϕ3=π/2 the Cl(2)-module Vcan be decomposed as V= l MV0. The above decompositions can be assumed to be orthogonal because the complex structures ¯ Piare orthogonal. This also implies that two different summands are H-orthogonal: if v, w ∈Vbelong to two different summands, hJkv, wi=hPkv, wi= 0 by the ¯ Pk-invariance. Finally, since ¯ Pileaves each factor Vr(r∈ {0,1}) invariant, we deduce that each Vrhas constant quaternionic K¨ahler angle (ϕ1, ϕ2, ϕ3). This leads us to state the following: Lemma 4.3. Let Vbe a real subspace of Hnof dimension 4l, with l∈N, and constant quaternionic K¨ahler angle (ϕ1, ϕ2, ϕ3). Assume that there exists a canonical basis {J1, J2, J3}of Jsuch that the K¨ahler angle of any non-zero v∈Vwith respect to Jiand Vis ϕi, for each i∈ {1,2,3}. Then, there is an H-orthogonal decomposition V= l M r=1 Vr, where each Vrhas dimension 4and Φ(Vr) = (ϕ1, ϕ2, ϕ3)as a subspace of Hn. Conversely, let Vbe a real subspace of Hngiven by an H-orthogonal direct sum V:= Ll r=1 Vr, where each Vrhas dimension 4, and Φ(Vr)=(ϕ1, ϕ2, ϕ3). Let {J1, J2, J3}be a canonical structure of Jsuch that every non-zero vector in Vrhas K¨ahler angle ϕiwith respect to Jiand Vr, for each i∈ {1,2,3}and each r∈ {1, . . . , l}. Then, Φ(V) = (ϕ1, ϕ2, ϕ3). Proof. The first assertion has been proved above under the assumption ϕ26=π/2. If ϕ2= π/2, the first claim follows from the classification of subspaces Vwith Φ(V) = (ϕ, π/2, π/2), ϕ∈[0, π/2] (cf. § 2.4 and [2, pp. 230-232]). In order to prove the converse, we first note that πV(HVr) = Vr, for each r∈ {1, . . . , l}. Indeed, for every v∈Vrand w∈Vs,r6=s,hπVJiv, wi=hJiv, wi= 0, where in the last equality we have used HVr⊥HVs. Hence, πVJ(Vr)⊂Vr, and since πV(Vr) = Vr, we deduce πV(HVr) = Vr. HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 17 Now let v=Pl r=1 vr∈V, with vr∈Vrfor each r∈ {1, . . . , l}. Denoting as usual Pi=πVJi, for each i∈ {1,2,3}, we have Lv(Ji, Jj) = hPiv, Pjvi= l X r,s=1hπVJivr, πVJjvsi= l X r=1 hπVJivr, πVJjvri = l X r=1 hPivr, Pjvri= l X r=1 cos2(ϕi)δijkvrk2= cos2(ϕi)δijkvk2, where in the third equality we have used πV(HVr) = Vrand Vr⊥Vsfor all r, s ∈ {1, . . . , l}, and in the fifth one we have used that the quaternionic K¨ahler angle of vrwith respect to Vrand {J1, J2, J3}is (ϕ1, ϕ2, ϕ3). Since v∈Vis arbitrary, by Lemma 2.2 we conclude that Φ(V)=(ϕ1, ϕ2, ϕ3).  5. Low dimensional subspaces with constant quaternionic K¨ ahler angle As a consequence of Proposition 3.2, we only have to study subspaces of dimensions 3 and multiples of 4. The latter can be reduced to studying subspaces of dimension 4 by virtue of Corollary 4.2 and Lemma 4.3. We will devote this section to the classification of (not necessarily protohomogeneous) real subspaces of dimensions k∈ {3,4}with constant quaternionic K¨ahler angle. The main tool that we will use in this section is the isospectrality of the K¨ahler angle map Ω introduced in (1). We start with a lemma that provides an appropriate basis of the subspace. Lemma 5.1. Let Vbe a real subspace of Hnof dimension k∈ {3,4}with Φ(V) = (ϕ1, ϕ2, ϕ3). Let e0∈Vbe a unit vector. Then, there exists a canonical basis {J1, J2, J3} of Jand vectors ei∈HnHe0,i∈ {1, . . . , k −1}, such that (2) cos(ϕi)Jie0+ sin(ϕi)Jiei, i ∈ {0, . . . , k −1}, constitute an R-orthonormal basis of V, where we put J0:= Id and ϕ0= 0. Moreover, for each i∈ {0, . . . , k −1}with ϕi6=π/2, we have ¯ Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei, where ¯ Pi=Pi/cos(ϕi) = πVJi/cos(ϕi). Finally, if ϕi= 0 we take ei= 0, whereas if ϕi>0, then eiis a unit vector. Proof. Let e0∈Vbe a unit vector. By Lemma 2.2, there is a canonical basis {J1, J2, J3} of Jsuch that e0has K¨ahler angle ϕiwith respect to Jifor i∈ {1,2,3}, and hPie0, Pje0i= cos2(ϕi)δij. In particular, h¯ Pie0,¯ Pje0i= 0 for any i,j∈ {0, . . . , k −1},i6=j, with ϕi, ϕj6=π/2. Fix i∈ {1,2,3}. If ϕi= 0, then we take ei= 0. Let us assume first that ϕi∈(0, π/2). By regarding Hnas a complex vector space C2nwith respect to the complex structure Ji, [2, Lemma 2] yields the existence of a unit vector ei∈HnspanR{e0, Jie0}satisfying ¯ Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei. 18 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ We have to see that ei∈HnHe0. Observe that HnspanR{e0, Jie0}coincides with the orthogonal sum (HnHe0)⊕spanR{Ji+1e0, Ji+2e0}, where indices are taken modulo 3. Let a, b ∈R. Then hei, aJi+1e0+bJi+2e0i=−1 sin(ϕi)hJi¯ Pie0+ cos(ϕi)e0, aJi+1e0+bJi+2e0i =1 sin(ϕi) cos(ϕi)(ahPie0, Pi+2e0i−bhPie0, Pi+1e0i)=0, where in the last equality we have used Lemma 2.2. Therefore, ei∈HnHe0. Now if ϕ2=π/2, subspaces Vwith Φ(V) = (ϕ, π/2, π/2), ϕ∈[0, π/2], are classified (see § 2.4) and they can be spanned by a basis as in the statement (see [2, p. 232] and note that the {ei}in the statement do not have to be H-orthonormal). Thus, we finally have to deal with the case k= 4, ϕ26=π/2, and ϕ3=π/2. Then, by the previous argument, there exists a unit vector v∈Hnsuch that {e0,¯ P1e0,¯ P2e0, v}is an R-orthonormal basis of V, where ¯ Pie0= cos(ϕi)Jie0+ sin(ϕi)Jiei,i∈ {1,2}. Recalling the definition of the K¨ahler angle map (1), we have tr(Ω(e0)) = 3 X i=1 hPie0, Pie0i= 3 X i=1 hPie0, e0i2+hPie0,¯ P1e0i2+hPie0,¯ P2e0i2+hPie0, vi2 = cos2(ϕ1) + cos2(ϕ2) + 3 X i=1 hJie0, vi2, where we have used Lemma 2.2 and ¯ Pie0=Pie0/cos(ϕi). Since Φ(V) = (ϕ1, ϕ2, π/2), the eigenvalues of Ω(e0) are cos2(ϕ1), cos2(ϕ2) and 0, and hence we deduce that v∈HnHe0. Thus, taking e3=−J3vyields the result.  Remark 5.2.Whenever ϕ1>0, the orthogonality of (2) yields hJ3e1, e2i= 0, and if k= 4, also hJ1e2, e3i=hJ2e3, e1i= 0. 5.1. Subspaces of dimension 3. In this subsection we classify 3-dimensional real subspaces of Hnwith constant quaternionic K¨ahler angle. Proposition 5.3. Let V⊂Hnbe a real subspace of dimension 3. Then, Vhas constant quaternionic K¨ahler angle if and only if Φ(V)=(ϕ, ϕ, π/2),ϕ∈[0, π/2], and for any unit e0∈V, there is a canonical basis {J1, J2, J3}of Jsuch that (3) {e0,cos(ϕ)J1e0+ sin(ϕ)J1e1,cos(ϕ)J2e0+ sin(ϕ)J2e2} is an orthonormal basis of V, where, if ϕ6= 0,e1, e2are unit vectors satisfying e1, e2∈ HnHe0,e2∈Hn(Im H)e1, and either he1, e2i= cos(ϕ)/(cos(ϕ)−1) with ϕ∈[π/3, π/2], or he1, e2i= cos(ϕ)/(cos(ϕ) + 1) with ϕ∈(0, π/2]. Proof. By Proposition 3.2, we have that ϕ1=ϕ2=ϕ∈[0, π/2] and ϕ3=π/2. Let us assume that Vis spanned by the basis described in Lemma 5.1 with k= 3. If ϕ= 0 or HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 19 ϕ=π/2 the claim follows from the classification of subspaces with constant quaternionic K¨ahler angle (0,0, π/2) or (π/2, π/2, π/2); see § 2.4. Thus, let us assume ϕ∈(0, π/2). Then, for each l∈ {1,2}and understanding the subscript l+ 1 ∈ {1,2}modulo 2, Ω( ¯ Ple0)ij =hPi¯ Ple0, Pj¯ Ple0i=hJi¯ Ple0, e0ihJj¯ Ple0, e0i+ 2 X r=1hJi¯ Ple0,¯ Pre0ihJj¯ Ple0,¯ Pre0i, =h¯ Ple0, Pie0ih¯ Ple0, Pje0i+hJi¯ Ple0,¯ Pl+1e0ihJj¯ Ple0,¯ Pl+1e0i, where in the second equality we have calculated the orthogonal projection of vectors onto Vby using the orthonormal basis {e0,¯ P1e0,¯ P2e0}of V. Hence, for l∈ {1,2}, using Lemma 5.1 we have (4) Ω( ¯ Ple0)ll = cos2(ϕ) + hel, Jl+1el+1i2sin4(ϕ), Ω( ¯ Ple0)l+1,l =he1, J1e2ihe1, J2e2isin4(ϕ), Ω( ¯ Ple0)l+1,l+1 =hel+1, Jleli2sin4(ϕ), Ω( ¯ Ple0)13 =he2, J2e1isin2(ϕ)(cos2(ϕ) + he1, e2isin2(ϕ)), Ω( ¯ Ple0)23 =−he1, J1e2isin2(ϕ)(cos2(ϕ) + he1, e2isin2(ϕ)), Ω( ¯ Ple0)33 = (cos2(ϕ) + he1, e2isin2(ϕ))2. Now, since Ω( ¯ Ple0) is symmetric with eigenvalues cos2(ϕ) (of multiplicity 2) and 0, by the min-max theorem, one obtains 0≤Ω( ¯ Ple0)ll ≤cos2(ϕ), l ∈ {1,2}. This implies he1, J2e2i=he2, J1e1i= 0, which together with Remark 5.2 yields e2∈ Hn(Im H)e1. Taking again into account the spectrum of Ω( ¯ P1e0), we have the following relation for its trace, 2 cos(ϕ)2= tr(Ω( ¯ P1e0)) = (cos2(ϕ) + he1, e2isin2(ϕ))2+ cos2(ϕ). From this and the fact that e1and e2are unit vectors, we deduce that either he1, e2i= cos(ϕ)/(cos(ϕ)−1) where ϕ∈[π/3, π/2) or he1, e2i= cos(ϕ)/(1 + cos(ϕ)) where ϕ∈ (0, π/2). This proves the necessity of the statement. For the converse we take an arbitrary unit vector v∈Vwhich we write as v=x0e0+x1cos(ϕ)J1e0+ sin(ϕ)J1e1+x2cos(ϕ)J2e0+ sin(ϕ)J2e2. Then, if ε∈ {±1}is such that he1, e2i= cos(ϕ)/(1 + εcos(ϕ)), we have Ω(v) = cos2(ϕ) x2 0+x2 1x1x2−εx0x2 x1x2x2 0+x2 2εx0x1 −εx0x2εx0x1x2 1+x2 2 . Since vis a unit vector, x2 0+x2 1+x2 2= 1, and it is now easy to see that Ω(v) has a double eigenvalue cos2(ϕ), and a simple eigenvalue 0.  20 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ Remark 5.4.We will denote by Vϕ +and Vϕ −any real subspace of Hnconstructed as in Proposition 5.3, depending on whether he1, e2i= cos(ϕ)/(cos(ϕ) + 1) for ϕ∈(0, π/2], or he1, e2i= cos(ϕ)/(cos(ϕ)−1) for ϕ∈[π/3, π/2], respectively. Note that the subspaces Vϕ ± can be constructed as subspaces of any Hnwith n≥3. One can easily check that the only one that fits into an H2is Vπ/3 −(but it cannot fit into H). Proposition 5.5. Let Vbe a subspace of Hnwith constant quaternionic K¨ahler angle and dimension 3. Then Vis protohomogeneous. Proof. We know from Proposition 5.3 that Φ(V)=(ϕ, ϕ, π/2). We can assume that ϕ∈(0, π/2) since, otherwise, Vis known to be protohomogeneous (see § 2.4). Let e0∈Vbe an arbitrary unit vector. By Lemma 2.2 there is a canonical basis {J1, J2, J3}of Jsuch that e0has K¨ahler angle ϕwith respect to J1and J2, and K¨ahler angle π/2 with respect to J3. In view of Lemma 5.1 and Proposition 5.3, let us consider the unit vectors ei∈HnHe0,i∈ {1,2}, given by (5) ei:= −(Ji¯ Pie0+ cos(ϕ)e0)/sin(ϕ), i ∈ {1,2}, where ¯ Pi:= πVJi/cos(ϕ). On the one hand, by (5) we have (6) he1, e2i=1 sin2(ϕ)hJ1¯ P1e0+ cos(ϕ)e0, J2¯ P2e0+ cos(ϕ)e0i =1 sin2(ϕ)hJ1¯ P1e0, J2¯ P2e0i−cos2(ϕ). On the other hand, again by Proposition 5.3, he1, e2ican take two possible values. We will first see that, given V,he1, e2iis independent of e0. Let S2denote the unit sphere of V. We define Θ: S2→Rby Θ(e0) = he1, e2i. We claim that Θ is well defined. Let {J0 1, J0 2, J3}be another canonical basis of Jsuch that e0has K¨ahler angle ϕwith respect to J0 i,i∈ {1,2}, and let e0 i:= −(J0 i¯ P0 ie0+ cos(ϕ)e0)/sin(ϕ) where ¯ P0 i:= πVJ0 i/cos(ϕ) for i∈ {1,2}. Then, there is θ∈[0,2π) such that J0 i= cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1 for i∈ {1,2}and subscripts modulo 2. Thus, (7) J0 i¯ P0 i= (cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)(cos(θ)¯ Pi+ (−1)i+1 sin(θ)¯ Pi+1) = cos2(θ)Ji¯ Pi+ sin2(θ)Ji+1 ¯ Pi+1 + (−1)i+1 cos(θ) sin(θ)(J1¯ P2+J2¯ P1). Consequently, using Equation (6) twice, and then (7), we get, after some calculations, he1, e2i−he0 1, e0 2i=1 sin2(ϕ)hJ1¯ P1e0, J2¯ P2e0i−hJ0 1¯ P0 1e0, J0 2¯ P0 2e0i= 0, which implies that Θ is well-defined. Now note that the assignment e0∈S27→ span{J1, J2} ∈ G2(J), where G2(J) is the Grassmannian of 2-planes of J∼ =R3, is continuous due to the continuous dependence of the quadratic form J∈J7→ Lv(J, J) = hPJv, PJvi ∈ Ron v. Hence, the map Θ is also continuous. But, as mentioned just after (6), Θ(S2) has at most two elements. Therefore, Θ is constant on S2. HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 21 Finally, we prove that Vis protohomogeneous. Let e0,e0 0be arbitrary unit vectors in V. Let {J1, J2, J3},{J0 1, J0 2, J0 3}be canonical bases of J, and e1, e2, e0 1, e0 2be unit vectors in V such that both (3), and (3) with e0 iinstead of eiand J0 iinstead of Ji, are orthonormal bases of V. Both sets of vectors {e0, e1, e2}and {e0 0, e0 1, e0 2}span a totally real subspace of Hn, and since Θ is constant, hei, eji=he0 i, e0 jifor all i, j ∈ {0,1,2}. It then follows that there exists an element T∈Sp(1)Sp(n) such that Tei=e0 ifor each i∈ {0,1,2}, and TJj=J0 jTfor each j∈ {1,2,3}. Thus, by (5) we get T¯ Pie0=¯ P0 ie0 0for i∈ {0,1,2}, where ¯ P0=¯ P0 0= Id. Therefore, Tis an element of Sp(1)Sp(n) such that TV =Vand Te0=e0 0. Since e0, e0 0are arbitrary, this proves that Vis protohomogeneous.  Finally we show that the two types of subspaces Vϕ +and Vϕ −introduced in Remark 5.4 are indeed inequivalent for ϕ6=π/2. Recall that Vϕ +is defined for all ϕ∈(0, π/2], but Vϕ − only for ϕ∈[π/3, π/2]. Proposition 5.6. Let ϕ∈[π/3, π/2]. Then there exists T∈Sp(1)Sp(n)such that TV ϕ += Vϕ −if and only if ϕ=π/2. Proof. If ϕ=π/2, then Vπ/2 +and Vπ/2 −are totally real, therefore equivalent. Let us assume that ϕ6=π/2 and that there is T∈Sp(1)Sp(n) such that TV ϕ +=Vϕ −. By applying an element of Sp(1)Sp(n) if necessary, we can assume that there is a unit vector e0∈Vϕ +∩Vϕ − and that e0has quaternionic K¨ahler angle (ϕ, ϕ, π/2) with respect to both Vϕ +and Vϕ −and a common canonical basis {J1, J2, J3}of J. Then, by Lemma 5.1 and Proposition 5.3, Vϕ ±is spanned by the basis {e0,¯ P± 1e0,¯ P± 2e0}, where ¯ P± i:= πVϕ ±Ji/cos(ϕ), i∈ {1,2}. Moreover, ¯ P± ie0= cos(ϕ)Jie0+ sin(ϕ)Jie± i,with he± 1, e± 2i=cos(ϕ) cos(ϕ)±1, and e± i∈HnHe0,i∈ {1,2}. By Proposition 5.5, we can assume that Te0=e0. Let J0=TJ3T−1∈J. Since J3e0∈HnVϕ +, we have J0e0=J0Te0=TJ3e0∈HnVϕ −. This implies TJ3=εJ3T, where ε∈ {−1,1}, because ±J3are the only complex structures in Jwhich send e0to HnVϕ −. Therefore, there exists θ∈[0,2π) such that (8) TJi=εi(cos(θ)Ji+ (−1)i+1 sin(θ)Ji+1)T, i ∈ {1,2},and TJ3=εJ3T. Using (8) and Te0=e0, we have (9) T¯ P+ 1e0= cos(ϕ)TJ1e0+ sin(ϕ)TJ1e+ 1 =εcos(ϕ)(cos(θ)J1e0+ sin(θ)J2e0) + εsin(ϕ)(cos(θ)J1Te+ 1+ sin(θ)J2Te+ 1). By Proposition 5.5, Vϕ ±is protohomogeneous, and note that SO(3) is the only connected subgroup of Sp(1)Sp(n)⊂SO(4n) that acts transitively and effectively on the unit sphere of Vϕ ±. Thus, we can assume that T¯ P+ 1e0=ε¯ P− 1e0, just by composing Twith some element in the isotropy of the action of SO(3) on Vϕ −at e0. But inserting (9) and ¯ P− 1e0= cos(ϕ)J1e0+sin(ϕ)J1e− 1into the equality T¯ P+ 1e0=ε¯ P− 1e0, and analyzing the He0and Hn He0components (note that e± 1∈HnHe0,Te0=e0, and Tpreserves H-orthonormality) we get θ= 0 and Te+ 1=e− 1. Moreover, by (8) we get TJi=εiJiT,i∈ {1,2,3}. 22 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ Since Te0=e0,T¯ P+ 1e0=ε¯ P− 1e0and TV ϕ +=Vϕ −, we must have T¯ P+ 2e0=±¯ P− 2e0. Then, inserting ¯ P± 2e0= cos(ϕ)J2e0+sin(ϕ)J2e± 2in the last equality, and using TJ2=J2T, we deduce that Te+ 2=e− 2. But this jointly with Te+ 1=e− 1yields a contradiction with the fact that Tis an orthogonal transformation of Hn, because he+ 1, e+ 2i 6=he− 1, e− 2ifor all ϕ6=π/2.  5.2. Subspaces of dimension four. The aim of this subsection is to classify 4-dimensional real subspaces of Hnwith constant quaternionic K¨ahler angle. We start by restricting our attention to subspaces with ϕ1= 0. Proposition 5.7. Let V⊂Hnbe a real subspace of dimension 4with constant quaternionic K¨ahler angle (0, ϕ2, ϕ3). Then, ϕ2=ϕ3∈[0, π/2]. Proof. First of all, if ϕ2= 0, then ϕ3= 0 by a combination of [2, Proposition 9] and the fact that subspaces with Φ(V) = (0,0, π/2) have dimension 3 (see § 2.4). Hence, let us assume that ϕ26= 0. Lemma 5.1 yields a basis {e0, J1e0, v2, v3}of V, where vi= cos(ϕi)Jie0+sin(ϕi)Jiei, for certain unit ei∈HnHe0,i∈ {2,3}. Therefore, a computation as in Equations (4), for each i∈ {2,3}, gives Ω(vi)11 =cos(ϕ2) cos(ϕ3) + he2, e3isin(ϕ2) sin(ϕ3)2, Ω(vi)22 = cos(ϕi)2+hJ3e3, e2i2sin2(ϕ2) sin2(ϕ3), Ω(vi)33 = cos(ϕi)2+hJ2e2, e3i2sin2(ϕ2) sin2(ϕ3). Hence, by the isospectrality of Ω, 0 = tr(Ω(v2)) −tr(Ω(v3)) = 2 cos2(ϕ2)−2 cos2(ϕ3), from where we conclude ϕ2=ϕ3. In view of Proposition 5.7, all real subspaces Vof Hnwith Φ(V) = (0, ϕ2, ϕ3) actually satisfy Φ(V) = (0, ϕ, ϕ). Note that such subspaces have been classified (see § 2.4). Thus, in the following results we will analyze the case ϕ1>0. We consider the basis of Vgiven in Lemma 5.1. Lemma 5.8. Let V⊂Hnbe a real subspace of dimension 4such that Φ(V)=(ϕ1, ϕ2, ϕ3) with ϕ1>0. For each i∈ {1,3}with ϕi6=π/2, we have hei, Jjeji= 0 for all j∈ {1,2,3}. Proof. According to Lemma 5.1, e0has K¨ahler angle ϕiwith respect to Vand Ji∈Jfor each i∈ {1,2,3}. Let us regard Hnas a complex Euclidean space C2nwhose complex structure is Ji, for i∈ {1,2,3}. By [16, Theorem 2.7] there is a non-empty finite subset Ψi⊂[0, π/2] such that V=Lϕ∈ΨiVi ϕis a C-orthonormal decomposition of Vand Vi ϕ⊂ C2nis a real subspace with constant K¨ahler angle ϕ∈Ψi. It follows that any non-zero v∈Vi ϕhas K¨ahler angle ϕwith respect to Vand Ji, and the minimum (resp. maximum) of Ψicoincides with the minimum (resp. maximum) K¨ahler angle of a non-zero vector v∈V with respect to Vand Ji. HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 23 We claim that ϕ1∈Ψ1. On the one hand, if there existed ϕ∈Ψ1such that ϕ<ϕ1, then there would be vectors in Vwhose K¨ahler angle with respect to Vand J1∈Jis ϕ<ϕ1, thus contradicting the minimality of ϕ1by Lemma 2.2. On the other hand, if ϕ > ϕ1for all ϕ∈Ψ1, then we would get a contradiction with the fact that e0has K¨ahler angle ϕ1 with respect to J1. Analogously, we get that ϕ3∈Ψ3. Now assume ϕ16=π/2. By [16, p. 1190–1191] and the discussion above, we have a decomposition V=Vϕ1⊕Vψ1into real subspaces of constant K¨ahler angle with respect to the complex structure J1, where ψ1∈Ψ1(the possibility ψ1=ϕ1is allowed). We also have that ¯ P1:= πVϕ1J1/cos(ϕ1) = πVJ1/cos(ϕ1)|Vϕ1defines a complex structure on Vϕ1. As e0∈ Vϕ1, we get Vϕ1= spanR{e0,¯ P1e0}. Moreover, CVϕ1⊥CVψ1, so Vψ1= spanR{¯ P2e0,¯ P3e0}, and for j∈ {2,3}, using Lemma 5.1, 0 = h¯ P1e0, J1¯ Pje0i= sin(ϕ1) sin(ϕj)he1, Jjeji. Since ϕ1>0, we get he1, Jjeji= 0. A similar argument works for ϕ3, if ϕ36=π/2.  Before addressing the classification, we state a lemma that refines [15, Lemma 5.1]. Lemma 5.9. Assume 0< ϕ1≤ϕ2≤ϕ3≤π/2, and let ε∈ {−1,1}. Then, there exists a subset {e1, e2, e3}of unit vectors of R3with inner products hei, ei+1i=εcos(ϕi+2)−cos(ϕi) cos(ϕi+1) sin(ϕi) sin(ϕi+1)for each i∈ {1,2,3} if and only if cos(ϕ1) + cos(ϕ2)−εcos(ϕ3)≤1. Furthermore, the subspace spanR{e1, e2, e3}has dimension 2if and only if cos(ϕ1) + cos(ϕ2) + εcos(ϕ3) = 1, and dimension 3otherwise. Proof. A subset {e1, e2, e3}of the Euclidean space R3satisfies the inner product relations in the statement if and only if the associated Gram matrix G= (hei, eji)1≤i,j≤3is positive semi-definite. This happens precisely when all principal minors of Gare non-negative; in this proof, by Gij we denote the matrix of order 2 resulting from deleting the i-th row and the j-th column of G. Let xi:= cos(ϕi) for each i∈ {1,2,3}. Hence, Gis positive semi-definite if and only if det Gii ≥0 for all i∈ {1,2,3}and det G≥0. We compute det(G) = (ε+x1−x2−x3)(−ε+x1+x2−x3)(−ε+x1−x2+x3)(ε+x1+x2+x3) (1 −x2 1)(1 −x2 2)(1 −x2 3). Taking into account that 1 > x1≥x2≥x3≥0, one can check that det G≥0 if and only if −1+x1+x2−εx3≤0. Similarly, det(Gii) = (1−x2 1−x2 2−x2 3+2εx1x2x3)/Qj6=i(1−x2 j), i∈ {1,2,3}. Hence, det(Gii)≥0 if and only if (10) 1 −x2 1−x2 2−x2 3+ 2εx1x2x3≥0. Now, if 1 > x1≥x2≥x3≥0, one can show that (10) holds provided that −1 + x1+x2− εx3≤0. This completes the proof of the first claim of the lemma. Assume that we are in the situation of the first assertion of the statement. Then {e1, e2, e3}spans a 3-dimensional subspace if and only if Gis positive definite, which in this situation amounts to det G > 0. This happens precisely when x1+x2−εx3<1. Hence, 24 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ the proof of the lemma will be complete if we show that spanR{e1, e2, e3}cannot have dimension 1. Assume this is the case. Then the rank of Gis 1. Hence x1= 1 −x2+εx3, and the minor det(G33) vanishes, i.e. 0 = det(G33) = −2(1 + x3) (1 + x2)(−2 + x2−x3). Therefore, x3=−ε, which yields a contradiction. This finishes the proof.  We are now in position to complete the description of 4-dimensional real subspaces of Hnwith constant quaternionic K¨ahler angle. Proposition 5.10. Let V⊂Hnbe a real subspace of dimension 4and e0∈Va unit vector. Then Vhas constant quaternionic K¨ahler angle Φ(V)=(ϕ1, ϕ2, ϕ3), with ϕ1>0, if and only if there is a canonical basis {J1, J2, J3}of J,ε∈ {−1,1}, and unit vectors e1, e2, e3∈HnHe0with ei∈Hn(Im H)ej,i,j∈ {1,2,3}, such that (i) 0 < ϕ1≤ϕ2≤ϕ3≤π/2, (ii) cos(ϕ1) + cos(ϕ2)−εcos(ϕ3)≤1, (iii) for all i∈ {1,2,3}and indices modulo 3, hei, ei+1i=εcos(ϕi+2)−cos(ϕi) cos(ϕi+1) sin(ϕi) sin(ϕi+1), (iv) {cos(ϕi)Jie0+ sin(ϕi)Jiei:i= 0,1,2,3}is an orthonormal basis of V, where for simplicity we put ϕ0:= 0 and J0:= Id. Moreover, if Vis as above, the K¨ahler angle of any non-zero v∈Vwith respect to Ji and Vis ϕi, for each i∈ {1,2,3}. Proof. In order to prove the necessity, let us assume that Vis spanned by the basis described in Lemma 5.1 with k= 4. Notice that ϕ2=π/2 implies Φ(V)=(ϕ, π/2, π/2); such subspaces are classified (see § 2.4), and [2, p. 232], together with some straightforward calculations, show that they can be spanned by a basis as above. Thus, we can suppose ϕ1, ϕ2∈(0, π/2). Let us first assume ϕ36=π/2. A long but elementary calculation, similar to the one used to obtain Equations (4), using the isospectrality of Ω, Remark 5.2 and Lemma 5.8, yields 3 Y j=1 cos2(ϕj) = det(Ω( ¯ Pie0)) = cos2(ϕi) 3 Y j=1 j6=i (cos(ϕi) cos(ϕj) + hei, ejisin(ϕi) sin(ϕj))2, for each i∈ {1,2,3}. This implies (11) cos2(ϕi+2) = (cos (ϕi) cos(ϕi+1) + hei, ei+1isin (ϕi) sin (ϕi+1)) 2, i ∈ {1,2,3}. Using (11), we can also calculate for i∈ {1,3} 3 X j=1 cos2(ϕj) = tr(Ω( ¯ Pie0)) = 3 X j=1 cos2(ϕj) + he2, Jieii2sin2(ϕi) sin2(ϕ2), HOMOGENEOUS AND INHOMOGENEOUS ISOPARAMETRIC HYPERSURFACES 25 which implies he2, Jieii= 0, i∈ {1,3}. This, along with Remark 5.2 and Lemma 5.8, shows that ei∈Hn(Im H)ej,i, j ∈ {1,2,3}. Furthermore, (11) gives rise to the two possible expressions for hei, ei+1iin the statement (corresponding to ε= 1 or ε=−1). Note that such expressions are incompatible for a fixed V, that is, if for some i∈ {1,2,3}we have hei, ei+1i=cos(ϕi+2)−cos(ϕi) cos(ϕi+1) sin(ϕi) sin(ϕi+1),hei+1, ei+2i=−cos(ϕi) + cos(ϕi+1) cos(ϕi+2) sin(ϕi+1) sin(ϕi+2), then one can check that det Ω((e0+¯ Pi+1e0)/√2)= 0, which gives a contradiction with the assumption ϕ36=π/2. Finally, the inequality in item (ii) of the statement follows from Lemma 5.9. Now assume that ϕ3=π/2. Let {e0,¯ P1e0,¯ P2e0, J3e3}be the orthonormal basis provided by Lemma 5.1. A similar computation as in (4), using the isospectrality of Ω and Lemma 5.8, yields 2 X i=1 cos2(ϕi) = tr(Ω( ¯ P1e0)) + tr(Ω( ¯ P2e0)) −tr(Ω(J3e3)) = cos2(ϕ1) + cos2(ϕ2) + 2 sin2(ϕ1) sin2(ϕ2)hJ1e1, e2i2 + 2 (cos(ϕ1) cos(ϕ2) + sin(ϕ1) sin(ϕ2)he1, e2i)2. Then, (12) he2, J1e1i= 0 and he1, e2i=−cot(ϕ1) cot(ϕ2). Also, using (12), if i∈ {1,2}we get 0 = det Ω1 √2e0+1 √2 ¯ Pie0=1 4he3, Jieii2cos2(ϕ1) cos2(ϕ2) sin2(ϕi). Thus, (13) he3, Jieii= 0, i ∈ {1,2}. Taking into account (12) and (13), we can calculate 2 X i=1 cos2(ϕi) = tr(Ω( ¯ P1e0)) = cos2(ϕ1) + he1, e3i2sin2(ϕ1), 2 X i=1 cos2(ϕi) = tr(Ω( ¯ P2e0)) = cos2(ϕ2) + sin2(ϕ2)(he2, e3i2+he2, J3e3i2), whence (14) he1, e3i=εcos(ϕ2)/sin(ϕ1) and cos2(ϕ1) = sin2(ϕ2)(he2, e3i2+he2, J3e3i2), for some ε∈ {−1,1}. Using these relations we compute 2 X i=1 cos2(ϕi) = tr Ω1 √2 ¯ P1e0+1 √2 ¯ P2e0= 2 X i=1 cos2(ϕi) + ε1 2he2, J3e3isin(2ϕ2), 32 J. C. D´ IAZ-RAMOS, M. DOM´ INGUEZ-V´ AZQUEZ, AND A. RODR´ IGUEZ-V´ AZQUEZ Furthermore, when n < k ≤2n, we have Φ(V) = (0, π/2, π/2), whereas when k≤n we have Φ(V) = (ϕ, π/2, π/2), for some ϕ∈[0, π/2]. (4) Case k= 3. By Proposition 3.2, Φ(V) = (ϕ, ϕ, π/2) for some ϕ∈[0, π/2]. 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Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973), 495–506. Department of Mathematics, University of Santiago de Compostela, Spain. Email address:[email protected] Department of Mathematics, University of Santiago de Compostela, Spain. Email address:[email protected] Department of Mathematics, University of Santiago de Compostela, Spain. Email address:[email protected]