j. differential geometry 63 (2003) 1-40

HOMOGENEOUS CODIMENSION ONE FOLIATIONS ON NONCOMPACT SYMMETRIC SPACES

JÜRGEN BERNDT & HIROSHI TAMARU

Abstract

We determine the isometric congruence classes of homogeneous Riemannian foliations of codimension one on connected irreducible Riemannian symmetric spaces of noncompact type. As an application we show that on each connected irreducible Riemannian symmetric space of noncompact type and rank greater than two there exist noncongruent homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.

1. Introduction

An action of a Lie group on a manifold is of cohomogeneity one if there exists an orbit of codimension one. Our motivation for this work is the classical problem: For a connected complete Riemannian manifold M, determine the space 𝔐 of all isometric cohomogeneity one actions on M modulo orbit equivalence. Two cohomogeneity one actions on M are orbit equivalent if there exists an isometry of M mapping the orbits of one of these actions onto the orbits of the other action.

For the sphere S^{n} with its standard metric Hsiang and Lawson [12] proved that every element in 𝔐 can be represented by the isotropy representation of an (n + 1)-dimensional Riemannian symmetric space of rank two. In successive work by Takagi [25], Uchida [27], D’Atri [8], Iwata [13, 14] and Kollross [16] the problem was solved for all connected, simply connected, irreducible Riemannian symmetric spaces of compact type. For some exceptional symmetric spaces the moduli space 𝔐 is an empty set, otherwise 𝔐 is a finite set and representatives for all actions are explicitly known. The main tool for proving these results is the classification of maximal closed connected subgroups of semisimple compact Lie groups.

Research supported by EPSRC Grant GR/M18355.

Received 05/08/2002.

For noncompact manifolds the methods employed by the above authors are not applicable. A major difficulty in the noncompact case arises from the fact that there can exist large families of nonconjugate subgroups of the isometry group having the same orbits. In general it is a difficult problem to decide about orbit equivalence for nonconjugate subgroups. The aim of this paper is to present a partial solution to this problem for Riemannian symmetric spaces of noncompact type.

Let M be a connected irreducible Riemannian symmetric spaces of noncompact type. General theory about cohomogeneity one actions (see e.g., [20] for the compact case and [3] for the general case) implies that any such action on M either induces a foliation on M or has exactly one singular orbit. This induces a disjoint union 𝔐 = 𝔐_{F} ∪𝔐_{S}, where 𝔐_{F} is the set of all homogeneous codimension one foliations on M modulo isometric congruence and 𝔐_{S} is the set of all connected normal homogeneous submanifolds with codimension ≥ 2 in M modulo isometric congruence. The main result of this paper is a complete description of 𝔐_{F}.

Denote by r the rank of M and by ℝP^{r−1} the (r − 1)-dimensional real projective space. To each ℓ ∈ ℝP^{r−1} we associate a homogeneous codimension one foliation 𝔉_{ℓ} on M all of whose leaves are isometrically congruent to each other, and to each i ∈ {1, …, r} we associate a homogeneous codimension one foliation 𝔉_{i} on M that has exactly one minimal leaf. The symmetry group Aut(DD) of the Dynkin diagram of the restricted root system Σ associated to M acts on a set of simple roots in Σ, which is a set of r elements and forms a basis of an r-dimensional Euclidean vector space. This action then induces canonically an action on ℝP^{r−1}, and thus we get an action of the finite symmetry group Aut(DD) on ℝP^{r−1} ∪ {1, …, r}.

Theorem. Let M be a connected irreducible Riemannian symmetric space of noncompact type and with rank r. The moduli space 𝔐_{F} of all noncongruent homogeneous codimension one foliations on M is isomorphic to the orbit space of the action of Aut(DD) on ℝP^{r−1} ∪ {1, …, r}:

𝔐_{F} ≅ (ℝP^{r−1} ∪ {1, …, r})/Aut(DD).

A remarkable consequence is that 𝔐_{F} depends only on the rank and on possible duality or triality principles on the symmetric space. For instance, for SO(17, ℂ)/SO(17), Sp(8, ℝ)/U(8), Sp(8, ℂ)/Sp(8), SO(16, ℍ)/U(16), SO(17, ℍ)/U(17), E_{8}^{8}/SO(16), E_{8}^{ℂ}/E_{8} and for the hyperbolic Grassmannians G_{8}^{∗}(ℝ^{n+16}) (n ≥ 1), G_{8}^{∗}(ℂ^{n+16}) (n ≥ 0), G_{8}^{∗}(ℍ^{n+16}) (n ≥ 0) the moduli space 𝔐_{F} of all noncongruent homogeneous codimension one foliations is isomorphic to ℝP^{7} ∪ {1, …, 8}. Another consequence is that on each of the hyperbolic spaces ℝH^{n}, ℂH^{n}, ℍH^{n} and 𝕆H^{2} there exist exactly two congruence classes of homogeneous codimension one foliations. One of them is given by the horosphere foliation, the other is not so well-known and has been studied in more detail by the first author in [4].

The construction of the model foliations 𝔉_{ℓ} and 𝔉_{i} is in fact elementary, they arise as orbits of codimension one subgroups of the solvable group in an Iwasawa decomposition of the isometry group. These model foliations have quite interesting geometric features. In the rank one case 𝔉_{ℓ} is a foliation by horospheres. For general rank the leaves of 𝔉_{ℓ} are still all congruent to each other (Proposition 3.1). If r ≥ 2, there exists an (r − 2)-dimensional family of harmonic foliations (i.e., all leaves are minimal) among the model foliations 𝔉_{ℓ}, and hence the corresponding projections F_{ℓ} : M → ℝ onto the space of leaves of 𝔉_{ℓ} are harmonic functions (Corollary 3.2). Each foliation 𝔉_{i} has exactly one minimal leaf (Corollary 4.5). Moreover, there exists a reflection of M in a totally geodesic submanifold leaving the unique minimal leaf invariant and interchanging the two leaves at any given positive distance from the minimal leaf (Proposition 4.6). A mean curvature argument shows that two such foliations are not isometrically congruent when they come from simple roots of different length. If α_{i} and α_{j} are simple roots with the same length, then there exists a bijective correspondence between the leaves of 𝔉_{i} and 𝔉_{j} such that corresponding leaves have the same principal curvatures, counted with multiplicities (Theorem 4.7). Moreover, we prove that 𝔉_{i} and 𝔉_{j} are congruent if and only if α_{i} and α_{j} are related by a Dynkin diagram symmetry, which implies (Corollary 4.9): On every connected irreducible Riemannian symmetric space of noncompact type and with rank r ≥ 3 there exist noncongruent homogeneous isoparametric systems for which corresponding leaves have the same principal curvatures, counted with multiplicities. Among the inhomogeneous isoparametric systems on spheres constructed by Ferus, Karcher and Münzner [9] from Clifford modules one can find systems with such an isospectral feature. The above examples show that this curious isospectral phenomenon also occurs within the class of homogeneous isoparametric systems.

The parameter space for the model foliations is ℝP^{r−1} ∪ {1, …, r}. The two major problems we have to solve is to determine the congruency classes among these model foliations and to show that any homogeneous codimension one foliation on M is congruent to one of these model foliations. The above mentioned geometric investigations provide partial answers to the first problem, but the result about the noncongruent homogeneous isoparametric systems indicates that it is impossible to solve it geometrically. We apply a conjugacy result of Alekseevsky [1] about completely solvable transitive groups of motions to reformulate the first problem in algebraic terms. One of the crucial steps in solving this algebraic problem is a rigidity result for Lie algebra isomorphisms of the nilpotent Lie algebra in an Iwasawa decomposition of 𝔤 (Theorem 3.4). This result, together with some structure theory for the solvable Lie algebras 𝔰_{ℓ}, enables us to decide the congruency problem for the foliations 𝔉_{ℓ} (Theorem 3.5). For the foliations 𝔉_{i} the algebraic problem is quite easy to handle when the multiplicity of the simple root is greater than one. In case of multiplicity one we introduce certain diagrams that allow us to deal with the congruency question in a similar fashion as Dynkin diagrams are used for semisimple complex Lie algebras (Theorem 4.8).

In order to prove that there are no other homogeneous codimension one foliations up to orbit equivalence we proceed as follows. First, using the Levi decomposition of a Lie group and a structure theorem by Malcev [19] about solvable Lie groups, we can restrict to connected solvable subgroups S of G^{o} acting on M freely and with cohomogeneity one. The Lie algebra 𝔰 of such a Lie group S is contained in a maximal solvable subalgebra of 𝔤 that contains, by a result of Mostow [21], a Cartan subalgebra 𝔥 of 𝔤. We prove that 𝔥 must be a maximally noncompact Cartan subalgebra of 𝔤, which implies that 𝔰 ⊂ 𝔱 + 𝔞 + 𝔫 for some suitable Iwasawa decomposition 𝔤 = 𝔨 + 𝔞 + 𝔫 of 𝔤 and a maximal abelian subalgebra 𝔱 in the centralizer of 𝔞 in 𝔨 (Proposition 5.2). The orthogonal projection 𝔰_{n} of 𝔰 onto 𝔞 + 𝔫 is a Lie subalgebra of 𝔞 + 𝔫 (Proposition 5.4). We classify all possible Lie subalgebras of 𝔞 + 𝔫 with codimension one (Lemma 5.3), which leads, up to orbit equivalence, to the model foliations constructed above. The final step is then to prove that S and the connected subgroup of AN with Lie algebra 𝔰_{n} induce isometrically congruent foliations (Theorem 5.5).

The paper is organized as follows. In the next section we present some basic material about the restricted root systems associated to Riemannian symmetric spaces of noncompact type. In Section 3 we construct the foliations 𝔉_{ℓ} and study the congruency problem for them, and in Section 4 we deal with the analogue for the foliations 𝔉_{i}. In Section 5 we prove that any homogeneous codimension one foliation on M is isometrically congruent to one of the foliations constructed in Sections 3 and 4.

We mention here that for the standard spaces of nonpositive constant curvature the relatively simple structure of the Gauss-Codazzi equations for submanifolds can be used to determine the entire moduli space 𝔐. For the Euclidean space ℝ^{n} it follows from work of Levi Civita [18] (for n = 3) and Segre [23] (in general) on isoparametric hypersurfaces that 𝔐 is a set of n points, one of which represents the subset 𝔐_{F}. Analogous work of E. Cartan [7] for the real hyperbolic space ℝH^{n} shows that for this space the moduli space 𝔐 consists of n + 1 points, two of which represent the subset 𝔐_{F}. An analogous approach for more general manifolds leads to hopeless calculations. We finally remark that the moduli space 𝔐_{S} is not known for any connected irreducible Riemannian symmetric space of noncompact type different from ℝH^{n}. Some recent work of the first author and Brück [5] on cohomogeneity one actions on the hyperbolic spaces ℂH^{n}, ℍH^{n} and 𝕆H^{2} indicates that the structure of this moduli space is far more complicated. In [6] the authors determined the subset of 𝔐_{S} that corresponds to totally geodesic singular orbits.

We would like to thank Dmitri Alekseevsky and Josef Dorfmeister for valuable comments and the Engineering and Physical Sciences Research Council (United Kingdom) for the financial support. We also thank the referee for pointing out to us a reference that shortened the proof of Lemma 4.1.

2. Preliminaries

For details about symmetric spaces and structure theory of semisimple Lie algebras we refer to the monographs [10], [11], [17] and [22]. For root systems Σ and corresponding sets {α_{1}, …, α_{r}} of simple roots we adopt the notation from [17].

Let M be a connected irreducible Riemannian symmetric space of noncompact type, n = dim M, r the rank of M, and denote by G the isometry group of M and by G^{o} its identity component. Let o ∈ M and K resp. K^{o} the isotropy subgroup of G resp. G^{o} at o. As M is simply connected, K^{o} is the identity component of K. We denote by 𝔤 and 𝔨 the Lie algebra of G and K, respectively, and by B the Killing form of 𝔤. If 𝔭 is the orthogonal complement of 𝔨 in 𝔤 with respect to B then 𝔤 = 𝔨 ⊕ 𝔭 is a Cartan decomposition of 𝔤. We identify 𝔭 with T_{o}M in the usual manner. If θ : 𝔤 → 𝔤 is the corresponding Cartan involution, we get a positive definite inner product on 𝔤 by ⟨X, Y ⟩ = −B(X, θY) for all X, Y ∈ 𝔤. We normalize the Riemannian metric on M such that its restriction to T_{o}M × T_{o}M = 𝔭 × 𝔭 coincides with ⟨·, ·⟩.

Let 𝔞 be a maximal abelian subspace in 𝔭 and denote by 𝔞^{∗} the dual vector space of 𝔞. For each λ ∈ 𝔞^{∗} we define 𝔤_{λ} = {X ∈ 𝔤 | ad(H)X = λ(H)X for all H ∈ 𝔞}. Since the linear transformations ad(H) : 𝔤 → 𝔤, H ∈ 𝔞, form a commuting family of selfadjoint transformations on 𝔤 with respect to ⟨·, ·⟩, they induce an eigenspace decomposition 𝔤 = 𝔤_{0} ⊕ ⨁_{λ∈Σ} 𝔤_{λ}, the so-called restricted root space decomposition of 𝔤 with respect to 𝔞. Here, Σ is the set of all nonzero λ ∈ 𝔞^{∗} for which 𝔤_{λ} is nontrivial. Each λ ∈ Σ is called a restricted root and the corresponding eigenspace 𝔤_{λ} is called a restricted root space. The eigenspace 𝔤_{0} with respect to 0 ∈ 𝔞^{∗} is given by 𝔤_{0} = C(𝔞; 𝔨) ⊕ 𝔞, where C(𝔞; 𝔨) is the centralizer of 𝔞 in 𝔨. The root system Σ is either reduced and then of type A_{r}, B_{r}, C_{r}, D_{r}, E_{6}, E_{7}, E_{8}, F_{4}, G_{2} or nonreduced and then of type BC_{r}.

For each λ ∈ 𝔞^{∗} let H_{λ} ∈ 𝔞 be the dual vector in 𝔞 with respect to the Killing form, that is, λ(H) = ⟨H_{λ}, H⟩ for all H ∈ 𝔞. Then we get an inner product on 𝔞^{∗}, which we also denote by ⟨·, ·⟩, by means of ⟨λ, μ⟩ = ⟨H_{λ}, H_{μ}⟩ for all λ, μ ∈ 𝔞^{∗}. We choose a set Λ = {α_{1}, …, α_{r}} of simple roots in Σ and denote the resulting set of positive restricted roots by Σ^{+}. Each root λ ∈ Σ can be written as λ = c_{1}α_{1} + ⋯ + c_{r}α_{r} with all integers c_{k} either all ≥ 0 or all ≤ 0. The sum c_{1} + ⋯ + c_{r} is called the level of the root λ. Let m be the level of the maximal root in Σ^{+}, that is, m = m_{1} + ⋯ + m_{r} such that m_{1}α_{1} + ⋯ + m_{r}α_{r} is the maximal root in Σ^{+}.

By Aut(DD) we denote the group of symmetries of the Dynkin diagram associated to Λ. Each symmetry P ∈ Aut(DD) can be linearly extended to a linear isometry of 𝔞^{∗}, which we also denote by P. Denote by Φ the linear isometry from 𝔞^{∗} to 𝔞 defined by Φ(λ) = H_{λ} for all λ ∈ 𝔞^{∗}. Then P̃ = Φ∘P ∘Φ^{−1} is a linear isometry of 𝔞 with P̃(H_{λ}) = H_{μ} if and only if P (λ) = μ, λ, μ ∈ 𝔞^{∗}. Since P is an orthogonal transformation, P̃ is just the dual map of P^{−1} : 𝔞^{∗} → 𝔞^{∗}. In this way each symmetry P ∈ Aut(DD) induces linear isometries of 𝔞^{∗} and 𝔞, both of which we will denote by P, since it will always be clear from the context which of these two we are using.

The choice of Λ induces a gradation 𝔤 = ⨁_{k=−m}^{m} 𝔤^{k} of 𝔤, where 𝔤^{k} is the linear subspace of 𝔤 spanned by all root spaces corresponding to roots of level k ∈ {−m, …, m}. This gradation is of type α_{0} (see [15]), i.e., 𝔤^{1} generates the subalgebra ⨁_{k=1}^{m} 𝔤^{k} and 𝔤^{−1} generates the subalgebra ⨁_{k=−m}^{−1} 𝔤^{k}.

For λ, μ ∈ Σ^{+}, λ and μ linearly independent, the μ-string containing λ is the sequence λ−pμ, …, λ, …, λ+qμ such that p and q are nonnegative integers, λ−pμ, …, λ+qμ ∈ Σ^{+}, and λ−(p+1)μ, λ+(q+1)μ ∉ Σ^{+}. Then we have 2⟨λ, μ⟩/⟨μ, μ⟩ = p − q. The integer p + q ∈ {0, 1, 2, 3} is called the length of the μ-string containing λ. It is worthwhile to point out at this point that this formula enables us to calculate the inner product between roots by using the length of strings involving them. In particular, the Dynkin diagram of the restricted root system can be constructed from the set Λ of simple roots just from the string relations between the simple roots. More precisely, assume that α and β are two simple roots and consider the α-string containing β and the β-string containing α, that is, β, …, β + q_{βα}α and α, …, α + q_{αβ}β. If q_{αβ} = q_{βα}, then connect α and β by q_{αβ} lines. If q_{αβ} ≠ q_{βα}, then connect α and β by max{q_{αβ}, q_{βα}} lines and draw an arrow from α to β if q_{αβ} > q_{βα} and from β to α if q_{βα} > q_{αβ}.

If we define 𝔫 = ⨁_{λ∈Σ^{+}} 𝔤_{λ} we get an Iwasawa decomposition 𝔤 = 𝔨 ⊕ 𝔞 ⊕ 𝔫 of 𝔤. The subalgebra 𝔫 of 𝔤 is an m-step nilpotent subalgebra, where m is the level of the maximal root in Σ^{+}. The Lie algebra 𝔫 has a natural gradation 𝔫 = 𝔫_{1} ⊕⋯⊕𝔫_{m}, where 𝔫_{k} = 𝔤^{k}. Note that 𝔫_{1} generates 𝔫. Moreover, 𝔞+𝔫 is a solvable subalgebra of 𝔤 with [𝔞+𝔫, 𝔞+𝔫] = 𝔫. The connected subgroups A, N, AN of G with Lie algebras 𝔞, 𝔫, 𝔞 + 𝔫, respectively, are simply connected and AN acts simply transitively on M. The symmetric space M is isometric to the connected, simply connected, solvable Lie group AN equipped with the left-invariant Riemannian metric that is induced from the inner product ⟨·, ·⟩. We denote by ∇ the Levi Civita connection of the solvable Lie group AN equipped with this left-invariant metric. Using the Koszul formula for the Levi Civita connection, and using the fact that the metric on AN is leftinvariant, we get 2⟨∇_{X}Y, Z⟩ = ⟨[X, Y], Z⟩ − ⟨[Y, Z], X⟩ + ⟨[Z, X], Y ⟩ for all X, Y, Z ∈ 𝔞 + 𝔫, where we regard elements in the Lie algebra 𝔞 + 𝔫 as left-invariant vector fields on AN.

3. The foliations 𝔉_{ℓ}

Let ℓ be a linear line in 𝔞. Then the orthogonal complement 𝔰_{ℓ} = (𝔞+𝔫)⊖ℓ = (𝔞⊖ℓ)+𝔫 of ℓ in 𝔞+𝔫 is a subalgebra of 𝔞+𝔫 of codimension one. Let S_{ℓ} be the connected Lie subgroup of AN with Lie algebra 𝔰_{ℓ}. Then the orbits of the action of S_{ℓ} on M form a homogeneous foliation 𝔉_{ℓ} on M of codimension one.

Let H_{ℓ} ∈ 𝔞 be a unit vector such that ℓ = ℝH_{ℓ}. From the above expression for the Levi Civita connection we get ∇_{H_{ℓ}}H_{ℓ} = 0. General theory about Riemannian foliations implies that the integral curves of H_{ℓ} are geodesics in AN and hence intersect each leaf of 𝔉_{ℓ}, and at the points of intersection it intersects perpendicularly. Let γ : ℝ → AN be the geodesic in AN with γ(0) = o and γ̇(0) = H_{ℓ}. Then γ(ℝ) ⊂ A, and γ intersects each leaf of 𝔉_{ℓ}. Moreover, as A is abelian, N ⊂ S_{ℓ} and AN = NA, we get γ(t)S_{ℓ} = S_{ℓ}γ(t) and hence S_{ℓ}·γ(t) = γ(t)(γ(t)^{−1}S_{ℓ}γ(t))·o = γ(t)S_{ℓ} · o for all t ∈ ℝ. This shows that each leaf of 𝔉_{ℓ} is obtained by a suitable left translation of S_{ℓ} · o in AN. In particular, all leaves of 𝔉_{ℓ} are congruent to each other under an isometry of M. In order to study the geometry of the foliation 𝔉_{ℓ} it is therefore sufficient to study the geometry of the leaf S_{ℓ} · o.

The vector H_{ℓ} is a unit normal vector of S_{ℓ} · o at o. We denote by A_{H_{ℓ}} the shape operator of S_{ℓ} · o at o with respect to H_{ℓ} and by h the second fundamental form of S_{ℓ} · o. Since ad(H_{ℓ}) is a selfadjoint endomorphism on 𝔤 with respect to ⟨·, ·⟩, the above expression for the Levi Civita connection and the Weingarten formula imply ⟨h(X, Y), H_{ℓ}⟩ = ⟨A_{H_{ℓ}}X, Y ⟩ = ⟨ad(H_{ℓ})X, Y ⟩ for all X, Y ∈ 𝔰_{ℓ} = T_{o}(S_{ℓ} · o). Therefore A_{H_{ℓ}} = ad(H_{ℓ})|𝔰_{ℓ}, and we have proved:

Proposition 3.1. Using the above notations, we have:

(1) All leaves of 𝔉_{ℓ} are isometrically congruent to each other.

(2) The shape operator A_{H_{ℓ}} of the leaf S_{ℓ} · o of 𝔉_{ℓ} through o is given by A_{H_{ℓ}} = ad(H_{ℓ})|𝔰_{ℓ}.

(3) The subspace 𝔞 ⊖ ℓ is a principal curvature space of S_{ℓ} · o with corresponding principal curvature 0, and for each λ ∈ Σ^{+} the root space 𝔤_{λ} is a principal curvature space of S_{ℓ} · o with corresponding principal curvature λ(H_{ℓ}).

(4) For the (constant) mean curvature μ_{ℓ} of each leaf of 𝔉_{ℓ} we have

μ_{ℓ} = \frac{1}{n − 1} ∑_{λ∈Σ^{+}}(dim 𝔤_{λ})λ(H_{ℓ}).

A more general treatment of foliations induced by subalgebras on Lie groups with left-invariant Riemannian metrics can be found in [26]. It is worthwhile to mention the following observation. If the rank of M is greater than one, then the unit sphere in 𝔞 is connected. If H_{ℓ} lies in the positive Weyl chamber C^{+} of 𝔞, then the mean curvature of S_{ℓ} · o is positive, whereas it is negative when we choose H_{ℓ} in the negative Weyl chamber C^{−} of 𝔞. For continuity reasons we therefore find on each great circle in the unit sphere in 𝔞 through H_{ℓ} ∈ C^{+} and −H_{ℓ} ∈ C^{−} a point H_{ℓ′} ∈ 𝔞 such that the corresponding orbit S_{ℓ′} · o is minimal. Since all leaves of 𝔉_{ℓ′} are isometrically congruent to each other we conclude:

Corollary 3.2. On each connected irreducible Riemannian symmetric space of noncompact type and rank r ≥ 2 there exists an (r − 2)- dimensional family of homogeneous harmonic foliations of codimension one.

A foliation is called harmonic if all its leaves are minimal submanifolds. It is known that a foliation on a Riemannian manifold is harmonic if and only if the canonical projection from the manifold onto the space of leaves of the foliation is a harmonic map. From the above corollary we thus get an (r − 2)-dimensional family of harmonic functions on the symmetric space M with the property that its level sets are homogeneous hypersurfaces.

Our next aim is to investigate when two such foliations 𝔉_{ℓ} and 𝔉_{ℓ′} are isometrically congruent. We start with some basic properties about the subalgebras 𝔰_{ℓ}.

Lemma 3.3. Let r ≥ 2. The following statements hold:

(1) The Lie algebra 𝔰_{ℓ} is completely solvable, that is, there exists a basis of 𝔰_{ℓ} such that ad(𝔰_{ℓ}) consists of upper triangular matrices with respect to that basis.

(2) The subalgebra

𝔠_{ℓ} = \begin{cases}(𝔞 ⊖ ℓ) ⊕ 𝔤_{λ} ⊕ 𝔤_{2λ} & if ℓ = ℝH_{λ} for some λ ∈ Σ^{+}, λ/2 ∉ Σ^{+} \\ 𝔞 ⊖ ℓ & otherwise \end{cases}

is a Cartan subalgebra of 𝔰_{ℓ}. Note that 𝔤_{2λ} is trivial if 2λ ∉ Σ^{+}.

(3) The set {Y ∈ 𝔰_{ℓ} | ad(Y)^{k} = 0 for some k ∈ ℕ} of all nilpotent elements in 𝔰_{ℓ} is equal to 𝔫.

Proof. (1): Consider the gradation 𝔰_{ℓ} = (𝔞 ⊖ ℓ) ⊕ 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m} of 𝔰_{ℓ}. Choose a basis of 𝔰_{ℓ} such that the first vectors in that basis are in 𝔫_{m}, the next ones in 𝔫_{m−1}, and so on, and the last vectors are in 𝔞 ⊖ ℓ. It then follows immediately that ad(𝔰_{ℓ}) consists of upper triangular matrices with respect to that basis.

(2): We first assume that ℓ = ℝH_{λ} for some positive root λ ∈ Σ^{+} with λ/2 ∉ Σ^{+}. It is clear that 𝔠_{ℓ} = (𝔞 ⊖ ℓ) ⊕ 𝔤_{λ} ⊕ 𝔤_{2λ} is a nilpotent subalgebra of 𝔰_{ℓ}. If in particular 𝔤_{2λ} is trivial, then 𝔠_{ℓ} is an abelian subalgebra of 𝔰_{ℓ}. We have to show that 𝔠_{ℓ} equals its own normalizer in 𝔰_{ℓ}. Let X be in the normalizer of 𝔠_{ℓ} in 𝔰_{ℓ}. We write X = X_{1} + X_{2} with X_{1} ∈ 𝔠_{ℓ} and X_{2} ∈ 𝔫 ⊖ (𝔤_{λ} ⊕ 𝔤_{2λ}) = 𝔰_{ℓ} ⊖ 𝔠_{ℓ}. Since [𝔠_{ℓ}, 𝔠_{ℓ}] ⊂ 𝔠_{ℓ} and [𝔠_{ℓ}, 𝔰_{ℓ} ⊖ 𝔠_{ℓ}] ⊂ 𝔰_{ℓ} ⊖ 𝔠_{ℓ} we have [𝔠_{ℓ}, X_{2}] = 0. We now decompose X_{2} into X_{2} = ∑_{μ∈Σ^{+}} X_{2}^{μ}. If X_{2} ≠ 0, there exists a root ν ∈ Σ^{+} such that X_{2}^{ν} ≠ 0 and the level of ν is less or equal than the level of all roots μ ∈ Σ^{+} for which X_{2}^{μ} ≠ 0. The orthogonal projection of [𝔠_{ℓ}, X_{2}] onto 𝔤_{ν} is equal to the orthogonal projection of [𝔞 ⊖ ℓ, X_{2}] onto 𝔤_{ν}. Since ν ∉ {λ, 2λ}, and hence H_{ν} ∉ ℝH_{λ}, the orthogonal projection of [𝔞 ⊖ ℓ, X_{2}] onto 𝔤_{ν} is nontrivial, which contradicts [𝔠_{ℓ}, X_{2}] = 0. Thus we must have X_{2} = 0, and it follows that 𝔠_{ℓ} is a Cartan subalgebra of 𝔰_{ℓ}.

Next, assume that ℓ ≠ ℝH_{λ} for all λ ∈ Σ^{+}. We have to show that the normalizer of the abelian subalgebra 𝔠_{ℓ} = 𝔞 ⊖ ℓ in 𝔰_{ℓ} is equal to 𝔠_{ℓ}. Let X be in that normalizer and write X = X_{1} + X_{2} with X_{1} ∈ 𝔠_{ℓ} and X_{2} ∈ 𝔫 = 𝔰_{ℓ} ⊖ 𝔠_{ℓ}. We now decompose X_{2} into X_{2} = ∑_{μ∈Σ^{+}} X_{2}^{μ}. Then [H, X] = [H, X_{2}] = ∑_{μ∈Σ^{+}} μ(H)X_{2}^{μ} ∈ 𝔠_{ℓ} = 𝔞 ⊖ ℓ for all H ∈ 𝔞 ⊖ ℓ. Since by assumption H_{ℓ} is not a multiple of any root vector H_{λ}, we have μ(𝔞 ⊖ ℓ) ≠ 0 for all μ ∈ Σ^{+}, which implies X_{2}^{μ} = 0 for all μ ∈ Σ^{+}. Thus X_{2} = 0, and it follows that 𝔠_{ℓ} = 𝔞 ⊖ ℓ is a Cartan subalgebra of 𝔰_{ℓ}.

(3): From the graded Lie algebra structure of 𝔫 it is clear that every element in 𝔫 is nilpotent. Let 0 ≠ H ∈ 𝔞 ⊖ ℓ and X ∈ 𝔫. Then there exists a simple root α ∈ Λ such that α(H) ≠ 0. Let Y ∈ 𝔤_{α} be a nonzero vector. We will now show by induction that (ad(H + X)^{k}Y)_{𝔤_{α}} = α(H)^{k}Y ≠ 0 for all positive integers k, where (·)_{𝔤_{α}} denotes the orthogonal projection onto 𝔤_{α}.

Since X ∈ 𝔫 and Y ∈ 𝔤_{α}, and as α is a simple root, we have [X, Y] ∈ 𝔫_{2} ⊕ ⋯ ⊕ 𝔫_{m}, and hence the 𝔤_{α}-component of ad(X)Y vanishes. It follows that (ad(H + X)Y)_{𝔤_{α}} = (ad(H)Y)_{𝔤_{α}} = α(H)Y ≠ 0, which proves the assertion for k = 1.

Now assume that the assertion holds for the positive integer k and consider the expression (ad(H + X)^{k+1}Y)_{𝔤_{α}} = (ad(H + X)ad(H + X)^{k}Y)_{𝔤_{α}}. Since ad(H + X)^{k}Y ∈ 𝔫, only the 𝔤_{α}-component of it contributes to a 𝔤_{α}-component in (ad(H + X)ad(H + X)^{k}Y)_{𝔤_{α}}, that is, (ad(H +X)^{k+1}Y)_{𝔤_{α}} = (ad(H +X)((ad(H +X)^{k})Y)_{𝔤_{α}})_{𝔤_{α}}. The assumption and the statement for k = 1 then imply (ad(H + X)^{k+1}Y)_{𝔤_{α}} = α(H)^{k}(ad(H + X)Y)_{𝔤_{α}} = α(H)^{k+1}Y ≠ 0. This concludes the induction and shows that H + X is not a nilpotent element in 𝔰_{ℓ}. q.e.d.

The next result provides a necessary and sufficient condition in order that a Lie algebra isomorphism of 𝔫 can map a root space 𝔤_{λ} onto another root space 𝔤_{μ}. This result is of interest in its own right and will be applied later to settle the congruency problem.

Theorem 3.4. Let λ, μ ∈ Σ^{+}. Then there exists a Lie algebra isomorphism F of 𝔫 with F (𝔤_{λ}) = 𝔤_{μ} if and only if there exists a symmetry P ∈ Aut(DD) with P (λ) = μ.

Proof. The if-part follows from standard theory of real semisimple Lie algebras. For the converse, let F be a Lie algebra isomorphism of 𝔫 with F (𝔤_{λ}) = 𝔤_{μ}. First of all we will show that without loss of generality we can assume that F preserves the gradation 𝔫 = 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m}. For X ∈ 𝔫 we denote by X_{i} the orthogonal projection of X onto 𝔫_{i}. We define a linear map F̃ : 𝔫 → 𝔫 , X ↦ ∑_{i=1}^{m}(F (X_{i}))_{i}. Consider the lower central series of 𝔫, that is, L_{0}(𝔫) := 𝔫, L_{1}(𝔫) := [𝔫, 𝔫], and L_{k}(𝔫) := [L_{k−1}(𝔫), 𝔫] = 𝔫_{k+1} ⊕ ⋯ ⊕ 𝔫_{m} (2 ≤ k ≤ m − 1). Since F is a Lie algebra isomorphism of 𝔫 it induces a vector space isomorphism of L_{k}(𝔫) for all k ∈ {0 …, m − 1}. From this fact it follows easily that F̃ is a vector space isomorphism of 𝔫. For X_{i} ∈ 𝔫_{i} and Y_{j} ∈ 𝔫_{j}, 1 ≤ i, j ≤ m, we have

F̃[X_{i}, Y_{j}] = (F [X_{i}, Y_{j}])_{i+j} = [F X_{i}, F Y_{j}]_{i+j}
= [(F X_{i})_{i} + ⋯ + (F X_{i})_{m}, (F Y_{j})_{j} + ⋯ + (F Y_{j})_{m}]_{i+j}

= [(F X_{i})_{i}, (F Y_{j})_{j}]_{i+j} = [(F X_{i})_{i}, (F Y_{j})_{j}] = [F̃X_{i}, F̃X_{j}] ,

which implies that F̃ is a Lie algebra isomorphism of 𝔫 with F̃(𝔤_{λ}) = 𝔤_{μ}. Thus we can assume from now on that F preserves the gradation 𝔫 = 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m}, that is, F (𝔫_{i}) = 𝔫_{i} for all i ∈ {1, …, m}.

Since F preserves the gradation of 𝔫 it is clear that the level m_{λ} of λ must be equal to the level m_{μ} of μ. For a linear subspace V of 𝔫 we denote by C(V; 𝔫_{k}) the centralizer of V in 𝔫_{k}. Obviously, when F preserves V then F preserves also C(V; 𝔫_{k}) and the subalgebra of 𝔫 generated by C(V; 𝔫_{k}). Of particular importance for us will be the subalgebra 𝔫(α̃) generated by C(𝔫_{m−1}; 𝔫_{1}). Note that 𝔫(α̃) is the subalgebra of 𝔫 generated by the root spaces of those simple roots in Λ that are not connected with the maximal root α̃ in the extended Dynkin diagram of Σ. Since F preserves 𝔫(α̃) we have either 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃) or 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫 ⊖ 𝔫(α̃). We now prove the theorem by considering the root systems case-by-case.

Σ = A_{r}, r ≥ 1: For r ∈ {1, 2} this is trivial. Assume that r ≥ 3. The subalgebra 𝔫(α̃) is of type A_{r−2} and generated by 𝔤_{α_{2}} ⊕⋯⊕𝔤_{α_{r−1}}. Here, and in the following, when we say this subalgebra is of type A_{r−2} we refer to the nilpotent part in the induced Iwasawa decomposition of the semisimple subalgebra of 𝔤 of type A_{r−2} determined by the simple roots α_{2}, …, α_{r−2}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), we can apply an inductive argument, because every symmetry of the Dynkin diagram of the A_{r−2}-type subalgebra can be extended to a symmetry of the Dynkin diagram of A_{r}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫⊖𝔫(α̃), we have {λ, μ} ∈ {α_{1} +⋯+α_{m_{λ}}, α_{r−m_{λ}+1} +⋯+α_{r}}, which implies that there exists a symmetry P ∈ Aut(DD) with P (λ) = μ.

Σ = B_{r}, r ≥ 2: For r = 2 this is trivial. In the case r = 3 the subalgebra 𝔫(α̃) is of type A_{1} ⊕ A_{1} and equal to 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}}. This implies that F cannot map 𝔤_{α_{2}} onto 𝔤_{α_{1}} or 𝔤_{α_{3}}, and vice versa. The subspace [𝔤_{α_{1}}, [𝔤_{α_{1}}, 𝔤_{α_{2}}]] is trivial, whereas the subspace [𝔤_{α_{3}}, [𝔤_{α_{3}}, 𝔤_{α_{2}}]] ⊂ 𝔤_{α_{2}+2α_{3}} is nontrivial, and since the projection of F (𝔤_{α_{2}}) onto 𝔤_{α_{2}} must be equal to 𝔤_{α_{2}}, this implies that F cannot map 𝔤_{α_{1}} onto 𝔤_{α_{3}}, and vice versa. Since C(𝔤_{α_{1}+α_{2}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} and C(𝔤_{α_{2}+α_{3}}; 𝔫_{1}) = 𝔤_{α_{2}}, the isomorphism F cannot map one of the two root spaces on level 2 onto the other one. Finally, C(𝔤_{α_{1}+α_{2}+α_{3}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕𝔤_{α_{2}} generates a subalgebra of type A_{2}, whereas C(𝔤_{α_{2}+2α_{3}}; 𝔫_{1}) = 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} generates a subalgebra of type B_{2}, which implies that F cannot map one of the two root spaces on level 3 onto the other one. This proves the assertion for B_{3}. Now let r ≥ 4. The subalgebra 𝔫(α̃) is the subalgebra of type A_{1} ⊕ B_{r−2} generated by 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ ⋯ ⊕ 𝔤_{α_{r}}. This implies F (𝔤_{α_{1}}) = 𝔤_{α_{1}}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), we can apply an inductive argument to see that λ = μ. If 𝔤_{λ}, 𝔤_{μ} ⊄ 𝔫(α̃), then both λ and μ have an α_{2}-component. On each level m_{λ} ∈ {2, …, 2r − 2} there are exactly two roots with an α_{2}- component, namely {α_{1} + ⋯ + α_{m_{λ}}, α_{2} + ⋯ + α_{m_{λ}+1}} if m_{λ} < r, {α_{1} +⋯+α_{r}, α_{2} +⋯+α_{r−1} +2α_{r}} if m_{λ} = r, and {α_{1} +⋯+α_{2r−m_{λ}} + 2α_{2r−m_{λ}+1} + ⋯ + 2α_{r}, α_{2} + ⋯ + α_{2r−m_{λ}−1} + 2α_{2r−m_{λ}} + ⋯ + 2α_{r}} if m_{λ} > r. In all cases the centralizer of the first root in 𝔫_{1} contains the invariant root space 𝔤_{α_{1}}, whereas the centralizer of the second root in 𝔫_{1} does not contain this invariant subspace. This implies the assertion, since on the levels 1 and 2r − 1 = m there is only one root with an α_{2}-component.

Σ = C_{r}, r ≥ 2: For r = 2 this is the same as for B_{2}. For r ≥ 3 we see that 𝔫(α̃) is a subalgebra of type C_{r−1} generated by 𝔤_{α_{2}} ⊕ ⋯ ⊕ 𝔤_{α_{r}}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), we can apply an inductive argument to show that λ = μ. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫 ⊖ 𝔫(α̃), then both λ and μ must have an α_{1}-component. Since on each level there exists only one root with an α_{1}-component this implies λ = μ.

Σ = D_{r}, r ≥ 3: For r = 3 this is the same as for A_{3}. If r = 4, then 𝔫(α̃) is the subalgebra of type A_{1} ⊕ A_{1} ⊕ A_{1} given by the root spaces 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}}. Thus the root space 𝔤_{α_{2}} cannot be mapped onto any other simple root space under F, and vice versa. On the other hand, the symmetry group of the Dynkin diagram of D_{4} is the symmetric group 𝔖_{3} of three letters, and any two simple roots different from α_{2} can be mapped to each other by such a symmetry. There are three roots on level 2 and three roots on level 3, and on each level any two of them can be mapped to each other by a symmetry in 𝔖_{3}. On higher levels there is only one root. Now assume that r ≥ 5. In this case 𝔫(α̃) is a subalgebra of type A_{1} ⊕ D_{r−2} generated by 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ ⋯ ⊕ 𝔤_{α_{r}}. In particular, this implies F (𝔤_{α_{1}}) = 𝔤_{α_{1}}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), we again want to apply an inductive argument. It is clear that every symmetry of the Dynkin diagram of D_{r−2} can be extended to a symmetry of the Dynkin diagram of D_{r} if r ≠ 6. We thus have to investigate the special case r = 6. We have C(𝔫_{7}; 𝔫_{1}) = 𝔤_{α_{2}} ⊕ 𝔤_{α_{4}} ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}}, which generates a subalgebra of type A_{1} ⊕ A_{3} and implies that F (𝔤_{α_{2}}) = 𝔤_{α_{2}}. Next, we have C(𝔫_{6}; 𝔫_{1}) = 𝔤_{α_{2}} ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}}, which is a subalgebra of type A_{1} ⊕ A_{1} ⊕ A_{1}. Since F preserves 𝔤_{α_{2}}, this implies F (𝔤_{α_{5}} ⊕ 𝔤_{α_{6}}) = 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}}, which shows that r = 6 does not cause any problems for the inductive argument. If 𝔤_{λ}, 𝔤_{μ} ⊄ 𝔫(α̃), then both λ and μ have an α_{2}-component. On each level m_{λ} ∈ {2, …, 2r − 5} there are exactly two roots with an α_{2}-component modulo the nontrivial Dynkin diagram symmetry, namely {α_{1} + ⋯ + α_{m_{λ}}, α_{2} + ⋯ + α_{m_{λ}+1}} if m_{λ} < r − 1, {α_{1} + ⋯ + α_{r}, α_{2} + ⋯ + 2α_{r−2} + α_{r−1} + α_{r}} if m_{λ} = r, and {α_{1} + ⋯ + α_{2r−m_{λ}−2} + 2α_{2r−m_{λ}−1} + ⋯ + 2α_{r−2} + α_{r−1} + α_{r}, α_{2} + ⋯ + α_{2r−m_{λ}−3} + 2α_{2r−m_{λ}−2} + ⋯ + 2α_{r−2} + α_{r−1} + α_{r}} if m_{λ} > r. In all cases the centralizer of the second root in 𝔫_{1} contains the invariant root space 𝔤_{α_{1}}, whereas the centralizer of the first root in 𝔫_{1} does not contain this invariant subspace. This implies the assertion, since on the levels 1, 2r − 4 and 2r − 3 = m there is only one root with an α_{2}-component.

Σ = BC_{r}, r ≥ 2: For r = 2 we have 𝔫(α̃) = 𝔤_{α_{2}} ⊕𝔤_{2α_{2}}, which readily implies that F cannot map any root space onto another root space. For r ≥ 3, 𝔫(α̃) is a subalgebra of type BC_{r−1} generated by 𝔤_{α_{2}} ⊕⋯⊕𝔤_{α_{r}}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), we can apply an inductive argument to show that λ = μ. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫 ⊖ 𝔫(α̃), then both λ and μ must have an α_{1}-component. Since on each level m_{λ} ∈ {1, …, 2r} there exists exactly one root with an α_{1}-component this implies λ = μ.

Σ = E_{6}: The root β_{10} = α_{1} + α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + α_{6} is the only root on level 10 and C(𝔤_{β_{10}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ ⋯ ⊕ 𝔤_{α_{6}}, which generates a subalgebra of type A_{5}. This shows that F leaves the subalgebra generated by 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}} ⊕ 𝔤_{α_{5}} invariant, which is a subalgebra of type A_{3}. Using 𝔫(α̃) for this A_{3}-type subalgebra implies F (𝔤_{α_{4}}) = 𝔤_{α_{4}}. The root β_{9} = α_{1} + α_{2} + 2α_{3} + 2α_{4} + 2α_{5} + α_{6} is the only root on level 9 and C(𝔤_{β_{9}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}}, which generates a subalgebra of type A_{2} ⊕ A_{1} ⊕ A_{2}. The A_{1}-part gives F (𝔤_{α_{2}}) = 𝔤_{α_{2}}. Taking the intersection of the (A_{2} ⊕ A_{1} ⊕ A_{2})-type subalgebra with the A_{3}-type subalgebra obtained in level 10 we see that F (𝔤_{α_{3}} ⊕ 𝔤_{α_{5}}) = 𝔤_{α_{3}} ⊕ 𝔤_{α_{5}}. The subalgebra 𝔫(α̃) is of type A_{5} and generated by 𝔤_{α_{1}} ⊕𝔤_{α_{3}} ⊕⋯⊕𝔤_{α_{6}}. If 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫(α̃), there exists a transformation in the symmetry group of the Dynkin diagram of A_{5} mapping λ to μ. The structure of the Dynkin diagram of E_{6} shows that this symmetry extends to a symmetry of the Dynkin diagram of E_{6}. Now assume that 𝔤_{λ}, 𝔤_{μ} ⊂ 𝔫 ⊖ 𝔫(α̃). From the structure of the root system E_{6}, and taking into account the symmetry group of the Dynkin diagram of E_{6}, it remains to show that the root spaces corresponding to the following roots cannot be mapped to each other under F:

on level 4: β_{4}^{1} = α_{1} + α_{2} + α_{3} + α_{4}, β_{4}^{2} = α_{2} + α_{3} + α_{4} + α_{5};

on level 5: β_{5}^{1} = α_{1} + α_{2} + α_{3} + α_{4} + α_{5}, β_{5}^{2} = α_{2} + α_{3} + 2α_{4} + α_{5};

on level 6: β_{6}^{1} = α_{1}+α_{2}+α_{3}+α_{4}+α_{5}+α_{6}, β_{6}^{2} = α_{1}+α_{2}+α_{3}+2α_{4}+α_{5};

on level 7: β_{7}^{1} = α_{1}+α_{2}+α_{3}+2α_{4}+α_{5}+α_{6}, β_{7}^{2} = α_{1}+α_{2}+2α_{3}+2α_{4}+α_{5}.

These are exactly all pairs of roots in Σ^{+} modulo Aut(DD)-congruence with an α_{1}-component. The assertion then follows by considering the following centralizers:

on level 4: 𝔤_{β_{4}^{1}} ⊂ C(𝔤_{α_{4}}; 𝔫_{4}), 𝔤_{β_{4}^{2}} ⊄ C(𝔤_{α_{4}}; 𝔫_{4});

on level 5: 𝔤_{β_{5}^{1}} ⊂ C(𝔤_{α_{2}}; 𝔫_{5}), 𝔤_{β_{5}^{2}} ⊄ C(𝔤_{α_{2}}; 𝔫_{5});

on level 6: 𝔤_{β_{6}^{1}} ⊄ C(𝔤_{α_{4}}; 𝔫_{6}), 𝔤_{β_{6}^{2}} ⊂ C(𝔤_{α_{4}}; 𝔫_{6});

on level 7: 𝔤_{β_{7}^{1}} ⊄ C(𝔤_{α_{3}} ⊕ 𝔤_{α_{5}}; 𝔫_{7}), 𝔤_{β_{7}^{2}} ⊂ C(𝔤_{α_{3}} ⊕ 𝔤_{α_{5}}; 𝔫_{7}), and the fact that all these centralizers are invariant under F.

Σ = E_{7}: The root β_{16} = α_{1} +2α_{2} +3α_{3} +4α_{4} +3α_{5} +2α_{6} +α_{7} is the only root on level 16 and C(𝔤_{β_{16}}; 𝔫_{1}) = 𝔤_{α_{2}} ⊕ ⋯ ⊕ 𝔤_{α_{7}}, which generates a subalgebra of type D_{6}. Applying the 𝔫(α̃)-method to this subalgebra shows that F leaves the subalgebra generated by 𝔤_{α_{2}} ⊕ ⋯ ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{7}} invariant, which is a subalgebra of type D_{4} ⊕A_{1}. This implies F (𝔤_{α_{7}}) = 𝔤_{α_{7}}. The root β_{15} = α_{1} + 2α_{2} + 2α_{3} + 4α_{4} + 3α_{5} + 2α_{6} + α_{7} is the only root on level 15 and C(𝔤_{β_{15}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} ⊕ 𝔤_{α_{4}} ⊕ ⋯ ⊕ 𝔤_{α_{7}}, which generates a subalgebra of type A_{1} ⊕ A_{5}. This implies F (𝔤_{α_{1}}) = 𝔤_{α_{1}}. Moreover, applying the 𝔫(α̃)-method to the subalgebra of type A_{5} implies that F leaves the subalgebra generated by 𝔤_{α_{4}} ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}} invariant, which is a subalgebra of type A_{3}. Applying once again the 𝔫(α̃)-method to this A_{3}-type subalgebra shows that F (𝔤_{α_{5}}) = 𝔤_{α_{5}}. The root β_{14} = α_{1} +2α_{2} +2α_{3} +3α_{4} +3α_{5} +2α_{6} +α_{7} is the only root on level 14 and C(𝔤_{β_{14}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{5}} ⊕ 𝔤_{α_{6}} ⊕ 𝔤_{α_{7}}, which generates a subalgebra of type A_{2} ⊕ A_{1} ⊕ A_{3}. The A_{1}-part implies F (𝔤_{α_{2}}) = 𝔤_{α_{2}}, and applying the 𝔫(α̃)-method to the A_{3}-part implies F (𝔤_{α_{6}}) = 𝔤_{α_{6}}. Taking the intersection of the D_{6}-type subalgebra obtained in level 16 and the A_{2}-type subalgebra obtained in level 14 implies F (𝔤_{α_{3}}) = 𝔤_{α_{3}}. On level 13 there are two roots, namely β_{13}^{1} = α_{1} + 2α_{2} + 2α_{3} + 3α_{4} + 2α_{5} + 2α_{6} + α_{7} and β_{13}^{2} = α_{1} + α_{2} + 2α_{3} + 3α_{4} + 3α_{5} + 2α_{6} + α_{7}, and C(𝔤_{β_{13}^{1}} ⊕ 𝔤_{β_{13}^{2}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}} ⊕ 𝔤_{α_{6}} ⊕ 𝔤_{α_{7}}, which generates a subalgebra of type A_{3} ⊕ A_{2}. Taking the intersection of the A_{3}-type subalgebra with the A_{5}-type subalgebra obtained in level 15 implies F (𝔤_{α_{4}}) = 𝔤_{α_{4}}. Hence all simple roots spaces are preserved by F, which implies F (𝔤_{λ}) = 𝔤_{λ} for all λ ∈ Σ^{+}.

Σ = E_{8}: The root β_{28} = 2α_{1} +3α_{2} +4α_{3} +6α_{4} +5α_{5} +4α_{6} +3α_{7} +α_{8} is the only root on level 28 and C(𝔤_{β_{28}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ ⋯ ⊕ 𝔤_{α_{7}}, which generates a subalgebra of type E_{7}. Since F preserves this subalgebra, the discussion in the previous case shows that F (𝔤_{α_{i}}) = 𝔤_{α_{i}} for all i ∈ {1, …, 7}. The root β_{27} = 2α_{1}+3α_{2}+4α_{3}+6α_{4}+5α_{5}+4α_{6}+2α_{7}+α_{8} is the only root on level 27 and C(𝔤_{β_{27}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕⋯⊕𝔤_{α_{6}} ⊕𝔤_{α_{8}}, which generates a subalgebra of type E_{6} ⊕ A_{1}. This shows F (𝔤_{α_{8}}) = 𝔤_{α_{8}}. Hence F preserves all simple root spaces, which implies F (𝔤_{λ}) = 𝔤_{λ} for all λ ∈ Σ^{+}.

Σ = F_{4}: The root β_{10} = α_{1} + 3α_{2} + 4α_{3} + 2α_{4} is the only root on level 10 and C(𝔤_{β_{10}}; 𝔫_{1}) = 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}}, which generates a subalgebra of type C_{3}. This subalgebra is invariant under F, and applying the 𝔫(α̃)-method to it shows that the subalgebra generated by 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} is invariant under F, which is a subalgebra of type B_{2}. Applying the 𝔫(α̃)-method to this B_{2}-type subalgebra implies that F (𝔤_{α_{2}}) = 𝔤_{α_{2}}. The root β_{9} = α_{1} + 2α_{2} + 4α_{3} + 2α_{4} is the only root on level 9 and C(𝔤_{β_{9}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}}, which generates a subalgebra of type A_{1} ⊕ A_{2}. This implies F (𝔤_{α_{1}}) = 𝔤_{α_{1}} and F (𝔤_{α_{3}} ⊕ 𝔤_{α_{4}}) = 𝔤_{α_{3}} ⊕ 𝔤_{α_{4}}. Since F leaves also 𝔤_{α_{2}} ⊕ 𝔤_{α_{3}} invariant, the latter statement implies F (𝔤_{α_{3}}) = 𝔤_{α_{3}}. Finally, the root β_{8} = α_{1} + 2α_{2} + 3α_{3} + 2α_{4} is the only root on level 8 and C(𝔤_{β_{8}}; 𝔫_{1}) = 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} ⊕ 𝔤_{α_{4}}, which generates a subalgebra of type A_{2} ⊕ A_{1}. This implies F (𝔤_{α_{4}}) = 𝔤_{α_{4}} and thus all simple root spaces are preserved by F, which implies F (𝔤_{λ}) = 𝔤_{λ} for all λ ∈ Σ^{+}.

Σ = G_{2}: The root β_{4} = 3α_{1} + α_{2} is the only root on level 4 and C(𝔤_{β_{4}}; 𝔫_{1}) = 𝔤_{α_{1}}, which implies F (𝔤_{α_{1}}) = 𝔤_{α_{1}}. The root β_{3} = 2α_{1} + α_{2} is the only root on level 3 and C(𝔤_{β_{3}}; 𝔫_{1}) = 𝔤_{α_{2}}, which implies F (𝔤_{α_{2}}) = 𝔤_{α_{2}}. Altogether this implies F (𝔤_{λ}) = 𝔤_{λ} for all λ ∈ Σ^{+}. q.e.d.

We now start to investigate the congruency problem for the foliations 𝔉_{ℓ}. We assume r ≥ 2 from now on. According to Proposition 3.1 all leaves of 𝔉_{ℓ} and all leaves of 𝔉_{ℓ′} are isometrically congruent to each other. Therefore the two leaves S_{ℓ} · o and S_{ℓ′} · o are isometrically congruent to each other. Since S_{ℓ} is connected, it follows from Lemma 3.3 that S_{ℓ} is completely solvable, that is, there exists a basis of 𝔰_{ℓ} such that Ad(S_{ℓ}) consists of upper triangular matrices with respect to that basis. Since any two connected completely solvable transitive groups of isometries on a Riemannian manifold are conjugate in the isometry group of that manifold (see [1]), we conclude that the Lie algebras 𝔰_{ℓ} and 𝔰_{ℓ′} are isomorphic.

Let F : 𝔰_{ℓ} → 𝔰_{ℓ′} be a Lie algebra isomorphism. Then F maps the Cartan subalgebra 𝔠_{ℓ} of 𝔰_{ℓ} onto a Cartan subalgebra of 𝔰_{ℓ′}. Since any two Cartan subalgebras of a solvable Lie algebra are conjugate to each other under an inner automorphism we can assume that F (𝔠_{ℓ}) = 𝔠_{ℓ′}. In view of Lemma 3.3 it is natural to distinguish the following cases:

Case 1. ℓ = ℝH_{λ} for some root λ ∈ Σ^{+} with 2λ ∈ Σ^{+}.

Case 2. ℓ = ℝH_{λ} for some root λ ∈ Σ^{+} with 2λ, \frac{1}{2}λ ∉ Σ^{+}.

Case 3. ℓ ≠ ℝH_{λ} for all λ ∈ Σ^{+}.

In the first case 𝔠_{ℓ} is a nilpotent and nonabelian Cartan subalgebra, in the second case 𝔠_{ℓ} is an abelian Cartan subalgebra containing nontrivial nilpotent elements, and in the third case 𝔠_{ℓ} is an abelian Cartan subalgebra without nontrivial nilpotent elements.

We start with Case 1. Assume that ℓ = ℝH_{λ} for some root λ ∈ Σ^{+} with 2λ ∈ Σ^{+}. From Lemma 3.3 we see that the Cartan subalgebra 𝔠_{ℓ} = (𝔞 ⊖ ℓ) ⊕ 𝔤_{λ} ⊕ 𝔤_{2λ} of 𝔰_{ℓ} is nilpotent and nonabelian. Thus the Cartan subalgebra 𝔠_{ℓ′} of 𝔰_{ℓ′} must also be nilpotent and nonabelian, and we get from Lemma 3.3 that there exists a root λ′ ∈ Σ^{+} with 2λ′ ∈ Σ^{+} such that ℓ′ = ℝH_{λ′}. The derived subalgebra of 𝔠_{ℓ} is 𝔤_{2λ} and must be mapped onto the derived subalgebra 𝔤_{2λ′} of 𝔠_{ℓ′} by F. We can now apply Theorem 3.4 and get a symmetry P ∈ Aut(DD) with P (2λ) = 2λ′ and hence also P (λ) = λ′. By construction, this implies then P (ℓ) = ℓ′.

Next, we consider Case 2. Let ℓ = ℝH_{λ} for some root λ ∈ Σ^{+} with 2λ, \frac{1}{2}λ ∉ Σ^{+}. Then the Cartan subalgebra 𝔠_{ℓ} of 𝔰_{ℓ} is abelian and dim 𝔠_{ℓ} = r − 1 + dim 𝔤_{λ}. Since F maps 𝔠_{ℓ} onto the Cartan subalgebra 𝔠_{ℓ′} of 𝔰_{ℓ′}, the latter one must also be abelian and of dimension greater than r − 1. According to Lemma 3.3 this can happen only when ℓ′ = ℝH_{λ′} for some root λ′ ∈ Σ^{+} with 2λ′, \frac{1}{2}λ′ ∉ Σ^{+}. The isomorphism F maps the Cartan subalgebra 𝔠_{ℓ} = (𝔞 ⊖ ℓ) ⊕ 𝔤_{λ} of 𝔰_{ℓ} to the Cartan subalgebra 𝔠_{ℓ′} = (𝔞⊖ℓ′)⊕𝔤_{λ′} of 𝔰_{ℓ′}. Moreover, F maps the set of nilpotent elements in 𝔰_{ℓ} onto the set of nilpotent elements in 𝔰_{ℓ′}, which for both subalgebras is equal to 𝔫. It follows that 𝔤_{λ} = 𝔠_{ℓ} ∩ 𝔫 is mapped onto 𝔤_{λ′} = 𝔠_{ℓ′} ∩ 𝔫 by F. From Theorem 3.4 we then conclude that there exists a symmetry P ∈ Aut(DD) with P (λ) = λ′ and hence also P (ℓ) = ℓ′.

We finally consider Case 3. Then 𝔠_{ℓ} = 𝔞 ⊖ ℓ is a Cartan subalgebra of 𝔰_{ℓ}, which is mapped by the isomorphism F to the Cartan subalgebra 𝔠_{ℓ′} = 𝔞 ⊖ ℓ′ of 𝔰_{ℓ′}. Using Lemma 3.3 we therefore must have ℓ′ ≠ ℝH_{λ} for all λ ∈ Σ^{+}. If λ ∈ Σ^{+} then λ|𝔠_{ℓ} is a root of 𝔰_{ℓ} with respect to the Cartan subalgebra 𝔠_{ℓ}. We denote by Σ_{ℓ}^{+} the set of these roots and by 𝔤_{λ}^{ℓ} the corresponding root spaces. The set Λ_{ℓ} = {α|𝔠_{ℓ} | α ∈ Λ} generates Σ_{ℓ}^{+} in a similar way as Λ generates Σ^{+}. For each λ ∈ Σ_{ℓ}^{+} we define a linear map F^{∗}λ : 𝔠_{ℓ′} → ℝ by F^{∗}λ(H) = λ(F^{−1}H) for all H ∈ 𝔠_{ℓ′}. Since [H, F X] = F [F^{−1}H, X] = λ(F^{−1}H)F X for all X ∈ 𝔤_{λ}^{ℓ}, we see that F^{∗}λ ∈ Σ_{ℓ′}^{+} and F (𝔤_{λ}^{ℓ}) = 𝔤_{F^{∗}λ}^{ℓ′} for all λ ∈ Σ_{ℓ}^{+}. Thus F^{∗} induces an isomorphism from the set Σ_{ℓ}^{+} onto the set Σ_{ℓ′}^{+} and maps root spaces of roots in Σ_{ℓ}^{+} onto root spaces of roots in Σ_{ℓ′}^{+}. Since F^{∗}(λ+μ) = F^{∗}λ+F^{∗}μ for all λ, μ ∈ Σ_{ℓ}^{+}, F^{∗} maps root strings to root strings, and hence induces an isomorphism between the sets Λ_{ℓ} and Λ_{ℓ′}.

It may happen that there exist two roots λ, μ ∈ Σ^{+} such that λ|𝔠_{ℓ} = μ|𝔠_{ℓ}. In this case 𝔤_{λ} ⊕ 𝔤_{μ} is the root space 𝔤_{λ}^{ℓ} = 𝔤_{μ}^{ℓ} corresponding to λ|𝔠_{ℓ} = μ|𝔠_{ℓ} ∈ Σ_{ℓ}^{+}, and we say that the two roots λ and μ collapse in 𝔰_{ℓ}. Note that if λ and μ collapse in 𝔰_{ℓ}, and if κ ∈ Σ^{+} such that κ + kλ, κ + kμ ∈ Σ^{+} for some k > 0, then also κ + kλ and κ + kμ collapse in 𝔰_{ℓ}. If λ ∈ Σ^{+} does not collapse in 𝔰_{ℓ} with any other root in Σ^{+} then 𝔤_{λ}^{ℓ} = 𝔤_{λ}.

If no two simple roots in Λ collapse in 𝔰_{ℓ}, then Λ_{ℓ} consists of r elements. Since F^{∗} maps the set Λ_{ℓ} isomorphically onto the set Λ_{ℓ′}, the latter set consists also of r elements and it follows that no two simple roots in Λ collapse in 𝔰_{ℓ′}. Since F^{∗} maps root strings to root strings we conclude that F^{∗} induces a Dynkin diagram symmetry P. Let k be an isometry in the normalizer of 𝔞 + 𝔫 in K such that the action of Ad(k)^{∗} on Λ coincides with the action of P. For all α ∈ Λ, X_{α} ∈ 𝔤_{α} and H′ ∈ 𝔠_{ℓ′} we have [H′, F X_{α}] = F [F^{−1}H′, X_{α}], as F is a Lie algebra isomorphism. The left-hand side of the previous equation is equal to (F^{∗}α)(H′)F X_{α} = (Ad(k)^{∗}α)(H′)F X_{α} = α(Ad(k)^{−1}H′)F X_{α}, and the right-hand side equals α(F^{−1}H′)F X_{α}. Comparing the two sides shows that α(Ad(k)^{−1}H′) = α(F^{−1}H′) for all α ∈ Λ and H′ ∈ 𝔠_{ℓ′}. This implies Ad(k)^{−1}|𝔠_{ℓ′} = F^{−1}|𝔠_{ℓ′} and hence Ad(k)|𝔠_{ℓ} = F |𝔠_{ℓ}. Then we have Ad(k)(𝔰_{ℓ}) = 𝔰_{ℓ′} and hence, since Ad(k) preserves the inner product on 𝔞, also Ad(k)(ℓ) = ℓ′. Thus P ∈ Aut(DD) is a symmetry with P (ℓ) = ℓ′.

Now assume that two simple roots α, β ∈ Λ collapse in 𝔰_{ℓ}. If r = 2, we must have ℓ = ℝ(H_{α_{1}} − H_{α_{2}}) = ℓ′. We therefore can assume r ≥ 3 from now on. Then Λ_{ℓ} consists of r − 1 elements, and since Λ_{ℓ′} is isomorphic to Λ_{ℓ}, we see that there exist two simple roots α′, β′ ∈ Λ that collapse in 𝔰_{ℓ′}. We claim that F (𝔤_{α} ⊕ 𝔤_{β}) = 𝔤_{α′} ⊕ 𝔤_{β′}. Assume that this is not true. Then there exists a simple root γ ∈ Λ \ {α′, β′} such that F (𝔤_{α} ⊕ 𝔤_{β}) = 𝔤_{γ}. By inspection of the possible root systems and the multiplicities of the roots (see [2] for a list of the multiplicities), we see that this is possible only in the following cases:

(1) Σ = B_{r} and M = SO^{o}(r + 2, r)/SO(r + 2) × SO(r);

(2) Σ = BC_{r} and M = SU(r + 2, r)/S(U(r + 2) × U(r));

(3) Σ = BC_{r} and M = Sp(r + 2, r)/Sp(r + 2) × Sp(r);

(4) Σ = F_{4} and M = E_{6}^{2}/SU(6) × SU(2).

Since α and β are on the same level it follows that whenever two roots in Σ^{+} collapse in 𝔰_{ℓ} they are on the same level in Σ^{+}. This implies that F preserves the gradation 𝔫 = 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m} of 𝔫. In Cases (1), (2) and (3) we must have γ = α_{r}. In Case (1) the centralizer of 𝔤_{α_{r}} in 𝔫_{1} is 𝔫_{1} ⊖ 𝔤_{α_{r−1}}, which is a linear subspace of codimension one in 𝔫_{1}. On the other hand, the orthogonal complement of the centralizer of 𝔤_{α} ⊕ 𝔤_{β} in 𝔫_{1} contains at least two simple root spaces and hence has a codimension greater than one in 𝔫_{1}. This excludes Case (1). In Cases (2) and (3) we consider derived subalgebras. Since F maps 𝔤_{α} ⊕ 𝔤_{β} isomorphically onto 𝔤_{α_{r}}, it also maps their derived subalgebras isomorphically onto each other. But [𝔤_{α_{r}}, 𝔤_{α_{r}}] = 𝔤_{2α_{r}} has odd dimension, whereas [𝔤_{α} ⊕ 𝔤_{β}, 𝔤_{α} ⊕ 𝔤_{β}] = [𝔤_{α}, 𝔤_{β}] = 𝔤_{α+β} has even dimension, which gives a contradiction. Finally, in Case (4), we must have γ ∈ {α_{3}, α_{4}}. We consider again centralizers to derive a contradiction. The centralizer of 𝔤_{α} ⊕ 𝔤_{β} in 𝔫_{1} is 𝔤_{α_{4}} and has dimension 2. On the other hand, the centralizer of 𝔤_{α_{3}} in 𝔫_{1} is 𝔤_{α_{1}} ⊕ 𝔤_{α_{3}}, which is 3-dimensional, and the centralizer of 𝔤_{α_{4}} in 𝔫_{1} is 𝔤_{α_{1}} ⊕ 𝔤_{α_{2}} ⊕ 𝔤_{α_{4}}, which is 4-dimensional. This shows that F cannot map 𝔤_{α} ⊕ 𝔤_{β} isomorphically onto 𝔤_{γ}. We thus have proved that F (𝔤_{α} ⊕ 𝔤_{β}) = 𝔤_{α′} ⊕ 𝔤_{β′}.

Since r ≥ 3, the sets {γ ∈ Σ^{+} | ⟨γ, α⟩ = 0} and {γ ∈ Σ^{+} | ⟨γ, β⟩ = 0} are nonempty and not identical. Thus there exists a root γ ∈ Σ^{+} with ⟨γ, α⟩ = 0 and ⟨γ, β⟩ ≠ 0. This means that γ ± α ∉ Σ^{+} and γ + β ∈ Σ^{+}. Assume that γ collapses in 𝔰_{ℓ} with some root μ ∈ Σ^{+}. Then there exist a root δ ∈ Σ^{+} and a positive integer k such that γ = δ + kα and μ = δ + kβ. Then γ − kα = δ ∈ Σ^{+}, which implies γ − α ∈ Σ^{+} by general properties of root strings. This contradicts γ − α ∉ Σ^{+}, and we conclude that γ does not collapse in 𝔰_{ℓ} with any root in Σ^{+}. We can characterize 𝔤_{α} by 𝔤_{α} = {X ∈ 𝔤_{α} ⊕ 𝔤_{β} | [X, 𝔤_{γ}] = 0}. Then F (𝔤_{α}) = {Y ∈ 𝔤_{α′} ⊕ 𝔤_{β′} | Y = F X and [X, 𝔤_{γ}] = 0 for some X ∈ 𝔤_{α} ⊕ 𝔤_{β}}. Since γ does not collapse in 𝔰_{ℓ}, there exists a root γ′ ∈ Σ^{+} such that F^{∗}γ|𝔠_{ℓ′} = γ′|𝔠_{ℓ′}, and we get F (𝔤_{γ}) = 𝔤_{γ′} and hence F (𝔤_{α}) = {Y ∈ 𝔤_{α′} ⊕ 𝔤_{β′} | [Y, 𝔤_{γ′}] = 0}. If α′ + γ′, β′ + γ′ ∈ Σ^{+}, then F (𝔤_{α}) = 0, which contradicts the fact that F is an isomorphism. If α′ + γ′, β′ + γ′ ∉ Σ^{+} then F (𝔤_{α}) = 𝔤_{α′} ⊕ 𝔤_{β′}, which is again a contradiction. We therefore must have either α′ + γ′ ∈ Σ^{+}, β′ + γ′ ∉ Σ^{+}, which implies F (𝔤_{α}) = 𝔤_{β′}, or α′ + γ′ ∉ Σ^{+}, β′ + γ′ ∈ Σ^{+}, which implies F (𝔤_{α}) = 𝔤_{α′}. We therefore can assume, without loss of generality, that F (𝔤_{α}) = 𝔤_{α′} and F (𝔤_{β}) = 𝔤_{β′}. From Theorem 3.4 we conclude that there exist P, Q ∈ Aut(DD) with P (α) = α′ and Q(β) = β′.

We claim that we can choose P = Q. If α or β is in the set Fix(Λ, Aut(DD)) of fixed points of the action of Aut(DD) on Λ, then this is clear. Now assume that α, β ∉ Fix(Λ, Aut(DD)). Then we have necessarily Σ ∈ {A_{r}, D_{r}, E_{6}}. If Σ = D_{4}, we have Aut(DD) = 𝔖_{3}, and for any two pairs of distinct roots not in Fix(Λ, Aut(DD)) there exists a symmetry in 𝔖_{3} mapping the first pair to the second one. This shows that we can choose P = Q if Σ = D_{4}. If Σ = D_{r}, r ≠ 4, the set Λ \ Fix(Λ, Aut(DD)) consists of exactly two points, which readily implies that we can choose P = Q. If Σ = E_{6}, the structure of the Dynkin diagram of E_{6} shows that Λ \ Fix(Λ, Aut(DD)) consists of two pairs of simple roots, where in each pair the roots form nontrivial strings with each other, but no nontrivial strings with the roots of the other pair. If {α, β} forms such a pair, the other pair consists of simple roots of Λ that do not collapse in 𝔰_{ℓ} and hence correspond to roots in Λ_{ℓ}. Since F preserves strings of roots in Λ_{ℓ}, we conclude that {α′, β′} must also be one of the two pairs. So the structure of the Dynkin diagram of E_{6} tells us that we must have P = Q. If α and β are in different pairs, a similar argument gives P = Q as well. Finally, consider the case Σ = A_{r}, in which case we have Aut(DD) = ℤ_{2}. If P ≠ Q, then either P or Q must be the identity, say Q. Then β = β′, and just contemplating about the structure of the Dynkin diagram of A_{r} and taking into account that string relations between roots in Λ \ {α, β} are preserved by F, shows that also P must be the identity, which is a contradiction. Thus either both P and Q are the identity, or both of them are different from the identity in Aut(DD). Altogether this implies that we can choose P = Q. Since ℓ = ℝ(H_{α} − H_{β}) and ℓ′ = ℝ(H_{α′} − H_{β′}), we can now conclude that there exists a symmetry P ∈ Aut(DD) with P (ℓ) = ℓ′.

Conversely, let ℓ, ℓ′ be two different lines and assume that there exists a symmetry P ∈ Aut(DD) with P (ℓ) = ℓ′. There exists an outer automorphism F of 𝔤 such that the induced action of F on Σ^{+} is equal to the action of P on Σ^{+}. In fact, F = Ad(k) for some k in the normalizer of 𝔞 + 𝔫 in K. By construction we have F (𝔰_{ℓ}) = 𝔰_{ℓ′}, which implies that k is an isometry of M mapping the leaves of the foliation 𝔉_{ℓ} onto the leaves of the foliation 𝔉_{ℓ′}. This finishes the investigations about the congruency of the foliations 𝔉_{ℓ}, and we summarize it in:

Theorem 3.5. Two foliations 𝔉_{ℓ} and 𝔉_{ℓ′} are isometrically congruent to each other if and only if there exists a symmetry P ∈ Aut(DD) with P (ℓ) = ℓ′.

4. The foliations 𝔉_{i}

The following lemma is crucial for showing that the foliations 𝔉_{i} are well-defined:

Lemma 4.1. Let α ∈ Λ be a simple root. For each unit vector ξ ∈ 𝔤_{α} the subspace 𝔰_{ξ} = 𝔞+(𝔫⊖ℝξ) is a subalgebra of 𝔞+𝔫. Moreover, if ξ, η are two unit vectors in 𝔤_{α}, then there exists an isometry k in the centralizer of 𝔞 in K^{o} such that Ad(k)(𝔰_{ξ}) = 𝔰_{η}.

Proof. The fact that 𝔰_{ξ} is a subalgebra of 𝔞 + 𝔫 follows immediately from elementary properties of root systems. For the congruency problem there is nothing to prove if dim 𝔤_{α} = 1, and for dim 𝔤_{α} > 1 this follows from Exercise 2 on p. 211 in [11] (see p. 566 for the solution). q.e.d.

The previous lemma implies that for each simple root α_{i} ∈ Λ we obtain a congruence class of homogeneous foliations of codimension one on M. More precisely, let α_{i} ∈ Λ, ξ ∈ 𝔤_{α_{i}} be a unit vector, and S_{ξ} be the connected Lie subgroup of AN with Lie algebra 𝔰_{ξ}. Then the orbits of the action of S_{ξ} on M form a homogeneous foliation 𝔉_{ξ} of codimension one on M. If η ∈ 𝔤_{α_{i}} is another unit vector, it follows from Lemma 4.1 that the induced foliation 𝔉_{η} is congruent to 𝔉_{ξ} under an isometry in the centralizer of 𝔞 in K^{o}. We denote by 𝔉_{i} a representative of this congruence class of homogeneous foliations of codimension one on M, that is, 𝔉_{i} = 𝔉_{ξ} for some unit vector ξ ∈ 𝔤_{α_{i}}.

Our next aim is to study the geometry of such a foliation 𝔉_{i} in more detail. Let α_{i} ∈ Λ be a simple root and ξ ∈ 𝔤_{α_{i}} be a unit vector such that 𝔉_{i} = 𝔉_{ξ}. We put H_{i} = H_{α_{i}}/|H_{α_{i}}| = H_{α_{i}}/|α_{i}|. The vectors ξ and H_{i} span a two-dimensional subalgebra of 𝔞 + 𝔫 with Lie bracket [H_{i}, ξ] = α_{i}(H_{i})ξ = |α_{i}|ξ. A straightforward calculation shows that the covariant derivatives of these left-invariant vector fields are given by ∇_{ξ}ξ = |α_{i}|H_{i}, ∇_{ξ}H_{i} = −|α_{i}|ξ, ∇_{H_{i}}H_{i} = 0 and ∇_{H_{i}}ξ = 0. Thus ξ and H_{i}, now considered as left-invariant vector fields, span an autoparallel subbundle of the tangent bundle of AN, and it follows that the orbit of the corresponding connected subgroup of AN through o is a totally geodesic real hyperbolic plane ℝH^{2} ⊂ AN = M. Let γ : ℝ → M be the geodesic in M with γ(0) = o and γ̇(0) = ξ. Clearly, this geodesic lies in the totally geodesic ℝH^{2}. Using again the Koszul formula for the Levi Civita connection of AN it is easy to see that the tangent vector field γ̇ of γ satisfies γ̇(t) = \frac{1}{cosh(|α_{i}|t)}ξ_{γ(t)} − tanh(|α_{i}|t)(H_{i})_{γ(t)} for all t ∈ ℝ, where we view ξ and H_{i} as left-invariant vector fields on AN.

Let t ∈ ℝ, g = γ(t) ∈ AN, and denote by I_{g^{−1}} the conjugation on G^{o} by g^{−1}. Clearly, I_{g^{−1}} leaves AN invariant, and hence I_{g^{−1}}(S_{ξ}) is also a subgroup of AN. Since I_{g^{−1}}(S_{ξ})·o = g^{−1}S_{ξ}g ·o = γ(t)^{−1}S_{ξ} ·γ(t), we see that the orbit of the action of I_{g^{−1}}(S_{ξ}) through o is the left translate from γ(t) to o of the orbit of the action of S_{ξ} through γ(t). Since γ̇(t) is a unit normal vector of S_{ξ} ·γ(t) at γ(t), and left translation L_{g^{−1}} with g^{−1} is an isometry, the vector ξ_{t} = L_{g^{−1}∗}γ̇(t) = \frac{1}{cosh(|α_{i}|t)}ξ_{o} − tanh(|α_{i}|t)(H_{i})_{o} is a unit normal vector of I_{g^{−1}}(S_{ξ}) · o at o. It follows that Ad(g^{−1})𝔰_{ξ} = 𝔰_{ξ_{t}}, or equivalently, Ad(g)𝔰_{ξ_{t}} = 𝔰_{ξ}, where 𝔰_{ξ_{t}} is the subalgebra of 𝔞 + 𝔫 given by 𝔰_{ξ_{t}} = (𝔞 + 𝔫) ⊖ ℝξ_{t}. Thus we have proved:

Lemma 4.2. Let 𝔉_{i} = 𝔉_{ξ} with some unit vector ξ ∈ 𝔤_{α_{i}}. Then the leaf of 𝔉_{i} at oriented distance t ∈ ℝ in direction of ξ is isometrically congruent the orbit S_{ξ_{t}} · o, where

ξ_{t} = \frac{1}{cosh(|α_{i}|t)}ξ − \frac{1}{|α_{i}|} tanh(|α_{i}|t)H_{α_{i}}

and S_{ξ_{t}} is the connected Lie subgroup of AN with Lie algebra (𝔞 + 𝔫) ⊖ ℝξ_{t}.

This lemma is very useful for studying the geometry of the leaves of 𝔉_{i}. Let A_{ξ_{t}} be the shape operator with respect to ξ_{t} of the orbit S_{ξ_{t}} · o at o. By means of the Weingarten formula for the shape operator and the Koszul formula for the Levi Civita connection of AN we have 2⟨A_{ξ_{t}}X, Y ⟩ = 2⟨∇_{X}Y, ξ_{t}⟩ = ⟨[X, Y], ξ_{t}⟩ − ⟨[Y, ξ_{t}], X⟩ + ⟨[ξ_{t}, X], Y ⟩. Since 𝔰_{ξ_{t}} is a subalgebra of 𝔞 + 𝔫, we have [X, Y] ∈ 𝔰_{ξ_{t}} and hence ⟨[X, Y], ξ_{t}⟩ = 0 for all X, Y ∈ 𝔰_{ξ_{t}}. Moreover, since the adjoint transformation ad(ξ_{t})^{∗} of ad(ξ_{t}) satisfies ad(ξ_{t})^{∗} = −ad(θξ_{t}), we have ⟨[Y, ξ_{t}], X⟩ = −⟨ad(ξ_{t})Y, X⟩ = −⟨ad(ξ_{t})^{∗}X, Y ⟩ = ⟨ad(θξ_{t})X, Y ⟩. Altogether we now easily get ⟨A_{ξ_{t}}X, Y ⟩ = \frac{1}{2}(⟨ad(ξ_{t})X, Y ⟩ + ⟨X, ad(ξ_{t})Y ⟩) = \frac{1}{2}⟨(ad(ξ_{t}) − ad(θξ_{t}))X, Y ⟩ for all X, Y ∈ 𝔰_{ξ_{t}} = T_{o}(S_{ξ_{t}} · o). Thus we have proved:

Lemma 4.3. Let 𝔉_{i} = 𝔉_{ξ} with some unit vector ξ ∈ 𝔤_{α_{i}}. The shape operator A_{ξ_{t}} with respect to ξ_{t} of the orbit S_{ξ_{t}} · o at o is given by

A_{ξ_{t}}X = [\frac{1}{2 cosh(|α_{i}|t)}(ξ − θξ) − \frac{1}{|α_{i}|} tanh(|α_{i}|t)H_{α_{i}} , X]_{𝔰_{ξ_{t}}}

for all X ∈ 𝔰_{ξ_{t}}, where the subscript 𝔰_{ξ_{t}} denotes the orthogonal projection onto 𝔰_{ξ_{t}}.

We will now investigate the principal curvatures of the leaves of the foliation 𝔉_{i}. Let 𝔉_{i} = 𝔉_{ξ} with some unit vector ξ ∈ 𝔤_{α_{i}}. We identify the leaf of 𝔉_{i} at distance t in direction ξ with S_{ξ_{t}} · o.

Proposition 4.4. Let 𝔉_{i} = 𝔉_{ξ} with some unit vector ξ ∈ 𝔤_{α_{i}}. For the principal curvatures of S_{ξ_{t}} · o we have:

(1) The (r − 1)-dimensional subspace 𝔞 ⊖ ℝH_{α_{i}} is invariant under A_{ξ_{t}} and the corresponding principal curvature is 0.

(2) The subspace (𝔤_{α_{i}} ⊖ ℝξ) ⊕ 𝔤_{2α_{i}} is invariant under A_{ξ_{t}} and the corresponding principal curvatures are −|α_{i}| tanh(|α_{i}|t) with multiplicity dim 𝔤_{α_{i}} − dim 𝔤_{2α_{i}} − 1 and

−\frac{3}{2}|α_{i}| tanh(|α_{i}|t) ± \frac{1}{2}|α_{i}|\sqrt{2 − tanh^{2}(|α_{i}|t)}

with multiplicity dim 𝔤_{2α_{i}}.

(3) The one-dimensional subspace (ℝH_{α_{i}} ⊕ ℝξ) ⊖ ℝξ_{t} is invariant under A_{ξ_{t}} with corresponding principal curvature −|α_{i}| tanh(|α_{i}|t).

(4) Define k in the normalizer N_{K}(𝔞) of 𝔞 in K by

k = Exp (\frac{π}{\sqrt{2}|α_{i}|}(ξ + θξ)) .

The subspace 𝔫⊖(𝔤_{α_{i}} ⊕𝔤_{2α_{i}}) is invariant under A_{ξ_{t}} and Ad(k) and we have A_{ξ_{t}}Ad(k)X = −Ad(k)A_{ξ_{t}}X for all X ∈ 𝔫 ⊖ (𝔤_{α_{i}} ⊕ 𝔤_{2α_{i}}). In particular, if A_{ξ_{t}}X = cX, then A_{ξ_{t}}Ad(k)X = −cAd(k)X.

Proof.

(1): For all H ∈ 𝔞 ⊖ ℝH_{α_{i}} we have [ξ, H] = −α_{i}(H)ξ = 0, [θξ, H] ∈ 𝔤_{−α_{i}} and [H_{α_{i}}, H] = 0, which implies A_{ξ_{t}}H = 0 by means of Lemma 4.3.

(2): We have ⟨[θξ, X], H⟩ = ⟨X, [H, ξ]⟩ = α_{i}(H)⟨X, ξ⟩ = 0 for all X ∈ 𝔤_{α_{i}} ⊖ ℝξ and H ∈ 𝔞. Since obviously [θξ, X] ∈ 𝔤_{0}, this implies [θξ, 𝔤_{α_{i}} ⊖ ℝξ]_{𝔰_{ξ_{t}}} = 0, and shows that (𝔤_{α_{i}} ⊖ ℝξ) ⊕ 𝔤_{2α_{i}} is invariant under A_{ξ_{t}}. The vector space 𝔤_{α_{i}} ⊖ ℝξ can be decomposed orthogonally into 𝔤_{α_{i}} ⊖ ℝξ = V_{1} ⊕ V_{2}, where V_{1} is the kernel of the linear map ad(ξ)|𝔤_{α_{i}} : 𝔤_{α_{i}} → 𝔤_{2α_{i}} and V_{2} is the image of the linear map ad(θξ)|𝔤_{2α_{i}} : 𝔤_{2α_{i}} → 𝔤_{α_{i}}. One can easily see that V_{1} consists of principal curvature vectors with principal curvature −|α_{i}| tanh(|α_{i}|t) and corresponding multiplicity dim 𝔤_{α_{i}} − dim 𝔤_{2α_{i}} − 1. Note that ad(ξ)|V_{2} : V_{2} → 𝔤_{2α_{i}} is an isomorphism because of [ξ, [θξ, Y]] = −[Y, [ξ, θξ]] = [Y, H_{α_{i}}] = −2|α_{i}|^{2}Y for all Y ∈ 𝔤_{2α_{i}}. Moreover, if Y ∈ 𝔤_{2α_{i}} is a unit vector, then X := \frac{1}{\sqrt{2}|α_{i}|}[θξ, Y] ∈ 𝔤_{α_{i}} is also a unit vector. It follows easily from Lemma 4.3 that the two-dimensional linear subspace ℝX ⊕ ℝY of (𝔤_{α_{i}} ⊖ ℝξ) ⊕ 𝔤_{2α_{i}} is invariant under A_{ξ_{t}} and has the matrix representation

−|α_{i}| tanh(|α_{i}|t) −\frac{|α_{i}|}{\sqrt{2} cosh(|α_{i}|t)}
−\frac{|α_{i}|}{\sqrt{2} cosh(|α_{i}|t)} −2|α_{i}| tanh(|α_{i}|t)

with respect to X and Y. From this one can calculate the eigenvalues directly.

(3): The subspace (ℝH_{α_{i}} ⊕ℝξ)⊖ℝξ_{t} is of dimension one. Since ξ and \frac{1}{|α_{i}|}H_{α_{i}} are orthonormal, the vector X := tanh(|α_{i}|t)ξ + \frac{1}{|α_{i}| cosh(|α_{i}|t)}H_{α_{i}} belongs to this subspace. A direct calculation using Lemma 4.3 shows that A_{ξ_{t}}X = −|α_{i}| tanh(|α_{i}|t)X.

(4): The subspace ℝH_{α_{i}} ⊕ ℝξ ⊕ ℝθξ is a subalgebra of 𝔤 isomorphic to 𝔰𝔩(2, ℝ). The isometry k = Exp (\frac{π}{\sqrt{2}|α_{i}|}(ξ + θξ)) is in N_{K}(𝔞) and satisfies Ad(k)(H_{α_{i}}) = −H_{α_{i}} and Ad(k)(ξ − θξ) = θξ − ξ. The significance of k is that Ad(k)|𝔞 is the reflection in the hyperplane 𝔞⊖ℝH_{α_{i}}. It is clear that Ad(k) preserves 𝔫⊖(𝔤_{α_{i}} ⊕𝔤_{2α_{i}}), and from (1),

(2) and (3) it follows that A_{ξ_{t}} preserves this subspace as well. Moreover, from the explicit expression of A_{ξ_{t}} in Lemma 4.3 we easily see that Ad(k) ∘ A_{ξ_{t}} = −Ad(k) ∘ A_{ξ_{t}} holds on 𝔫 ⊖ (𝔤_{α_{i}} ⊕ 𝔤_{2α_{i}}). q.e.d.

This proposition shows that the sum of the principal curvatures on 𝔫 ⊖ (𝔤_{α_{i}} ⊕ 𝔤_{2α_{i}}) is zero. This makes it easy to calculate the mean curvature of each leaf of the foliation 𝔉_{i}. The following corollary is a direct consequence of Proposition 4.4:

Corollary 4.5. The (constant) mean curvature μ_{t} of the leaf of 𝔉_{i} at distance t in direction ξ satisfies

μ_{t} = −\frac{|α_{i}| tanh(|α_{i}|t)}{n − 1}(dim 𝔤_{α_{i}} + 2 dim 𝔤_{2α_{i}}).

Therefore the leaf of 𝔉_{i} through o, i.e., the orbit S_{ξ}·o, is the only minimal leaf of 𝔉_{i}. This minimal leaf is an austere submanifold, i.e., if c is a principal curvature then also −c is a principal curvature with the same multiplicity.

Each foliation 𝔉_{i} has a kind of reflective symmetry. Let 𝔉_{i} = 𝔉_{ξ} with a unit vector ξ ∈ 𝔤_{α_{i}}. Consider the gradation 𝔤 = ⨁_{k=−m}^{m} 𝔤^{k} of 𝔤 and the subalgebra 𝔥 = ⨁_{k even} 𝔤^{k} of 𝔤. Let H be the connected Lie subgroup of G with Lie algebra 𝔥. Since 𝔥 is invariant under the Cartan involution θ, the orbit Q = H · o is a totally geodesic submanifold of M. The subspace ν_{o}Q = ⨁_{k odd} 𝔭_{k} ⊂ 𝔭 with 𝔭_{k} = (𝔤^{k} ⊕ 𝔤^{−k}) ∩ 𝔭 is the normal space of Q at o. It is easy to see that ν_{o}Q is a Lie triple system in 𝔭, which implies that for each normal space of Q there exists a totally geodesic submanifold of M that is tangent to that normal space. This implies that the reflection Φ_{Q} of M in Q is an isometry. By construction, Ad(Φ_{Q})X = X if X ∈ ⨁_{k even} 𝔤^{k}, and Ad(Φ_{Q})X = −X if X ∈ ⨁_{k odd} 𝔤^{k}. Thus 𝔰_{ξ} is invariant under Ad(Φ_{Q}) and it follows that Φ_{Q} maps S_{ξ} · o onto itself. Since Ad(Φ_{Q})ξ = −ξ, we then conclude that Φ_{Q} interchanges the two orbits at any positive distance to the minimal orbit. It is worthwhile to point out that the submanifold Q is independent of the choice of the simple root α_{i} and the choice of the unit vector ξ ∈ 𝔤_{α_{i}}. We thus have proved:

Proposition 4.6. There exists a totally geodesic submanifold Q of M such that the reflection Φ_{Q} of M in Q is an isometry that leaves the minimal leaf of 𝔉_{i} invariant and interchanges the two leaves in 𝔉_{i} at any given positive distance to the minimal leaf.

Next we shall compare the principal curvatures of the leaves of 𝔉_{i} and 𝔉_{j}. As Corollary 4.5 shows, the mean curvature of the leaves of 𝔉_{i} depends only on the length of the root α_{i} and the distance t to the unique minimal leaf of 𝔉_{i}. This is also true for the principal curvatures, as the following result shows:

Theorem 4.7. If α_{i}, α_{j} ∈ Λ with |α_{i}| = |α_{j}|, then the foliations 𝔉_{i} and 𝔉_{j} have the same principal curvatures, counted with multiplicities. More precisely, let ξ ∈ 𝔤_{α_{i}} and η ∈ 𝔤_{α_{j}} be unit vectors. The geodesic γ_{ξ} : ℝ → M with γ_{ξ}(0) = o and γ̇_{ξ}(0) = ξ intersects each leaf of 𝔉_{ξ} and γ̇_{ξ}(t) is a unit normal vector of the leaf 𝔉_{ξ}(t) of 𝔉_{ξ} through γ_{ξ}(t), which by homogeneity of the leaves naturally extends to a unit normal vector field ξ^{(t)} on the leaf 𝔉_{ξ}(t) (and analogously for η). Then the principal curvatures, counted with multiplicities, of 𝔉_{ξ}(t) with respect to ξ^{(t)} and of 𝔉_{η}(t) with respect to η^{(t)} coincide.

Proof. Let ξ ∈ 𝔤_{α_{i}} and η ∈ 𝔤_{α_{j}} be unit vectors. We have to show that for each t ∈ ℝ the orbits S_{ξ_{t}} · o and S_{η_{t}} · o have the same principal curvatures, counted with multiplicities. Since α_{i} and α_{j} have the same length there exists an isometry k ∈ N_{K}(𝔞) for which the induced action Ad(k)^{∗} of Ad(k) on Σ maps α_{i} to α_{j}, and hence Ad(k)ξ ∈ 𝔤_{α_{j}}. If dim 𝔤_{α_{j}} > 1, we can assume Ad(k)ξ = η by Exercise 2 on p. 211 in [11]. If dim 𝔤_{α_{j}} = 1, we can assume Ad(k)ξ = η by Proposition 4.6. Note that, by construction, Ad(k)H_{α_{i}} = H_{α_{j}} and Ad(k)ξ_{t} = η_{t}. The map Ad(k) induces a linear isometry from (𝔞 ⊕ 𝔤_{α_{i}} ⊕ 𝔤_{2α_{i}}) ⊖ ℝξ_{t} onto (𝔞 ⊕ 𝔤_{α_{j}} ⊕ 𝔤_{2α_{j}}) ⊖ ℝη_{t}, and using Lemma 4.3 it is easy to see that Ad(k) ∘ A_{ξ_{t}} = A_{η_{t}} ∘ Ad(k) holds on this subspace. Let λ ∈ Σ^{+} so that λ − α_{i} ∉ Σ^{+}, and let V_{λ,α_{i}} be the linear subspace of 𝔫 spanned by the root spaces of the roots in the α_{i}-string of λ. It follows from Lemma 4.3 that V_{λ,α_{i}} is invariant under A_{ξ_{t}}. If Ad(k)^{∗}λ ∈ Σ^{+}, then, using again Lemma 4.3, it follows that Ad(k) ∘ A_{ξ_{t}} = A_{η_{t}} ∘ Ad(k) holds on V_{λ,α_{i}}. If Ad(k)^{∗}λ ∈ Σ^{−} = Σ \ Σ^{+}, we define k′ = Exp (\frac{π}{\sqrt{2}|α_{j}|}(η + θη)) (see Proposition 4.4 (4)) and a linear isometry F from V_{λ,α_{i}} into 𝔫 by F = Ad(k′) ∘ θ ∘ Ad(k)|V_{λ,α_{i}}. Then, by construction, F (H_{α_{i}}) = H_{α_{j}}, F (ξ) = η, F (θξ) = θη, and hence also F (ξ_{t}) = η_{t}. A straightforward calculation shows that F ∘ A_{ξ_{t}} = A_{η_{t}} ∘ F holds on V_{λ,α_{i}}. Since 𝔫 ⊖ (𝔤_{α_{i}} ⊕ 𝔤_{2α_{i}}) is spanned by subspaces of the form V_{λ,α_{i}} this finishes the proof. q.e.d.

We will now investigate the congruence problem for the foliations 𝔉_{i}. First, let α, β ∈ Λ be two distinct simple roots and ξ ∈ 𝔤_{α}, η ∈ 𝔤_{β} be two unit vectors. Assume that 𝔉_{ξ} and 𝔉_{η} are isometrically congruent. It is clear that an isometry mapping 𝔉_{ξ} to 𝔉_{η} maps the unique minimal leaf S_{ξ} · o of 𝔉_{ξ} to the unique minimal leaf S_{η} · o of 𝔉_{η}. Thus there exists an isometry k ∈ K with k(S_{ξ} ·o) = S_{η} ·o. It is also clear that k preserves the mean curvature of each leaf as well as the distance of any leaf to the minimal leaf. Therefore, using the explicit expression in Corollary 4.5 of the mean curvature of the leaves, we necessarily have |α| = |β|. We conclude that if α and β are simple roots with different length, then the foliations 𝔉_{ξ} and 𝔉_{η} cannot be isometrically congruent.

We now assume that α and β have the same length and that S_{ξ}·o and S_{η} · o are congruent under an isometry of M. Our next aim is to prove that there exists a symmetry of the Dynkin diagram of the restricted root system of M mapping α to β. First of all, a similar proof as in Lemma 3.3 shows that S_{ξ} is completely solvable, that is, Ad(S_{ξ}) can be represented by upper triangular matrices in a suitable basis of the Lie algebra 𝔰_{ξ}. By assumption, S_{ξ} · o and S_{η} · o are isometric to each other. Since any two completely solvable transitive groups of isometries on a Riemannian manifold are conjugate to each other in the full isometry group of that manifold (see, e.g., [1]), we conclude that the Lie algebras 𝔰_{ξ} and 𝔰_{η} are isomorphic.

Let F : 𝔰_{ξ} → 𝔰_{η} be a Lie algebra isomorphism. It is easy to see that 𝔞 is a Cartan subalgebra of both 𝔰_{ξ} and 𝔰_{η}. Since any two Cartan subalgebras of a solvable Lie algebra are conjugate to each other under an inner automorphism we can assume without loss of generality that F (𝔞) = 𝔞. If dim 𝔤_{α} > 1, then Σ^{+} is the set of roots of 𝔰_{ξ} with respect to 𝔞, and if dim 𝔤_{α} = 1, then Σ^{+} \ {α} is the set of roots of 𝔰_{ξ} with respect to 𝔞. It follows that if dim 𝔤_{α} > 1, then also dim 𝔤_{β} > 1, and if dim 𝔤_{α} = 1, then also dim 𝔤_{β} = 1. We will now distinguish the two cases when dim 𝔤_{α} > 1 and dim 𝔤_{α} = 1.

Case 1. dim 𝔤_{α} > 1. Then we get [F (H), F (X_{λ})] = F ([H, X_{λ}]) = λ(H)F (X_{λ}) for all H ∈ 𝔞 and X_{λ} ∈ 𝔤_{λ} ∩ 𝔰_{ξ}, λ ∈ Σ^{+}. Note that 𝔤_{λ} ∩ 𝔰_{ξ} = 𝔤_{λ} for all λ ∈ Σ^{+} \ {α} and 𝔤_{α} ∩ 𝔰_{ξ} = 𝔤_{α} ⊖ ℝξ. Thus, for all λ ∈ Σ^{+} \ {α}, F maps the root space 𝔤_{λ} onto the root space 𝔤_{F^{∗}(λ)} ∩ 𝔰_{η} with F^{∗}(λ) = λ ∘ F^{−1}|𝔞. Moreover, F maps 𝔤_{α} ⊖ ℝξ onto 𝔤_{F^{∗}(α)} ∩ 𝔰_{η}, and consequently F^{∗} induces a permutation of the roots in Σ^{+}. Let λ, μ ∈ Λ be simple roots and let λ, …, λ+qμ be the μ-string containing λ. Since F^{∗} : Σ^{+} → Σ^{+} is a bijection, and as F^{∗}(λ+μ) = F^{∗}(λ)+F^{∗}(μ), the roots F^{∗}(λ), …, F^{∗}(λ) + qF^{∗}(μ) form the F^{∗}(μ)-string containing F^{∗}(λ). It follows that F^{∗} preserves the edge and arrow relations between the vertices in the Dynkin diagram of Σ, that is, F^{∗} is a symmetry of the Dynkin diagram of Σ. In particular, F^{∗} maps simple roots to simple roots, that is, F^{∗}(Λ) = Λ. Taking into account the multiplicities of simple roots it now follows easily that F (𝔤_{α} ⊖ℝξ) = 𝔤_{β} ⊖ℝη, and hence F^{∗}(α) = β. We thus have proved that if dim 𝔤_{α} > 1, and if S_{ξ} · o is isometric to S_{η} · o, then there exists a symmetry of the Dynkin diagram of the restricted root system Σ mapping the simple root α to the simple root β.

Case 2. dim 𝔤_{α} = 1. Recall that in this case the set of roots of 𝔰_{ξ} and 𝔰_{η} with respect to the Cartan subalgebra 𝔞 is Σ^{+}\{α} and Σ^{+}\{β}, respectively. As in the previous case we can construct a bijection F^{∗} of Σ^{+} \ {α} onto Σ^{+} \ {β} preserving the string relations between roots. Our aim now is to show that there exists a symmetry of the Dynkin diagram of Σ mapping α to β.

We can assume that 2α ∉ Σ^{+}, because α is a simple root and if 2α ∈ Σ^{+} we have Σ = BC_{r}, but if Λ ⊂ BC_{r} there exists only one simple root α ∈ Λ with 2α ∈ Σ^{+} and the length of it is different from the length of all other simple roots in Λ.

We now define Λ_{α} = {γ + qα | γ ∈ Λ \ {α} , γ + qα ∈ Σ^{+} , q ∈ {0, 1, 2, 3}}. First of all, from the structure of 𝔫 as a graded Lie algebra 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m} with 𝔫_{1} = 𝔤_{α_{1}} ⊕ ⋯ ⊕ 𝔤_{α_{r}} it is clear that 𝔫 ⊖ 𝔤_{α} is a graded Lie algebra generated by Gen(𝔫 ⊖ 𝔤_{α}) = ⨁_{γ∈Λ_{α}} 𝔤_{γ}. Moreover, if we denote by m_{α} the coefficient of α in the maximal root α̃ in Σ^{+}, we see that 𝔫 ⊖ 𝔤_{α} is an (m − m_{α})-step nilpotent Lie algebra.

We will now associate to each such subalgebra 𝔫 ⊖ 𝔤_{α} a diagram in the following manner. Each vertex v ∈ Λ_{α} with 2v ∉ Σ^{+} is represented by [figure], and each vertex v ∈ Λ_{α} with 2v ∈ Σ^{+} is represented by [figure]. Then connect two vertices with each other if one of the following cases occurs:

[figure: v_{1}; v_{2}] if v_{1} + v_{2} ∈ Σ^{+}, 2v_{1} + v_{2}, v_{1} + 2v_{2} ∉ Σ^{+};
[figure: v_{1}; v_{2}] if v_{1} + v_{2}, v_{1} + 2v_{2} ∈ Σ^{+}, 2v_{1} + v_{2}, v_{1} + 3v_{2} ∉ Σ^{+};
[figure: v_{1}; v_{2}] if v_{1} + v_{2}, v_{1} + 2v_{2}, v_{1} + 3v_{2} ∈ Σ^{+}, 2v_{1} + v_{2} ∉ Σ^{+};
[figure: v_{1}; v_{2}] if v_{1} + v_{2} ∈ Σ^{+};
[figure: v_{1}; v_{2}] if v_{1} + v_{2} ∈ Σ^{+}.

In the case [figure]we have 2v_{1}+v_{2}, v_{1}+2v_{2} ∈ Σ^{+} and 3v_{1}+v_{2}, v_{1}+3v_{2} ∉ Σ^{+}, which explains the two lines and the two arrows. In the case [figure] we have v_{1} + 2v_{2}, 2v_{1} + v_{2} ∉ Σ^{+}, which explains that there is only one line. It follows from general properties of root systems that v_{1}+v_{2} ∉ Σ^{+} if none of the above cases occurs. For i ≤ j we define

[figure: v_{i}; v_{j}] = \begin{cases}[figure: v_{i}] & if i = j \\ [figure: v_{i}; v_{i+1}] , …, [figure: v_{j−1}; v_{j}] & if i < j. \end{cases}

Since F maps the Cartan subalgebra 𝔞 of 𝔰_{ξ} onto the Cartan subalgebra 𝔞 of 𝔰_{η}, it induces a transformation from Λ_{α} onto Λ_{β} such that the string relations among the elements in Λ_{α} are the same as the string relations among the corresponding elements in Λ_{β}. It follows that the diagrams of 𝔫⊖𝔤_{α} and 𝔫⊖𝔤_{β} are isomorphic. Thus the diagrams provide us with a simple tool for deciding nonisomorphy of two Lie algebras 𝔰_{ξ} and 𝔰_{η}.

By taking into account the following two observations we can reduce the number of diagrams that have to be drawn. Firstly, we must have dim Gen(𝔫 ⊖ 𝔤_{α}) = dim Gen(𝔫 ⊖ 𝔤_{β}). It is quite easy to determine dim Gen(𝔫⊖𝔤_{α}) from the Dynkin diagram. First determine the number all vertices that are not connected with α. For each other vertex add the number two if it is connected with α by one line or by more lines with the arrow pointing away from α, and add the number three or four if it is connected with α by two or three lines with the arrow pointing towards α. Secondly, the number of steps of 𝔫 ⊖ 𝔤_{α} and 𝔫 ⊖ 𝔤_{β} must be equal, that is, m_{α} = m_{β}. Taking into account these two simple criteria and the symmetries of the Dynkin diagram we are left with the following cases:

1. Σ = A_{r}, r ≥ 5 and α, β ∈ \{α_{2}, …, α_{s}\} with s = [\frac{r+1}{2}]; 2. Σ = B_{r}, r ≥ 4 and α, β ∈ {α_{2}, …, α_{r−1}};

3. Σ = C_{r}, r ≥ 5 and α, β ∈ {α_{2}, …, α_{r−2}};

4. Σ = D_{r}, r ≥ 5 and α, β ∈ {α_{1}, α_{r}}, and r ≥ 6 and α, β ∈ {α_{2}, …, α_{r−3}};

5. Σ = E_{7} and α, β ∈ {α_{1}, α_{2}} or α, β ∈ {α_{3}, α_{5}};

6. Σ = E_{8} and α, β ∈ {α_{1}, α_{8}} or α, β ∈ {α_{3}, α_{6}};

7. Σ = BC_{r}, r ≥ 4 and α, β ∈ {α_{1}, …, α_{r−2}}.

It is a straightforward but lengthy exercise to draw all the relevant diagrams. We omit the list of all diagrams here and just illustrate it by discussing Case 3:

a) 𝔫 ⊖ 𝔤_{α_{2}}:

[figure: α_{1}; α_{2} + α_{3}; α_{4}; α_{1} + α_{2}; α_{3}; α_{r−1}; α_{r}.]

b) 𝔫 ⊖ 𝔤_{α_{i}} , 3 ≤ i ≤ r − 3 , r ≥ 6:

[figure: α_{1}; α_{i−2}; α_{i−1} + α_{i}; α_{i+1}; α_{i+2}; α_{i−1}; α_{i} + α_{i+1}; α_{r−1}; α_{r}.]

c) 𝔫 ⊖ 𝔤_{α_{r−2}}:

[figure: α_{1}; α_{r−4}; α_{r−3} + α_{r−2}; α_{r−1}; α_{r}.; α_{r−3}; α_{r−2} + α_{r−1}]

By comparing these diagrams it is now easy to conclude that there exists a symmetry P ∈ Aut(DD) with P (α) = β. The other cases can be settled in a similar fashion.

Conversely, let α, β ∈ Λ and ξ ∈ 𝔤_{α} and η ∈ 𝔤_{β} be unit vectors. Assume that there exists a symmetry P ∈ Aut(DD) with P (α) = β. Then there exists an isometry k in the normalizer of 𝔞 + 𝔫 in K such that the induced action of F = Ad(k) ∈ Aut(𝔤) on Σ^{+} is equal to the action of P on Σ^{+}. Then F (𝔰_{ξ}) = 𝔰_{η′} for some unit vector η′ ∈ 𝔤_{β}. Using Lemma 4.1 we see that the foliations 𝔉_{ξ} and 𝔉_{η} are isometrically congruent. Thus we have proved:

Theorem 4.8. Two foliations 𝔉_{i} and 𝔉_{j} are isometrically congruent to each other if and only if there exists a symmetry P ∈ Aut(DD) with P (α_{i}) = α_{j}.

This result has a remarkable consequence. A smooth function f : M → ℝ is isoparametric if there exist smooth real-valued functions a, b such that ||grad f||^{2} = (a ∘ f)f and Δf = (b ∘ f)f. The interest in such functions stems from geometrical optics. The first condition means that the level sets of f are equidistant, and then the second condition is equivalent to the constancy of the mean curvature of the level sets. The level sets of an isoparametric function form an isoparametric system on M. Clearly, the level sets of a cohomogeneity one action form an isoparametric system, where the corresponding isoparametric function is just the natural projection onto the orbit space. Such isoparametric systems are called homogeneous. From Theorems 4.7 and 4.8 we conclude:

Corollary 4.9. On every connected irreducible Riemannian symmetric space of noncompact type and rank ≥ 3 there exist homogeneous isoparametric systems with the same principal curvatures, counted with multiplicities.

Ferus, Karcher and Münzner [9] discovered such a phenomenon for inhomogeneous isoparametric systems on spheres, but apparently it was not clear whether homogeneous isoparametric systems can be distinguished in general by their “spectral data”. We illustrate the corollary by an example.

Example. Let M be the set of all Euclidean structures on ℝ^{4}. Then M can be realized as the 9-dimensional noncompact symmetric space SL(4, ℝ)/SO(4) of rank three. We get an Iwasawa decomposition of SL(4, ℝ) by choosing for A the diagonal (4 × 4)-matrices with determinant one, and for N the upper triangular (4 × 4)-matrices for which each element on the diagonal is equal to one. Then

M = AN = \{(\begin{matrix}x_{11} & x_{12} & x_{13} & x_{14} \\ 0 & x_{22} & x_{23} & x_{24} \\ 0 & 0 & x_{33} & x_{34} \\ 0 & 0 & 0 & x_{44} \end{matrix}) |x_{ij} ∈ ℝ , x_{11}x_{22}x_{33}x_{44} = 1\} .

The Lie algebra 𝔞 of A is the 3-dimensional abelian Lie algebra of all diagonal (4 × 4)-matrices with real coefficients and trace zero, and the Lie algebra 𝔫 of N is the 6-dimensional 3-step nilpotent Lie algebra of all upper triangular (4 × 4)-matrices with real coefficients. We define three vectors H_{1}, H_{2}, H_{3} ∈ 𝔞 by

H_{1} =(\begin{matrix}1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), H_{2} =(\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & −1 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}), H_{3} =(\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & −1 \end{matrix}),

and denote by α_{1}, α_{2}, α_{3} the dual one-forms on 𝔞 with respect to the Killing form of 𝔰𝔩(4, ℝ). Then Σ^{+} = {α_{1}, α_{2}, α_{3}, α_{1} + α_{2}, α_{2} + α_{3}, α_{1} + α_{2} + α_{3}} is the set of positive restricted roots of 𝔰𝔩(4, ℝ) with respect to 𝔞 and Λ = {α_{1}, α_{2}, α_{3}} is the corresponding set of simple roots. The resulting restricted root system is A_{3} and its Dynkin diagram is [figure: α_{1}; α_{2}; α_{3}]. The simple root spaces 𝔤_{α_{1}}, 𝔤_{α_{2}}, 𝔤_{α_{3}} are spanned by

(\begin{matrix}0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}) , (\begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \end{matrix}) ,

respectively. We define three subgroups S_{1}, S_{2}, S_{3} of AN by

S_{1} = \{(\begin{matrix}x_{11} & 0 & x_{13} & x_{14} \\ 0 & x_{22} & x_{23} & x_{24} \\ 0 & 0 & x_{33} & x_{34} \\ 0 & 0 & 0 & x_{44} \end{matrix}) | x_{ij} ∈ ℝ , x_{11}x_{22}x_{33}x_{44} = 1\} ,

S_{2} = \{(\begin{matrix}x_{11} & x_{12} & x_{13} & x_{14} \\ 0 & x_{22} & 0 & x_{24} \\ 0 & 0 & x_{33} & x_{34} \\ 0 & 0 & 0 & x_{44} \end{matrix}) | x_{ij} ∈ ℝ , x_{11}x_{22}x_{33}x_{44} = 1\} ,

S_{3} = \{(\begin{matrix}x_{11} & x_{12} & x_{13} & x_{14} \\ 0 & x_{22} & x_{23} & x_{24} \\ 0 & 0 & x_{33} & 0 \\ 0 & 0 & 0 & x_{44} \end{matrix}) | x_{ij} ∈ ℝ , x_{11}x_{22}x_{33}x_{44} = 1\} .

The corresponding subalgebras of 𝔞 + 𝔫 are 𝔰_{i} = (𝔞 + 𝔫) ⊖ 𝔤_{α_{i}}. Since the Dynkin diagram has only one nontrivial symmetry, we see that the homogeneous isoparametric systems induced by the actions of S_{1} and S_{3} are congruent under an isometry of M. On the other hand, the homogeneous isoparametric systems induced by the actions of S_{1} and S_{2} are not congruent under an isometry of M, but have the same principal curvatures, counted with multiplicities.

5. The classification

By means of Theorems 3.5 and 4.8 we have solved the congruency problem on the parameter space ℝP^{r−1} ∪ {1, …, r}. It remains to show that any homogeneous cohomogeneity one foliation on M is isometrically congruent to a foliation constructed from the parameter space.

Let M be a connected irreducible Riemannian symmetric space of noncompact type and 𝔉 a homogeneous foliation of codimension one on M. This means that there exists a connected subgroup S of G^{o} such that the leaves of 𝔉 coincide with the orbits of the action of S on M. Let S = L · R be a Levi decomposition of S, where L and R are the semisimple and solvable factor. Moreover, let L = L_{K} · L_{AN} be an Iwasawa decomposition of L, where L_{K} and L_{AN} are the compact and solvable factor. Then S = L · R = L_{K} · (L_{AN} · R). The compact group L_{K} has a fixed point o in M by Cartan’s fixed point theorem. It follows that the solvable subgroup L_{AN} · R of S is transitive on S · o, the leaf of 𝔉 through o, and consequently also on every leaf of 𝔉. Without loss of generality we can therefore assume that S is solvable.

Since S is solvable, we can decompose it into S = T · B with a compact subgroup T and a normal k-solvable subgroup B (see e.g., [19]). The latter means that there exists a compact normal subgroup B_{K} of B such that B/B_{K} is simply connected. The subgroup B_{K} is then maximal compact in B and contained in the center of B. An analogous argument as above shows that already B is transitive on each leaf of 𝔉. Since B is a subgroup of the isometry group of M, it acts effectively on M. Since the leaves of 𝔉 are of codimension one, B must also act effectively on each leaf of 𝔉. The isotropy subgroup of B at a point in M is compact and hence contained in some conjugate of B_{K}, and therefore lies in the center of B. Because of effectivity of B it follows that at every point the isotropy subgroup of B is trivial. Thus B acts simply transitively on each orbit. Without loss of generality we can therefore assume from now on that S is solvable and acts simply transitively on each orbit of 𝔉. This shows that the classification up to isometric congruence of all homogeneous foliations of codimension one on M is equivalent to the classification up to orbit equivalence of all (n − 1)-dimensional connected solvable subgroups of G^{o} acting freely on M.

Since the Lie algebra 𝔰 of S is solvable, there exists a maximal solvable subalgebra of 𝔤 containing 𝔰. The maximal solvable subalgebras of real semisimple Lie algebras have been classified by Mostow [21]. Any maximal solvable subalgebra of 𝔤 contains a Cartan subalgebra 𝔥 of 𝔤. The Cartan subalgebra 𝔥 is the direct sum of its toroidal part 𝔱 and its vector part 𝔞. Here, the toroidal part 𝔱 consists of all X ∈ 𝔥 for which the eigenvalues of ad(X) are purely imaginary, and the vector part 𝔞 consists of all X ∈ 𝔥 for which the eigenvalues of ad(X) are real. There exists a Cartan decomposition 𝔤 = 𝔨 + 𝔭 of 𝔤 such that 𝔱 ⊂ 𝔨 and 𝔞 ⊂ 𝔭. Since 𝔞 is abelian it induces a root space decomposition 𝔤 = 𝔤_{0} ⊕ ⨁_{λ∈Σ} 𝔤_{λ} of 𝔤, where Σ denotes the set of roots in the dual vector space 𝔞^{∗}. An element H ∈ 𝔞 is called regular if α(H) ≠ 0 for all λ ∈ Σ. The set of all regular elements in 𝔞 forms an open and dense subset of 𝔞. We choose a regular element H ∈ 𝔞 and define Σ^{+} = {λ ∈ Σ | λ(H) > 0} and 𝔫 = ⨁_{λ∈Σ^{+}} 𝔤_{λ}. Then 𝔥 + 𝔫 is a maximal solvable subalgebra of 𝔤. Conversely, as was shown by Mostow [21], any maximal solvable subalgebra of the real semisimple Lie algebra 𝔤 arises in this way.

Now let 𝔥 + 𝔫 be a maximal solvable subalgebra of 𝔤 that contains 𝔰. Note that for a Cartan subalgebra 𝔥 the vector part 𝔞 is in general not a maximal abelian subspace of 𝔭. If 𝔞 is maximal abelian, then 𝔥 is called a maximally noncompact Cartan subalgebra. In this case Σ is a root system and the above decomposition of 𝔤 is the restricted root space decomposition of 𝔤.

Lemma 5.1. If 𝔥 + 𝔫 is a maximal solvable subalgebra of 𝔤 with 𝔰 ⊂ 𝔥 + 𝔫, then 𝔥 is a maximally noncompact Cartan subalgebra of 𝔤.

Proof. Let HN and AN be the connected subgroup of G^{o} with Lie algebra 𝔥 + 𝔫 and 𝔞 + 𝔫, respectively. The orbits of the actions of HN and AN on M coincide. Therefore, in order that S has orbits of codimension one, we necessarily need that the dimension of 𝔞 + 𝔫 is at least n − 1. Assume that 𝔥 is not maximally noncompact. Then the vector part 𝔞 of 𝔥 is strictly contained in some maximal abelian subspace 𝔞̃ in 𝔭. Let 𝔤 = 𝔤̃_{0} ⊕ ⨁_{λ̃∈Σ̃} 𝔤̃λ̃ be the restricted root space decomposition of 𝔤 with respect to 𝔞̃, where Σ̃ is the corresponding set of restricted roots. By construction, the restriction to 𝔞 of any root in Σ̃ is either trivial or a root in Σ. This implies 𝔤_{0} = 𝔤̃_{0} ⊕ ⨁_{λ̃∈Σ̃,λ̃|𝔞=0} 𝔤̃λ̃ and 𝔤_{λ} = ⨁_{λ̃∈Σ̃,λ̃|𝔞=λ} 𝔤̃λ̃ for all λ ∈ Σ. We choose a set of simple

roots in Σ̃ such that λ̃(H) ≥ 0 for any such simple root λ̃, and define 𝔫̃ = ⨁_{λ̃∈Σ̃^{+}} 𝔤̃λ̃ where Σ̃^{+} denotes the resulting set of positive restricted roots in Σ̃. Clearly, we have 𝔫 ⊂ 𝔫̃. Since 𝔞 is strictly contained in 𝔞̃, the inequality dim(𝔞+𝔫) ≥ n−1 and the equality dim(𝔞̃ + 𝔫̃) = n imply that 𝔫 = 𝔫̃ and dim 𝔞 = dim 𝔞̃ − 1. According to Sugiura [24, Theorem 5], the one-dimensional normal space of 𝔞 in 𝔞̃ is contained in a wall of a Weyl chamber in 𝔞̃. But this implies that there exists a root in Σ̃^{+} whose restriction to 𝔞 is trivial, which contradicts the equality 𝔫 = 𝔫̃. Thus 𝔥 is a maximally noncompact Cartan subalgebra of 𝔤. q.e.d.

A maximally noncompact Cartan subalgebra 𝔥 of a real semisimple Lie algebra 𝔤 is of the form 𝔥 = 𝔱 + 𝔞, where 𝔞 is a maximal abelian subspace in the vector part 𝔭 of some Cartan decomposition 𝔤 = 𝔨 + 𝔭 and 𝔱 is a maximal abelian subalgebra in the centralizer of 𝔞 in 𝔨. Any two maximally noncompact Cartan subalgebras of a real semisimple Lie algebra are conjugate to each other. Thus we have proved:

Proposition 5.2. Let S be a connected solvable closed subgroup of G^{o} that acts freely on M with codimension one orbits. Then there exists a Cartan decomposition 𝔤 = 𝔨 + 𝔭, a restricted root space decomposition 𝔤 = 𝔤_{0} ⊕⨁_{λ∈Σ} 𝔤_{λ} with respect to a maximal abelian subspace 𝔞 in 𝔭, and a set Λ of simple roots in Σ, such that the Lie algebra 𝔰 of S satisfies 𝔰 ⊂ 𝔱 + 𝔞 + 𝔫, where 𝔱 is a maximal abelian subalgebra in the centralizer 𝔪 of 𝔞 in 𝔨 and 𝔫 = ⨁_{λ∈Σ^{+}} 𝔤_{λ}, where Σ^{+} denotes the set of positive restricted roots determined by Λ. In this situation 𝔥 = 𝔱 + 𝔞 is a Cartan subalgebra and 𝔥 + 𝔫 is a maximal solvable subalgebra of 𝔤.

We may assume from now on that 𝔰 is contained in a maximal solvable subalgebra of the form 𝔱 + 𝔞 + 𝔫 as described in Proposition 5.2. We denote by 𝔰_{c} the image of 𝔰 under the canonical projection from 𝔱 + 𝔞 + 𝔫 onto the compact part 𝔱. Analogously, 𝔰_{n} denotes the image of 𝔰 under the canonical projection from 𝔱 + 𝔞 + 𝔫 onto the noncompact part 𝔞 + 𝔫. Since dim 𝔰 = n − 1 and since S acts freely on M we also have dim 𝔰_{n} = n − 1. Thus there exists a unit vector ξ ∈ 𝔞 + 𝔫 such that 𝔰_{n} = (𝔞 + 𝔫) ⊖ ℝξ is the orthogonal complement of ℝξ in 𝔞 + 𝔫.

Lemma 5.3. We have ξ ∈ 𝔞, or there exists a simple root α ∈ Λ such that ξ ∈ ℝH_{α} + 𝔤_{α}.

Proof. We decompose ξ into ξ = ξ_{0} + ∑_{λ∈Σ^{+}} ξ_{λ} with ξ_{0} ∈ 𝔞 and ξ_{λ} ∈ 𝔤_{λ}. We have to show that if ξ_{λ} ≠ 0 for some λ ∈ Σ^{+}, then λ ∈ Λ, ξ_{0} ∈ ℝH_{λ}, and ξ_{μ} = 0 for all μ ∈ Σ^{+} \ {λ}. Assume that ξ_{λ} ≠ 0 for some λ ∈ Σ^{+}. Without loss of generality we can assume that λ is not the sum of two roots in {μ ∈ Σ^{+} | ξ_{μ} ≠ 0}.

We first show that λ ∈ Λ. Assume that λ ∉ Λ. Consider the natural gradation 𝔫 = 𝔫_{1} ⊕ ⋯ ⊕ 𝔫_{m} of 𝔫 generated by 𝔫_{1} = ⨁_{μ∈Λ} 𝔤_{μ}. Since λ ∉ Λ, there exist vectors X_{μ} ∈ 𝔤_{μ} and Y_{ν} ∈ 𝔤_{γ}, where μ, ν ∈ Σ^{+} with μ + ν = λ, such that ξ_{λ} = ⨁_{μ,ν∈Σ^{+},μ+ν=λ}[X_{μ}, Y_{ν}]. The assumption that λ is not the sum of two roots in {μ ∈ Σ^{+} | ξ_{μ} ≠ 0} ensures that these vectors X_{μ} and Y_{ν} are in 𝔰_{n}. Thus there exist vectors S_{μ}, T_{ν} ∈ 𝔱 such that S_{μ} + X_{μ} and T_{ν} + Y_{ν} are in 𝔰. Therefore we also have [S_{μ}, Y_{ν}] − [T_{ν}, X_{μ}] + [X_{μ}, Y_{ν}] = [S_{μ} + X_{μ}, T_{ν} + Y_{ν}] ∈ 𝔰. But since the derived subalgebra of 𝔱 + 𝔞 + 𝔫 is contained in 𝔫, we must have [𝔰, 𝔰] ⊂ 𝔫 and hence [S_{μ}, Y_{ν}] − [T_{ν}, X_{μ}] + [X_{μ}, Y_{ν}] = [S_{μ} + X_{μ}, T_{ν} + Y_{ν}] ∈ 𝔰_{n}. As 𝔱 is contained in 𝔪, the centralizer of 𝔞 in 𝔨, each element in 𝔱 leaves each root space invariant. Thus [S_{μ}, Y_{ν}] ∈ 𝔤_{ν} ⊂ 𝔰_{n} and [T_{ν}, X_{μ}] ∈ 𝔤_{μ} ⊂ 𝔰_{n}. Altogether this implies that ξ_{λ} = ⨁_{μ,ν∈Σ^{+},μ+ν=λ}[X_{μ}, Y_{ν}] = ⨁_{μ,ν∈Σ^{+},μ+ν=λ}([S_{μ} + X_{μ}, T_{ν} + Y_{ν}] − [S_{μ}, Y_{ν}] + [T_{ν}, X_{μ}]) ∈ 𝔰_{n}, which is a contradiction to ξ_{λ} ≠ 0. Thus we conclude that λ ∈ Λ.

We next show that ξ_{0} ∈ ℝH_{λ}. Assume that ξ_{0} ∉ ℝH_{λ}. Then there exists a vector H ∈ 𝔞 that is perpendicular to ξ_{0} and satisfies λ(H) ≠ 0. Since H is perpendicular to ξ_{0}, we have H ∈ 𝔰_{n}, and hence there exists a vector S ∈ 𝔱 such that S + H ∈ 𝔰. The vector ξ_{0} + xξ_{λ} with x = −|ξ_{0}|^{2}/|ξ_{λ}|^{2} < 0 is perpendicular to ξ and hence in 𝔰_{n}. Thus there exists a vector T ∈ 𝔱 such that T + ξ_{0} + xξ_{λ} ∈ 𝔰. We conclude that x[S, ξ_{λ}] + xλ(H)ξ_{λ} = [S + H, T + ξ_{0} + xξ_{λ}] ∈ 𝔰_{n}. Since ad(S) is a skewsymmetric transformation, we know that [S, ξ_{λ}] is perpendicular to ξ_{λ} and hence in 𝔰_{n}. Since also x ≠ 0 and λ(H) ≠ 0 we conclude that ξ_{λ} ∈ 𝔰_{n}, which is a contradiction. It follows that ξ_{0} ∈ ℝH_{λ}.

In the next step we show that ξ_{μ} = 0 for all μ ∈ Σ^{+} \ {λ} (if Σ is a reduced root system) resp. μ ∈ Σ^{+} \ {λ, 2λ} (if Σ is not reduced). Assume that there exists such a root μ with ξ_{μ} ≠ 0. Then there exists a vector H ∈ 𝔞 with λ(H) = 0 and μ(H) ≠ 0. Since ξ_{0} ∈ ℝH_{λ} and λ(H) = 0, we see that H is perpendicular to ξ_{0} and hence in 𝔰_{n}. Thus there exists a vector S ∈ 𝔱 such that S +H ∈ 𝔰. The vector ξ_{λ}+yξ_{μ} with y = −|ξ_{λ}|^{2}/|ξ_{μ}|^{2} < 0 is perpendicular to ξ and hence in 𝔰_{n}. Thus there exists a vector T ∈ 𝔱 such that T +ξ_{λ} +yξ_{μ} ∈ 𝔰. Then [S, ξ_{λ}]+y[S, ξ_{μ}]+ λ(H)ξ_{λ} + yμ(H)ξ_{μ} = [S + H, T + ξ_{λ} + yξ_{μ}] ∈ 𝔰_{n}. Since [S, ξ_{λ}] ∈ 𝔤_{λ} is perpendicular to ξ_{λ} and [S, ξ_{μ}] ∈ 𝔤_{μ} is perpendicular to ξ_{μ}, we see that [S, ξ_{λ}] and [S, ξ_{μ}] are in 𝔰_{n}. Since also λ(H) = 0, y ≠ 0 and μ(H) ≠ 0, we therefore get ξ_{μ} ∈ 𝔰_{n}, which is a contradiction. It follows that ξ_{μ} = 0 for all μ ∈ Σ^{+} \ {λ, 2λ}.

If Σ is a reduced root system we have just finished the proof of Lemma 5.3. Now let us assume that Σ is not reduced. We have to show that ξ_{2λ} = 0. Assume that ξ_{2λ} ≠ 0. Then the vectors H_{λ} + xξ_{λ} and ξ_{λ} + zξ_{2λ} with x = −⟨H_{λ}, ξ_{0}⟩/|ξ_{λ}|^{2} and z = −|ξ_{λ}|^{2}/|ξ_{2λ}^{2}| < 0 are in 𝔰_{n}. Hence there exist vectors S, T ∈ 𝔱 such that S+H_{λ}+xξ_{λ} and T +ξ_{λ}+zξ_{2λ} are in 𝔰. Now we get [S, ξ_{λ}]+z[S, ξ_{2λ}]+|H_{λ}|^{2}ξ_{λ} +2z|H_{λ}|^{2}ξ_{2λ} −x[T, ξ_{λ}] = [S + H_{λ} + xξ_{λ}, T + ξ_{λ} + zξ_{2λ}] ∈ 𝔰_{n}. Since [S, ξ_{λ}], [S, ξ_{2λ}] and [T, ξ_{λ}] are in 𝔰_{n}, we get that ξ_{λ} +2zξ_{2λ} ∈ 𝔰_{n}. But this implies 0 = ⟨ξ, ξ_{λ} +2zξ_{2λ}⟩ = |ξ_{λ}|^{2} + 2z|ξ_{2λ}|^{2} = −|ξ_{λ}|^{2}, which is a contradiction. Thus we must have ξ_{2λ} = 0, by which the proof of Lemma 5.3 is finished. q.e.d.

Proposition 5.4. Let 𝔰 be an (n − 1)-dimensional linear subspace of 𝔥 + 𝔫 with 𝔰 ∩ 𝔱 = 0.

(1) 𝔰_{n} is a subalgebra of 𝔞 + 𝔫 if and only if there exists a unit vector ξ in 𝔞 or in ℝH_{α} + 𝔤_{α} for some simple root α ∈ Λ such that 𝔰_{n} = (𝔞 + 𝔫) ⊖ ℝξ;

(2) 𝔰 is a subalgebra of 𝔥 + 𝔫 if and only if 𝔰_{n} is a subalgebra of 𝔞 + 𝔫 and the linear map L : 𝔰_{n} → 𝔱 defined by LX + X ∈ 𝔰 for all X ∈ 𝔰_{n} satisfies [LX, Y]+[X, LY] ∈ 𝔰_{n} and L([LX, Y]+[X, LY]+ [X, Y]) = 0 for all X, Y ∈ 𝔰_{n}.

Proof. If 𝔰_{n} is a subalgebra of 𝔞 + 𝔫, then it is also a subalgebra of 𝔥 + 𝔫, and the noncompact part (𝔰_{n})_{n} of 𝔰_{n} is 𝔰_{n} itself. It follows from Lemma 5.3 that there exists a unit vector ξ in 𝔞 or in ℝH_{α} + 𝔤_{α} for some simple root α ∈ Λ such that 𝔰_{n} = (𝔞 + 𝔫) ⊖ ℝξ. Conversely, since the derived subalgebra of 𝔞 + 𝔫 is 𝔫, and considering the gradation 𝔫 = 𝔫_{1} ⊕⋯⊕𝔫_{m}, where 𝔫_{1} = ∑_{α∈Λ} 𝔤_{α}, it is easy to see that (𝔞+𝔫)⊖ℝξ is a Lie subalgebra of 𝔞 + 𝔫 for any such vector ξ. This proves Part (1).

Since 𝔰∩𝔱 = 0, it is clear that L : 𝔰_{n} → 𝔱 is a well-defined linear map and that 𝔰 = {LX+X | X ∈ 𝔰_{n}}. The subspace 𝔰 is a subalgebra of 𝔥+𝔫 if and only if [LX, Y] + [X, LY] + [X, Y] = [LX + X, LY + Y] ∈ 𝔰 for all X, Y ∈ 𝔰_{n}. But this is equivalent to saying that for all X, Y ∈ 𝔰_{n} there exists a vector Z ∈ 𝔰_{n} such that [LX, Y] + [X, LY] + [X, Y] = LZ + Z. The vector on the left-hand side of this equation is in 𝔫. Thus 𝔰 is a subalgebra of 𝔥 + 𝔫 if and only if for all X, Y ∈ 𝔰_{n} there exists a vector Z ∈ 𝔰_{n} with LZ = 0 such that [LX, Y] + [X, LY] + [X, Y] = Z. If 𝔰 is a subalgebra of 𝔥 + 𝔫, then 𝔰_{n} is a subalgebra of 𝔞 + 𝔫 according to Lemma 5.3 and Part (1) of this proposition. Thus, if 𝔰 is a subalgebra of 𝔥+𝔫 then [LX, Y]+[X, LY] ∈ 𝔰_{n} and L([LX, Y]+[X, LY]+[X, Y]) = 0 for all X, Y ∈ 𝔰_{n}. Conversely, if 𝔰_{n} is a subalgebra of 𝔞 + 𝔫 and if L satisfies the previous two conditions, the last equivalence statement implies that 𝔰 is a subalgebra. q.e.d.

According to Proposition 5.4, 𝔰_{n} is a subalgebra of 𝔞 + 𝔫. Let S_{n} be the connected subgroup of AN with Lie algebra 𝔰_{n}. Every element in s ∈ S can be written in the form s = tan with t ∈ T, a ∈ A and n ∈ N, where T, A, N is the connected Lie subgroup of G^{o} with Lie algebra 𝔱, 𝔞, 𝔫, respectively. Based on Lemma 5.3 we distinguish three cases, namely Case 1. ξ ∈ 𝔞. Case 2. ξ ∈ 𝔤_{α} for some simple root α ∈ Λ. Case 3. ξ = aH_{α} + X_{α} for some 0 ≠ a ∈ ℝ, 0 ≠ X_{α} ∈ 𝔤_{α} and some simple root α ∈ Λ.

Case 1. ξ ∈ 𝔞. Then we have 𝔰_{n} = (𝔞 ⊖ ℝξ) ⊕ 𝔫, where 𝔞 ⊖ ℝξ denotes the orthogonal complement of ℝξ in 𝔞. Since 𝔰_{c} is contained in the centralizer 𝔪 of 𝔞 in 𝔨, it is clear that 𝔰_{c} leaves each root space 𝔤_{λ}, λ ∈ Σ, invariant. Thus we have [𝔰_{c}, 𝔫] ⊂ 𝔫. Moreover, 𝔰_{c} ⊂ 𝔱 and 𝔱 + 𝔞 is abelian, which shows that [𝔰_{c}, 𝔞 ⊖ ℝξ] = 0. Therefore, if tan ∈ S, we have at = ta and tn = n′t for some n′ ∈ N. We thus get tan · o = atn · o = an′t · o = an′ · o ∈ S_{n} · o, which shows that S · o ⊂ S_{n} · o. Since both orbits S · o and S_{n} · o have the same dimension and are connected and complete, we conclude S · o = S_{n} · o. This shows that the actions of S and S_{n} are orbit-equivalent, and hence the foliation 𝔉 is just the foliation 𝔉_{ξ}.

Case 2. ξ ∈ 𝔤_{α} for some simple root α ∈ Λ. In this case we have 𝔰_{n} = 𝔞 ⊕ (𝔫 ⊖ ℝξ) = 𝔞 ⊕ (𝔤_{α} ⊖ ℝξ) ⊕ ⨁_{λ∈Σ^{+}\{α}} 𝔤_{λ}. Let λ ∈ Σ^{+} \ {α} and X ∈ 𝔤_{λ}. Using Proposition 5.4 we get 0 = L([LH_{λ}, X] + [H_{λ}, LX] + [H_{λ}, X]) = L({ad(LH_{λ}) + λ(H_{λ})I_{λ}}X), where I_{λ} denotes the identity transformation on 𝔤_{λ}. But ad(LH_{λ}) is a skewsymmetric transformation on 𝔤_{λ} and λ(H_{λ}) is nonzero. Thus the transformation ad(LH_{λ}) + λ(H_{λ})I_{λ} : 𝔤_{λ} → 𝔤_{λ} is an isomorphism, and we conclude that L𝔤_{λ} = 0. Next, for X ∈ 𝔤_{α} ⊖ ℝξ we have [LH_{α}, X] = [LH_{α}, X] + [H_{α}, LX] ∈ 𝔰_{n} ∩ 𝔤_{α} by means of Proposition 5.4, and hence [LH_{α}, X] ∈ 𝔤_{α} ⊖ ℝξ. An analogous argument as above implies L(𝔤_{α} ⊖ ℝξ) = 0. Altogether we conclude that L(𝔫 ⊖ ℝξ) = 0. Therefore we have 𝔰_{c} = L𝔰_{n} = L𝔞, and using Proposition 5.4 we get [𝔰_{c}, 𝔫⊖ℝξ] = [L𝔞, 𝔫⊖ℝξ] ⊂ 𝔰_{n} ∩ 𝔫 = 𝔫 ⊖ ℝξ. As in Case 1 we can now deduce that S · o = S_{n} · o and hence 𝔉 is equal to 𝔉_{ξ}.

Case 3. ξ = aH_{α} + X_{α} for some 0 ≠ a ∈ ℝ, 0 ≠ X_{α} ∈ 𝔤_{α} and some simple root α ∈ Λ. We put X = X_{α}/|X_{α}|, define a subalgebra 𝔰_{X} of 𝔞+𝔫 by 𝔰_{X} = 𝔞⊕(𝔫⊖ℝX), and denote by S_{X} the corresponding connected Lie subgroup of AN. As was shown in the proof of Lemma 4.2 there exists an isometry g ∈ AN with Ad(g)(𝔰_{n}) = 𝔰_{X}. The foliations on M that are induced by the actions of S and I_{g}(S) are obviously isometrically congruent. We will now show that the foliations that are induced by the actions of I_{g}(S) and S_{X} are equal. Clearly, since g ∈ AN, we have Ad(g)𝔞 ⊂ 𝔞 + 𝔫, Ad(g)𝔫 = 𝔫 and Ad(g)(𝔞 + 𝔫) = 𝔞 + 𝔫. Furthermore, Ad(g)(𝔥 + 𝔫) = Ad(g)𝔥 + 𝔫 is a maximal solvable subalgebra of 𝔤 and Ad(g)𝔥 = Ad(g)𝔱 + Ad(g)𝔞 is a Cartan subalgebra of 𝔤. We define a linear map L_{g} : 𝔰_{X} → Ad(g)𝔱 by L_{g} = Ad(g)LAd(g)^{−1}, where L : 𝔰_{n} → 𝔱 is the linear map defined by LY + Y ∈ 𝔰 for all Y ∈ 𝔰_{n}. It can be easily verified that L_{g} satisfies the two conditions in Proposition 5.4. For the Lie algebra Ad(g)𝔰 of I_{g}(S) we then get Ad(g)𝔰 = {Ad(g)(LY + Y) | Y ∈ 𝔰_{n}} = {L_{g}Z + Z | Z ∈ Ad(g)𝔰_{n} = 𝔰_{X}}. This implies that (Ad(g)𝔰)_{n} = 𝔰_{X}, and from Case 2 we conclude that the foliations that are induced by the actions of I_{g}(S) and S_{X} are equal. Altogether we now see that 𝔉 is isometrically congruent to 𝔉_{X}. This concludes Case 3.

Since all Cartan decompositions 𝔤 = 𝔨 + 𝔭 of 𝔤, all maximal abelian subspaces 𝔞 of 𝔭, and all choices of sets Λ of simple roots in the corresponding set Σ of restricted roots are conjugate by inner automorphisms of 𝔤 we thus have proved:

Theorem 5.5. Let 𝔉 be a homogeneous foliation of codimension one on M. Then 𝔉 is isometrically congruent to one of the model foliations 𝔉_{ℓ} or 𝔉_{i}.

The main theorem now follows from Theorems 3.5, 4.8 and 5.5.

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