Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 36 (1995), No. 2, 155-170.

Shape Operators of Orbits of Isotropy Subgroups in Riemannian Symmetric Spaces

of the Compact Type ^{∗}

László Verhóczki

Department of Geometry, Technical University, Egry J. ut 1, 1521 Budapest, Hungary

Abstract. In the present paper orbits of isotropy subgroups are discussed in symmetric spaces of the compact type. Shape operators of the orbits are described by using the root space decompositions of orthogonal symmetric Lie algebras. Furthermore, some properties of totally geodesic orbits are established. MSC 1991: 53C35, 53C40, 57S15.

1. Introduction

Isometries of symmetric spaces form relatively large Lie groups. In order to study the properties of isometric actions on Riemannian manifolds we can take the actions of the subgroups of these Lie groups as models. It seems to be interesting to consider those socalled polar isometric actions which admit orthogonally transversal submanifolds (in other words, sections). For example, regarding a symmetric space, the action of the isotropy subgroup at a given point admits sections, since the maximal flat totally geodesic submanifolds containing the point intersect all the orbits orthogonally.

The first studies on the subject mentioned above were made by R. Bott and H. Samelson in the famous paper [3]. Later such special actions have been studied in several papers by L. Conlon, J. Szenthe, R. S. Palais and C. L. Terng (see [6], [18] and [16]). It was proved that a section is always a totally geodesic submanifold in the ambient space. Furthermore, it was shown that the principal orbits of such an action are isoparametric submanifolds, that is, their normal bundles have trivial holonomy group and they have constant principal curvatures with respect to parallel normal vector fields (see [18] and [16]). An isometric action is said to be hyperpolar if it admits locally Euclidean sections. Hyperpolar actions on symmetric spaces have been discussed in the papers [8] and [9].

∗Partially supported by Hungarian Nat. Found. for Sci. Research Grant No. T 7351 (1993).

0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix}1995 Heldermann Verlag, Berlin

In the present paper we study orbits of isotropy subgroups in symmetric spaces of the compact type. Concerning basic properties of these symmetric spaces, we refer to the book [10] of S. Helgason. Our purpose is to describe shape operators of these orbits and to select the totally geodesic ones. In the discussions we use the method of root space decompositions of orthogonal symmetric Lie algebras. Clearly, our discussions can be accomplished in symmetric spaces of the noncompact type, too. Since they are diffeomorphic with Euclidean spaces, orbits of isometric actions on them can be handled more easily. Furthermore, it can be seen that in symmetric spaces of the noncompact type isotropy subgroups have no totally geodesic orbit. Considering these facts, we discuss the compact case in this paper.

First we shall prove some statements concerning totally geodesic submanifolds of rank one in symmetric spaces of the compact type. Then the orbits of a fixed isotropy subgroup will be characterized by using the restricted root system of the ambient space. The main result of this paper is the presentation of the principal curvatures of each orbit with respect to any normal direction (see Theorem 1). By this statement we can give a necessary and sufficient condition for an orbit to be a totally geodesic one (see Theorem 2). This means that all the totally geodesic orbits can be found by regarding the weight lattice of the restricted root system of the ambient symmetric space. We acknowledge that this result can be obtained from the criterion given by R. Hermann in [11], too (see Remark 3). We shall show that the totally geodesic orbits are reflective submanifolds (see Theorem 3). Finally, as a corollary of Theorem 1 we obtain that all the orbits are curvature-adapted submanifolds (see Corollary 3).

Throughout this paper N denotes a Riemannian globally symmetric space with the metric tensor ⟨ , ⟩ and with the Levi-Civita connection ˜∇. The exponential map in N defined on the tangent bundle T N will be denoted by Exp and the curvature tensor by R. Concerning submanifolds, the notions and facts used here can be found in the books [14] and [17]. Considering a connected submanifold M in N, we always take the inherited Riemannian metric and the induced connection on it. The normal vector bundle of a given submanifold M will be denoted by ν(M). Let X, ξ be vector fields on M such that X is tangential and ξ is normal to M. Then the corresponding Weingarten equation is given by

˜∇_{X}ξ = −A_{ξ}(X) + D_{X}ξ

where A_{ξ} denotes the shape operator of M with respect to ξ and D denotes the induced normal connection in ν(M). If u is a unit vector in the normal subspace ν_{p}M (p ∈ M), then the eigenvalues of A_{u} are called principal curvatures of M at p with respect to the direction of u.

Taking a vector u in the tangent space T_{p}N, consider the endomorphism R_{u} : T_{p}N → T_{p}N defined by R_{u}(v) = R(v, u)u for v ∈T_{p}N. This self-adjoint map is called the curvature endomorphism with respect to u. Clearly, if u, v are two orthonormal vectors in T_{p}N, then ⟨R_{u}(v), v⟩ gives the sectional curvature of the plane spanned by u and v. The submanifold M is said to be curvature-adapted if T_{p}M is an invariant subspace of R_{u} and if the restriction of R_{u} to T_{p}M commutes with A_{u} for any p∈M and u∈ν_{p}M. This concept was introduced by J. Berndt and L. Vanhecke in the paper [1].

2. Totally geodesic submanifolds of rank one in symmetric spaces of the compact type

In this chapter some auxiliary statements are presented which will be used in the proofs of the main results. First let us recall some basic facts concerning symmetric spaces (see further [10]). Let G be the identity component of the Lie group formed by the isometries of the symmetric space N. Let us fix a point o in N and denote by K the compact isotropy subgroup of G at o. It is well-known that N can be identified with the coset space G|K. The identification is given by the diffeomorphism π : G|K → N where π(gK) = g(o) for gK ∈ G|K. Considering the canonical left actions of G on G|K and on N, π is an equivariant map since π(h(gK)) = h(π(gK)) holds for h∈G and gK ∈G|K.

Denoting by r_{o} the reflection of N on the point o, take the automorphism σ of G defined by the assignment σ(g) = r_{o}gr_{o} for any g ∈ G. This induces an involutive automorphism σ of the Lie algebra 𝔤 of G. Considering the eigenspaces of σ with respect to the eigenvalues 1 and −1, we obtain the direct sum

𝔤 = 𝔨 + 𝔭,

where 𝔨 coincides with the Lie algebra of the isotropy subgroup K. Since 𝔭 can be regarded as the tangent space of G|K at K, the tangent linear map T π of π presents an isomorphism between 𝔭 and T_{o}N. Furthermore, denoting by exp the exponential map of 𝔤 into G, the equality

exp(Y)(o) = Exp(T π(Y))

is valid for any Y ∈𝔭. Hereafter T_{o}N and 𝔭 are considered to be identical. Concerning the curvature tensor at o, it is known that for any vectors v_{1}, v_{2}, v_{3} in 𝔭

R(v_{1}, v_{2})v_{3} = −[ [v_{1}, v_{2}] v_{3}]  (1)

is valid where [ , ] denotes the bracket operation in 𝔤.

As usual, let Ad : G → Gl(𝔤) denote the adjoint representation of the Lie group G on 𝔤. It is well-known that the subspace 𝔭 is invariant with respect to the subgroup Ad_{K}.

Using normal coordinate systems and Jacobi vector fields along geodesics, the following well-known assertion can easily be proved.

Remark 1. Let N^{1}, N^{2} be locally symmetric spaces and p_{1}, p_{2} be points in N^{1}, N^{2}, respectively. If there is an isometric linear map φ : T_{p_{1}}N^{1} → T_{p_{2}}N^{2} such that φ ∘ R_{u}^{1} = R_{φ(u)}^{2} ∘ φ holds for any u∈T_{p_{1}}N^{1}, then N^{1} and N^{2} are locally isometric.

Henceforth we assume that N is a Riemannian symmetric space of the compact type. Therefore the pair (𝔤, σ) presents an orthogonal symmetric Lie algebra of the compact type (see Chapter VII in [10]). In order to characterize the totally geodesic submanifolds of rank one in N we apply the so-called root space decomposition of 𝔤.

Regarding the complexification 𝔤^{C} of 𝔤, let us take the dual orthogonal symmetric Lie algebra (𝔤̃, θ) of the noncompact type, where 𝔤̃ = 𝔨 + i𝔭 and θ is defined by the equality θ(Z + iY) = Z − iY for Z ∈ 𝔨, Y ∈ 𝔭. Let 𝔭_{0} be a fixed maximal Abelian subspace in 𝔭.

Clearly, 𝔭̃_{0} = i𝔭_{0} is also a maximal Abelian subspace in 𝔭̃ = i𝔭. Considering a linear form α on the vector space 𝔭̃_{0}, take the subspace

𝔤̃_{α} = {X ∈𝔤̃ | [H̃, X] = α(H̃)X for any H̃ ∈𝔭̃_{0}}.

A dual vector α (α ≠ 0) in 𝔭̃_{0}^{∗} is said to be a restricted root of 𝔤̃ with respect to 𝔭̃_{0} if 𝔤̃_{α} ≠ 0. Let △ denote the set of all the restricted roots. It is known that

𝔤̃ = 𝔤̃_{0} + ∑_{α∈△} 𝔤̃_{α}  (2)

gives a decomposition of the real semisimple Lie algebra 𝔤̃, furthermore, 𝔤̃_{0} = 𝔨_{0} + 𝔭̃_{0} is a direct sum with 𝔨_{0} = 𝔤̃_{0} ∩ 𝔨 (see [10; pp. 263-264]).

Considering two subpaces 𝔞, 𝔟 in 𝔤̃, let [𝔞, 𝔟] denote the subspace in 𝔤̃ generated by the vectors [X, Y] where X ∈𝔞 and Y ∈𝔟. Concerning the above decomposition, for any α and β in △ ∪ {0} we have

θ(𝔤̃_{α}) = 𝔤̃_{−α},  [𝔤̃_{α}, 𝔤̃_{β}] ⊂ 𝔤̃_{α+β}.  (3)

Let B denote the Killing form of the complex Lie algebra 𝔤^{C}. Obviously, the restrictions of B to 𝔤 and 𝔤̃ give the Killing forms of these real semisimple Lie algebras. Let us introduce the symmetric bilinear form D on 𝔤̃ defined by the equality D(X_{1}, X_{2}) = −2B(X_{1}, θ(X_{2})) for X_{1}, X_{2} ∈𝔤̃. It is well-known that the restriction of B to 𝔭̃ (respectively to 𝔨) is positive (respectively negative) definite. This fact implies that D presents a positive definite bilinear form on 𝔤̃. Furthermore, it can be seen that (2) gives an orthogonal decomposition of 𝔤̃ with respect to D.

For each root α ∈ △ we can define the vector Ã_{α} in 𝔭̃_{0} which is determined by the

equality B(Ã_{α}, H̃) = α(H̃) for H̃ ∈𝔭̃_{0}. The number |α| = \sqrt{B(Ã_{α}, Ã_{α})} is called the length of the root α.

Regarding an element α in △, let m_{α} denote the dimension of the subspace 𝔤̃_{α} which is called the multiplicity of the root α. Let us take an orthonormal basis X_{α,1}, …, X_{α,m_{α}} in 𝔤̃_{α} with respect to D. Further on we use the vectors

Z_{α,j} = X_{α,j} + θ(X_{α,j}),  Ỹ_{α,j} = X_{α,j} − θ(X_{α,j})  (j = 1, …, m_{α}).

Denoting by R the set of real numbers, let us consider the subspaces

𝔨_{α} = ∑_{j=1}^{m_{α}} RZ_{α,j},  𝔭̃_{α} = ∑_{j=1}^{m_{α}} RỸ_{α,j}.

It is obvious that the equalities 𝔨_{α} = 𝔨_{−α}, 𝔭̃_{α} = 𝔭̃_{−α}, 𝔤̃_{α} + 𝔤̃_{−α} = 𝔨_{α} + 𝔭̃_{α} (α∈△) are valid.

Now we can return to the original orthogonal symmetric Lie algebra (𝔤, σ) of the compact type. Regarding an element v in 𝔤, ad v will denote the endomorphism defined by ad v(w) = [v, w] for w ∈ 𝔤. In this case we shall use the vectors A_{α} = iÃ_{α}, Y_{α,j} = iỸ_{α,j} and the subspaces 𝔭_{α} = i𝔭̃_{α} (α ∈ △) in the calculations. It is well-known that the Killing form of 𝔤 is negative definite. The choice of the vectors X_{α,j} implies that Y_{α,1}, … Y_{α,m_{α}} (respectively Z_{α,1}, … Z_{α,m_{α}}) form an orthonormal basis in 𝔭_{α} (respectively in 𝔨_{α}) with respect to −B. Obviously, the elements of △ obtain purely imaginary values on 𝔭_{0}. Considering the above vectors, it can easily be seen that the equalities

[H, Z_{α,j}] = −α(iH)Y_{α,j},  [H, Y_{α,j}] = α(iH)Z_{α,j}  (4)

hold for any H ∈𝔭_{0}.

Endow the dual space 𝔭̃_{0}^{∗} with an arbitrary lexicographic ordering and denote by △^{+} the set of positive roots with respect to it. Using the equalities (2) and (3) and making computations with the basis vectors Z_{α,j}, Y_{α,j} (α ∈ △^{+}), the following statement can be proved (see [10; pp. 335-336]).

Lemma 1. The equalities

𝔨 = 𝔨_{0} + ∑_{α∈△^{+}} 𝔨_{α},  𝔭 = 𝔭_{0} + ∑_{α∈△^{+}} 𝔭_{α}  (5)

present orthogonal decompositions with respect to B. Furthermore, the relations

[𝔨_{α}, 𝔨_{β}] ⊂ 𝔨_{α+β} + 𝔨_{α−β}, [𝔨_{α}, 𝔭_{β}] ⊂ 𝔭_{α+β} + 𝔭_{α−β}, [𝔭_{α}, 𝔭_{β}] ⊂ 𝔨_{α+β} + 𝔨_{α−β}  (6)

hold for any α, β ∈△ ∪ {0}.

Hereafter we assume that α is an element of △^{+}. Now let us take the following simple lemma which will also be used later.

Lemma 2. The equality [Z_{α,j}, Y_{α,j}] = A_{α} holds. Furthermore, [Z_{α,j}, Y_{α,l}] ∈ 𝔭_{2α} and [Z_{α,j}, Y_{α,l}] + [Z_{α,l}, Y_{α,j}] = 0 are valid provided that j ≠ l.

Proof. By definition the equality

[Z_{α,j}, Y_{α,j}] = i([θ(X_{α,j}), X_{α,j}] − θ [θ(X_{α,j}), X_{α,j}])

holds, therefore the relations (3) imply that [Z_{α,j}, Y_{α,j}] is an element of 𝔭_{0}. Considering the equalities (4), we obtain that

B(H, [Z_{α,j}, Y_{α,j}]) = B([H, Z_{α,j}], Y_{α,j}) = α(iH) = B(H, A_{α})

is valid for any H ∈ 𝔭_{0}. It follows from this the validity of [Z_{α,j}, Y_{α,j}] = A_{α}. The second assertion can be proved in analogous way.

Let Z_{G} denote the centre of the Lie group G. It is known that an element g of G is contained in Z_{G} if and only if Ad_{g} = id holds where id denotes the identity map of 𝔤. The central elements in the subgroup exp(𝔭_{0}) can be found by the following statement.

Lemma 3. Let H be an element of 𝔭_{0}. Then Ad_{exp(H)} = id is valid if and only if α(H)∈2Zπi for any α∈△.

Proof. Clearly, the map Ad_{exp(H)} = e^{ad H} (H ∈𝔭_{0}) leaves the subspace 𝔨_{0} + 𝔭_{0} pointwise fixed. Hence, consider the proof of Lemma 6.5 of Chapter VII in [10] and use the pairs of the vectors Z_{α,j}, Y_{α,j} (α∈△^{+}) with the equalities (4).

By Lemma 3 the central elements in exp(𝔭_{0}) define a lattice in 𝔭_{0}. In the following chapter we shall see that all the totally geodesic orbits of K can be found by using this lattice.

Let us take now the Riemannian metric ⟨ , ⟩ on the tangent space T_{o}N = 𝔭. Clearly, △ uniquely decomposes as the union of irreducible root systems △^{i} (i = 1, … s). By these component root systems we obtain a decomposition 𝔭_{0} = ∑_{i=1}^{s}𝔭_{0}^{i} where the subspace 𝔭_{0}^{i} is generated by the vectors A_{α} (α∈△^{i}). Considering the subspaces 𝔭^{i} = 𝔭_{0}^{i} + ∑_{α∈△^{i}}𝔭_{α}, it is clear that

𝔭 = ∑_{i=1}^{s}𝔭^{i}

presents an orthogonal decomposition with respect to B and ⟨ , ⟩ (see Chapter VIII, Proposition 5.2 in [10]). Since the action of Ad_{K} on 𝔭^{i} is irreducible, the equality

⟨ , ⟩|_{𝔭^{i}×𝔭^{i}} = −c_{i}^{2} B|_{𝔭^{i}×𝔭^{i}}

holds for the restricted forms with a positive number c_{i}. Further on we use the notation c_{α} (α∈△^{+}) where c_{α} = c_{i} if α∈△^{i}.

By the orthonormal bases Y_{α,j} and Z_{α,j} (j = 1, …, m_{α}) with respect to −B we can define the isomorphism μ : 𝔭_{α} → 𝔨_{α} where μ(Y_{α,j}) = Z_{α,j}. Obviously, the equality μ(i(X_{α} − θ(X_{α}))) = X_{α} + θ(X_{α}) is valid for any X_{α} ∈𝔤̃_{α}.

In the following we need the unit vectors a_{α} = \frac{1}{c_{α} |α|}A_{α} (α ∈ △^{+}). If y_{α} is a unit vector in 𝔭_{α} with respect to ⟨ , ⟩, then it can be shown by Lemma 2 that the equalities

[Z_{α}, a_{α}] = −|α| y_{α},  [Z_{α}, y_{α}] = |α| a_{α}  (7)

are valid with Z_{α} = c_{α} · μ(y_{α}).

It is known that if lα is also a restricted root for an element α in △ with a number l, then l may be equal only to ±\frac{1}{2}, ±1, ±2. A root α ∈ △ is said to be unmultipliable if 2α∉△, that is, if m_{2α} = 0 holds.

Lemma 4. Let K^{α} be the connected Lie subgroup in K the Lie algebra of which coincides with 𝔨^{α} = 𝔨_{0} +𝔨_{α} +𝔨_{2α}. Then the orbit Ad_{K^{α}}(a_{α}) = ∪_{k∈K^{α}}Ad_{k}(a_{α}) coincides with the unit sphere around 0 in the subspace 𝔮_{α} = RA_{α} + 𝔭_{α} + 𝔭_{2α}.

Proof. First remark that by Lemma 1 𝔨^{α} is a subalgebra in k and 𝔮_{α} is an invariant subspace of the subgroup Ad_{K^{α}} in Ad_{G}. The elements of Ad_{K^{α}} are automorphisms of 𝔤, therefore Ad_{K^{α}}(a_{α}) is contained in the unit sphere around 0 in 𝔮_{α}. Considering the equalities (7), it can be shown that

Ad_{exp(tZ_{α})}(a_{α}) = e^{ad tZ_{α}}(a_{α}) = cos(|α|t) a_{α} − sin(|α|t) y_{α}  (8)

is valid for any unit vector y_{α} ∈𝔭_{α} with t∈R and Z_{α} = c_{α} · μ(y_{α}).

Let us take an arbitrary unit vector y = e_{0}a_{α} + e_{1}y_{α} + e_{2}y_{2α} in 𝔮_{α} (e_{0}, e_{1}, e_{2} ∈R). It is not difficult to show that y is included in Ad_{K^{α}}(a_{α}). Obviously, there are numbers t_{1}, t_{2} such that cos(2 |α|t_{2}) · cos(|α|t_{1}) = e_{0}, − sin(|α|t_{1}) = e_{1} and − sin(2 |α|t_{2}) · cos(|α|t_{1}) = e_{2} hold. The relation [𝔨_{2α}, 𝔭_{α}] ⊂ 𝔭_{α} implies that 𝔭_{α} is an invariant subspace with respect to Ad_{K^{2α}}. Let us regard the vectors Z_{2α} = c_{α} ·μ(y_{2α}), ŷ_{α} = Ad_{exp(−t_{2}Z_{2α})}(y_{α}) and Ẑ_{α} = c_{α} · μ(ŷ_{α}). Using the equality (8), we obtain that Ad_{k_{2}k_{1}}(a_{α}) = y holds where k_{1} = exp(t_{1}Ẑ_{α}) and k_{2} = exp(t_{2}Z_{2α}).

Regarding a subspace 𝔞 in 𝔭, Exp(𝔞) is a totally geodesic submanifold in N if and only if 𝔞 is a Lie triple system in 𝔭. It is known that N contains totally geodesic submanifolds of constant curvature (see Chapter VII, Theorem 11.1 in [10]).

Proposition 1. The subspace 𝔮_{α} = RA_{α} + 𝔭_{α} + 𝔭_{2α} is a Lie triple system in 𝔭. The corresponding totally geodesic submanifold Q_{α} = Exp(𝔮_{α}) is a compact symmetric space of rank one. Furthermore, if 2α ∉ △ and G is simply connected, then Q_{α} coincides with the sphere of constant curvature κ = \begin{matrix}|α| & ^{2} \\ c_{α} & ^{2} \end{matrix} = α(ia_{α})^{2}.

Proof. Using Lemma 2 and the relations (6), it can be seen that 𝔮_{α} is a Lie triple system in 𝔭. Since Q_{α} is totally geodesic in N, Q_{α} is also a compact symmetric space. Hence, the sectional curvatures in Q_{α} can be expressed by using the equality (1). Considering (4), we obtain that the curvature endomorphism R_{a_{α}} = −(ad a_{α})^{2} on 𝔮_{α} has the eigenvalues 0, \frac{|α|^{2}}{c_{α}^{2}} , 4\frac{|α|^{2}}{c_{α}^{2}} with the eigenspaces RA_{α}, 𝔭_{α}, 𝔭_{2α}. Regarding a unit vector y in 𝔮_{α}, by Lemma 4 there exists an element k ∈ K^{α} such that Ad_{k}(𝔮_{α})⊂𝔮_{α} and Ad_{k}(a_{α}) = y hold for the automorphism Ad_{k} on 𝔤. This fact implies that the restriction of the endomorphism R_{y} to 𝔮_{α} has the same eigenvalues with the same multiplicities. Therefore all the flat totally geodesic submanifolds in Q_{α} are one-dimensional.

Suppose that 2α∉△ is valid. Then Q_{α} is a space of constant curvature κ = \frac{|α|^{2}}{c_{α}^{2}} (see Remark 1). This means that Q_{α} is a sphere or a real projective space. It is known that the geodesic γ : R → Q_{α} defined by γ(t) = Exp(tA_{α}) = exp(tA_{α})(o) is periodic. If in addition the Lie group G is simply connected, then exp(tA_{α})∈K holds if and only if t∈ \frac{2π}{|α|^{2}} Z where Z denotes the set of integers (see Chapter VII, Theorem 8.5 in [10]). Therefore the length of the closed geodesic γ is equal to 2π \frac{c_{α}}{|α|}. This immediately implies that Q_{α} is a sphere, which completes the proof.

Regarding an element α of △^{+}, let us take the sphere S^{t} = { w ∈ 𝔮_{α} | |w| = t } of radius t in 𝔮_{α} (t > 0). It is important for us to describe the shape operators of the geodesic sphere Exp(S^{t}) around o in Q_{α}. In the proof of Proposition 1 we have seen that if y is a unit vector in 𝔮_{α}, then the eigenvalues of R_{y} restricted to 𝔮_{α} are 0, κ, 4κ where κ = α(ia_{α})^{2}. Denote by V_{1} and V_{2} the eigenspaces corresponding to κ_{1} = κ and κ_{2} = 4κ, respectively. Remark that the multiplicity of κ_{2} = 4κ may be equal to 0, 1, 3, 7. Clearly, this ”eigenvalue” should be omitted if 2α∉△, that is, if V_{2} = 0 holds.

For brevity, denote by p the point Exp(ty). Regarding the geodesic γ : R → Q_{α} defined by γ(τ) = Exp(τy), let V̂_{1}, V̂_{2} be the subspaces in T_{p}Q_{α} obtained by parallel displacements of V_{1}, V_{2} along γ. Let us denote by γ̇(t) the tangent vector of γ at t and by Â_{γ̇(t)} the shape operator of Exp(S^{t}) in Q_{α} with respect to γ̇(t). It can easily be seen that the following statement is valid for each symmetric space of the compact type with rank one (for details see [20]).

Lemma 5. If t ∉ \frac{1}{2\sqrt{κ}}Zπ holds, then Exp(S^{t}) is a hypersurface in Q_{α} and its shape

operator Â_{γ̇(t)} has the eigenvalues

λ_{1}(t) = −\sqrt{κ} cot(\sqrt{κ} t), λ_{2}(t) = −2\sqrt{κ} cot(2\sqrt{κ} t)

with the corresponding eigenspaces V̂_{1} and V̂_{2}. Furthermore, if t ∈ \frac{1}{2\sqrt{κ}}Zπ is valid, then Exp(S^{t}) is a totally geodesic submanifold in Q_{α}.

3. Orbits of isotropy subgroups in symmetric spaces of the compact type

In this chapter N always denotes a symmetric space of the compact type. Our purpose is to present the shape operators of the orbits of the isotropy subgroup K at a fixed point o. It is known that the compact Lie group K is not necessarily connected, therefore K may have disconnected orbits. However, it is trivial that the connected components of each disconnected orbit of K are isometric to one another. All the statements of this chapter remain valid if instead of the orbits of K we consider the orbits of its identity component. On the other hand, we have to remark that K is connected if N coincides with a compact semisimple Lie group or if G is simply connected (see Chapter VII, Theorem 8.2 in [10]).

Henceforth, for simplicity we assume that the orbits of K are connected. In the following let ρ : K × N → N denote the canonical isometric action of K on N. Consider an arbitrary vector w in 𝔭 and the point p = Exp(w). Regarding the trivial equality (k exp(w) k^{−1})K = exp(Ad_{k}(w)) K and the identification π, we obtain that

k(p) = Exp(Ad_{k}(w))  (9)

holds for any k ∈K.

Concerning orbits of differentiable actions, basic notions and facts we shall use in the following can be found in the book [13]. Let us take take the flat totally geodesic submanifold P_{0} = Exp(𝔭_{0}) in N. It can be proved that P_{0} intersects each orbit of K and T_{p}P_{0} ⊂ ν_{p}K(p) is valid for any point p of P_{0} (see Lemma 6.3 of Chapter V in [10]). By definition this means that P_{0} is a section (orthogonally transversal submanifold) of the action ρ. Hence, it can be seen that the principal orbits of ρ are isoparametric submanifolds in N, that is, their normal bundles have trivial holonomy group and they have constant principal curvatures with respect to parallel normal vector fields (see [16]).

The singular orbits of ρ can be found by regarding the diagram of N in 𝔭_{0} (see Chapter VII, Proposition 3.2 in [10]). The statement presented below gives the isotropy subgroups of K at the points of P_{0}.

Proposition 2. Let w be an element of 𝔭_{0} and let △_{w}^{Z} denote the set of those roots α in △^{+} which possess the property α(w) ∈ Zπi. Let K_{p} denote the isotropy subgroup of K at the point p = Exp(w). Then the Lie algebra of K_{p} coincides with

𝔨_{p} = 𝔨_{0} + ∑_{α∈△_{w}^{Z}} 𝔨_{α}.  (10)

Proof. Let us take the identification π : G|K → N where π(exp(w) K) = p. Since π is an equivariant map with respect to the action of G, it can be seen that K_{p} can be expressed in the form

K_{p} = { k ∈K | exp(−w) · k · exp(w)∈K }.

Since we have the equalities

\begin{aligned}exp(−w) exp(tC) exp(w) & = exp(Ad_{exp(−w)}(tC)) \\ & = exp(e^{−ad w}(tC)) \end{aligned}

for any C ∈𝔨 and t∈R, the Lie algebra of K_{p} coincides with

𝔨_{p} = { C ∈𝔨 | e^{−ad w}(C)∈𝔨 }.

Using the equalities (4), it can be shown that

e^{−ad w}(Z_{α}) = cos(α(iw)) Z_{α} + sin(α(iw)) Y_{α}  (11)

is true where Z_{α} ∈𝔨_{α} and Y_{α} = μ^{−1}(Z_{α}) (α∈△^{+}).

Considering a vector C in 𝔨, by Lemma 1 it can uniquely be expressed in the form

C = H̄ + ∑_{α∈△^{+}} Z_{α}

where H̄ ∈ 𝔨_{0} and Z_{α} ∈ 𝔨_{α}. Using the equality (11) and e^{−ad w}(H̄) = H̄, it can be seen that the vector e^{−ad w}(C) is included in 𝔨 if and only if either Z_{α} = 0 or α(iw)∈Zπ holds for any α∈△^{+}. Therefore we obtain that the Lie algebra 𝔨_{p} has the expression (10). The proof is completed.

Considering a point p in P_{0}, take a vector w in 𝔭_{0} such that Exp(w) = p holds. Denote by u the unit vector indicating the direction of w, that is, w = d · u is valid with d = |w|. Let △_{w}^{+} denote the set of those roots in △^{+} which are not contained by △_{w}^{Z}. Let us consider the subspace 𝔪 = ∑_{α∈△_{w}^{+}} 𝔭_{α} and the normal geodesic γ : [0, d] → N defined by γ(t) = Exp(tu), which joins the points o, p.

Proposition 3. The tangent space of the orbit M = K(p) at p coincides with the parallel displacement of 𝔪 along the geodesic γ.

Proof. Let 𝔭^{w} denote the tangent space of 𝔭 at w. Clearly, 𝔭^{w} can canonically be identified with 𝔭. Let 𝔭_{α}^{w}, y_{α}^{w} denote the subspace and the vector in 𝔭^{w} which correspond to 𝔭_{α}, y_{α} (α∈△^{+}), respectively.

Obviously, w can be expressed in the form w = ⟨a_{α}, w⟩a_{α} +b where b∈𝔭_{0} and α(b) = 0 hold. Considering the equality (8), for any unit vector y_{α} we have

Ad_{exp(tZ_{α})}(w) = b + ⟨a_{α}, w⟩(cos(|α|t) a_{α} − sin(|α|t) y_{α})  (12)

where Z_{α} = c_{α}·μ(y_{α}). The equality (12) implies that the tangent space of the orbit Ad_{K}(w) at w coincides with ∑_{α∈△_{w}} 𝔭_{α}^{w} where △_{w} = { α∈△^{+} | α(w) ≠ 0 }.

Regarding (9), the tangent linear map T_{w}Exp maps T_{w}Ad_{K}(w) onto T_{p}K(p). Let X̂_{α} denote the parallel vector field along γ defined by X̂_{α}(γ(0)) = y_{α} (α ∈ △_{w}). Let us consider the geodesic variation V (t, τ) = Exp(t(u + τy_{α})) (t ∈ [0, d], τ ∈ (−ε, ε)) of γ. It is well-known that the transversal vector field

Ŷ_{α}(γ(t)) = t · T_{tu}Exp(y_{α}^{tu})  (13)

of V along γ is a Jacobi vector field such that the initial conditions Ŷ_{α}(γ(0)) = 0 and ˜∇_{u}Ŷ_{α} = y_{α} hold (see e.g. [7; pp. 132] and [14; pp. 64]). By the equalities (1) and (4) R_{u}(y_{α}) = α(iu)^{2}y_{α} is valid. Therefore using the well-known presentation of Jacobi vector fields along geodesics (see [7; pp. 131]), it can be seen that

Ŷ_{α}(γ(t)) = \frac{1}{|α(iu)|} sin(|α(iu)|t) X̂_{α}(γ(t))  (14)

holds. The equalities (13) and (14) immediately imply

T_{w}Exp(y_{α}^{w}) = \frac{1}{|α(iw)|} sin(|α(iw)|) X̂_{α}(p).  (15)

This means that the vector X̂_{α}(p) (α∈△_{w}) is contained in T_{p}M if and only if α(iw)∉Zπ, which completes the proof.

Remark 2. By the relations (6) the subspace 𝔮̂^{α} = 𝔭_{0} + 𝔭_{α} + 𝔭_{2α} is a Lie triple system in 𝔭 for any α∈△^{+}. Take the totally geodesic submanifold Q̂^{α} = Exp(𝔮̂^{α}) in N. Considering the eigenvalues and the eigenspaces of the curvature endomorphisms of Q̂^{α} at o, by Remark 1 we obtain that Q̂^{α} is locally isometric with the product symmetric space R^{n−1} × Q_{α}, where n denotes the rank of N.

Proposition 4. Let b be an element of 𝔭_{0} such that ⟨a_{α}, b⟩ = 0 is valid for a given α∈△^{+}. Then Q_{α}^{b} = Exp(b + 𝔮_{α}) is a totally geodesic submanifold in N. Furthermore, the map φ : Q_{α} → Q_{α}^{b} defined by the assignment φ(Exp(y)) = Exp(b + y) (y ∈ 𝔮_{α}) is an isometry.

Proof. Let us take a vector y in 𝔮_{α} = RA_{α} + 𝔭_{α} + 𝔭_{2α}. The equality [y, b] = 0 implies that Rb + Ry is an Abelian subalgebra in 𝔤. It follows from this that exp(b + y) = exp(b)exp(y) is valid. Therefore the equality

exp(−b − y_{2})exp(b + y_{1}) = exp(−y_{2})exp(y_{1})

holds for any y_{1}, y_{2} ∈𝔮_{α}. Regarding the identification π, it is clear that Exp(v_{1}) = Exp(v_{2}) holds for two vectors v_{1}, v_{2} ∈𝔭 if and only if the condition exp(−v_{2})exp(v_{1})∈K is satisfied. The above assertions show that the map φ is well-defined.

Take a real number t such that Exp(ta_{α}) is not a conjugate point of o in Q_{α}. Notice that this condition is satisfied if t ∉ \frac{c_{α}}{2 |α|}Zπ. Assume that w coincides with one of the

vectors ta_{α}, ta_{α} + b. Preserving the notation of the proof of Proposition 3, let X̂ be the parallel vector field along γ defined by X̂(γ(0)) = a_{α}. It is trivial that T_{w}Exp(a_{α}^{w}) = X̂(γ(d)) holds. Considering the equality (15), the latter one and the definition of φ, we obtain that the tangent linear map T_{Exp(ta_{α})}φ is isometric. Lemma 4 and the equality (9) imply that K^{α}(Exp(Ra_{α})) = Q_{α} is valid. Since the maps k ∘ φ and φ ∘ k on Q_{α} coincide for any k ∈K^{α}, it can be seen that T_{q}φ is also isometric for any q ∈Q_{α}.

Finally, it is obvious that the reflection of N on the point Exp(\frac{b}{2}) carries Q_{α} into Q_{α}^{b}. Hence, Q_{α}^{b} is also a totally geodesic submanifold in N.

In order to get the principal curvatures of the orbits we can use the following statement. The straightforward proof is omitted.

Lemma 6. Let M be a submanifold in a Riemannian manifold N and T be a totally geodesic submanifold intersecting M in a connected submanifold L (L = M ∩T). Regarding L as a submanifold in T, let ν^{T} (L) denote the normal bundle of L in T. Let ξ be a smooth section of the vector bundle ν^{T} (L) and let A_{ξ}^{L} denote the corresponding shape operator of L in T.

Assume that ν_{p}^{T} L ⊂ ν_{p}M holds for any point p∈L. Then T_{p}L is an invariant subspace of A_{ξ(p)} for any p∈L, furthermore, the restriction of A_{ξ(p)} to T_{p}L coincides with A_{ξ(p)}^{L}.

Since exp(𝔭_{0}) is an Abelian subgroup in G, it can easily be seen that P_{0} is a flat torus.

Lemma 7. Let p be an arbitrary point in P_{0}. Let γ_{1}, γ_{2} : [0, 1] → P_{0} be two piecewise smooth curves which join the points o, p and consist of geodesic segments. Let E_{α}^{1}(p), E_{α}^{2}(p) (α ∈ △^{+}) denote the subspaces in T_{p}N obtained by parallel displacement of 𝔭_{α} along γ_{1}, γ_{2}, respectively. Then E_{α}^{1}(p) = E_{α}^{2}(p) holds.

Proof. Since P_{0} is a flat torus, each vector v in 𝔭_{0} uniquely determines a parallel vector v(p) in T_{p}P_{0}. Let us fix a vector v in 𝔭_{0} such that the numbers β(iv)^{2} (β ∈ △^{+}∪{0}) are different. By the equalities (1) and (4) 𝔭_{α} is an eigenspace of the curvature endomorphism R_{v} with the eigenvalue α(iv)^{2}. It is known that the parallel displacements along geodesics preserve the equalities concerning the Riemannian curvature tensor R. This means that E_{α}^{1}(p), E_{α}^{2}(p) (α ∈ △^{+}) are also eigenspaces of R_{v(p)} with the same eigenvalue α(iv)^{2}, which completes the proof.

Let w be a vector in 𝔭_{0} and let us consider the orbit M = K(p) where Exp(w) = p. Since the flat torus P_{0} is a section of ρ, its tangent space T_{p}P_{0} is included in the normal subspace ν_{p}M. Applying the earlier statements, the shape operators of M at p can explicitly be described as follows.

Let γ : [0, 1] → N be the geodesic defined by γ(t) = Exp(tw). By Proposition 3 the tangent space T_{p}M coincides with the parallel displacement of the subspace 𝔪 = ∑_{α∈△_{w}^{+}} 𝔭_{α} along γ where △_{w}^{+} = { α ∈ △^{+} | α(w) ∉ Zπi }. Denote by E_{α}(p) that subspace in T_{p}M which is obtained by the parallel displacement of 𝔭_{α} (α∈△_{w}^{+}) along γ.

Theorem 1. Let v be an element of 𝔭_{0} and let v(p) denote the corresponding parallel vector in T_{p}P_{0}. Then the eigenspaces of the shape operator A_{v(p)} of M = K(p) are E_{α}(p) (α∈△_{w}^{+}) with the corresponding eigenvalues

λ_{α} = −α(iv) · cot(α(iw)).  (16)

Proof. Let α be an arbitrary element of △_{w}^{+}. Let us consider the orthogonal decomposition w = ⟨a_{α}, w⟩a_{α} + b where α(b) = 0. Using Lemma 4 and the equality (12), we obtain that the orbit Ad_{K^{α}}(w) coincides with the set b + S_{α} where S_{α} denotes the sphere of radius r = \frac{c_{α}}{|α|}|α(iw)| around 0 in 𝔮_{α}, that is, S_{α} = \{ y ∈ 𝔮_{α} | |y| = \frac{c_{α}}{|α|}|α(iw)| \}. Regarding Proposition 4, it is clear that Exp(Ad_{K^{α}}(w)) coincides with the geodesic sphere of radius r around q = Exp(b) in the totally geodesic submanifold Q_{α}^{b}.

Take the totally geodesic submanifold Q̂^{α} in N (see Remark 2). It is an important fact for us that by the equality (9) the relation K^{α}(p) = Exp(Ad_{K^{α}}(w)) holds. Hence, K^{α}(p) is included in M ∩ Q̂^{α}.

First assume that 2α(w) ∉ Zπi or 2α ∉ △ holds. In this case, since the orthogonal decomposition

T_{p}Q̂^{α} = T_{p}P_{0} + E_{α}(p) + E_{2α}(p)

is valid, locally Q̂^{α} intersects M =K(p) in an (m_{α} + m_{2α})–dimensional submanifold. This implies that K^{α}(p) coincides with that component of M ∩ Q̂^{α} which contains p. Using Propositions 1 and 4, we obtain that Q_{α}^{b} is a totally geodesic submanifold of rank one and has the minimal sectional curvature κ_{1} = \frac{|α|^{2}}{c_{α}^{2}} . If 2α ∈ △ is valid, then Q_{α}^{b} is not of

constant curvature and its maximal sectional curvature is equal to κ_{2} = 4\frac{|α|^{2}}{c_{α}^{2}} .

It can be seen that in order to get the principal curvatures of M at p we can apply Lemma 6 with T = Q̂^{α} and L = K^{α}(p). Obviously, the points q and p are joined by a periodic geodesic γ̂_{2} in Q_{α}^{b} which is defined by γ̂_{2}(t) = Exp(b + ta_{α}). By Proposition 4 T_{o}φ(𝔭_{α}) and T_{o}φ(𝔭_{2α}) are eigenspaces of the curvature endomorphism R_{a_{α}(q)} with the eigenvalues κ_{1}, κ_{2}. Since K^{α}(p) is just the geodesic sphere of radius r around q in Q_{α}^{b}, Lemma 5 implies that the parallel displacements of these subspaces along γ̂_{2} to p are eigenspaces of the shape operator A_{a_{α}(p)} with the corresponding eigenvalues

λ_{1} = − sgn(⟨a_{α}, w⟩) \sqrt{κ_{1}} cot(\sqrt{κ_{1}} · r) = \frac{|α|}{c_{α}} cot(α(iw)), λ_{2} = 2 \frac{|α|}{c_{α}} cot(2 · α(iw)).

However, by Lemma 7 these eigenspaces are just E_{α}(p) and E_{2α}(p), respectively, since the tangent linear map T_{o}φ coincides with the parallel displacement of 𝔮_{α} along the geodesic γ̂_{1} : [0, 1] → N where γ̂_{1}(t) = Exp(tb). Concerning the principal curvatures λ_{1}, λ_{2}, remark that the equality α(ia_{α}) = −\frac{|α|}{c_{α}} is valid.

Clearly, if u ∈ 𝔭_{0} is normal to a_{α}, then A_{u(p)} vanishes on E_{α}(p) + E_{2α}(p), since Q_{α}^{b} is totally geodesic in N. The above assertions imply that E_{α}(p) is an eigenspace of A_{v(p)} with the eigenvalue (16) for any v ∈𝔭_{0}.

Finally, let us consider that special case when α(w) ∈ \frac{1}{2}(2Z + 1)πi and 2α ∈ △ are valid. Then using Lemma 5 and the analogous procedure, it can be seen that E_{α}(p) is an eigenspace of A_{v(p)} with the eigenvalue 0. Therefore the proof is completed.

Regarding the orbits of K as submanifolds, we can use the fact that K acts transitively on them. If the orbit K(p) is maximal dimensional, then T_{p}P_{0} coincides with ν_{p}K(p). Hence, Theorem 1 implies the following statement.

Corollary 1. If K(p) is a principal orbit, then its shape operators commute.

Using Theorem 1, the totally geodesic orbits of K can be selected. The following theorem can be obtained from the criterion given by R. Hermann in [11], too (see Remark 3).

Theorem 2. The submanifold M = K(p) is totally geodesic in N if and only if α(w) ∈ \frac{1}{2}Zπi holds for each α∈△^{+}.

Proof. By Theorem 1 it is evident that the given condition is necessary. Conversely, assume that α(w) ∈ \frac{1}{2}Zπi holds for each α ∈ △^{+}. Let Exp^{M} denote the restriction of Exp to the normal bundle ν(M). By the tangent linear maps of the isometries K acts on ν(M), too. It is clear that Exp^{M} ∘T k = k∘Exp^{M} holds for any k ∈K. Since P_{0} intersects all the orbits of ρ, ∪_{k∈K}k(P_{0}) = N is valid. The restriction of Exp^{M} to a suitable tubular neighborhood of M in ν(M) is a diffeomorphism. The above facts imply that ∪_{k∈K_{p}}T k(T_{p}P_{0}) = ν_{p}M holds where K_{p} denotes the isotropy subgroup of K at p. Hence, if v(p) is an arbitrary vector in ν_{p}M, then there exists an isometry k in K_{p} such that T k(v(p)) ∈ T_{p}P_{0} is valid. Since A_{v(p)} and A_{T k(v(p))} have the same eigenvalues, by Theorem 1 we obtain that A_{v(p)} vanishes on T_{p}M, which completes the proof.

Remark 3. In the paper [11] R. Hermann has studied totally geodesic orbits of polar isometric actions on Riemannian symmetric spaces. Without using root space decompositions, he has given a criterion for an orbit to be totally geodesic. By Theorem 2.2 in [11] the orbit K(h(o)) is totally geodesic for an element h ∈ exp(𝔭_{0}) if and only if Ad_{h^{4}} = id holds. However, Lemma 3 implies that h = exp(w) satisfies the above condition if and only if α(w)∈ \frac{1}{2}Zπi is valid for any root α in △.

Regarding Theorem 2, we can state the following assertion.

Corollary 2. If N is irreducible and not of constant curvature, then all the totally geodesic orbits of K are singular.

Remark 4. Evidently, we call a point p antipodal to o if there is a closed geodesic (circle) in N which contains o and p as opposite points. It is known that if p is an antipodal point to o, then the orbit K(p) is totally geodesic and called a polar of o (see [4]). It is natural to raise the question: in which symmetric spaces are there totally geodesic orbits of K which are not polars of o.

Assume that the Lie group G is simply connected. Let us take the lattice

Λ_{1} = \{ w ∈𝔭_{0} | α(w)∈ \frac{1}{2}Zπi for any α∈△^{+} \}

presenting the totally geodesic orbits. Furthermore, let Λ_{2} denote the lattice in 𝔭_{0} generated by the vectors \frac{π}{|α|^{2}} A_{α} (α ∈ △^{+}) which is a sublattice of Λ_{1}. Using Theorem 8.5 of Chapter VII in [10], we obtain that Exp(w) is an antipodal point to o or coincides with o if and only if w ∈Λ_{2}. In this case there are totally geodesic orbits consisting of points not antipodal to o if and only if Λ_{2} ≠ Λ_{1}.

Let N be a simply connected symmetric space of type II. This means that N coincides with such a compact Lie group the Lie algebra of which is a compact real form in a simple complex Lie algebra. By the above results the isotropy subgroup has totally geodesic orbits which are not polars of o in the following ”symmetric spaces” (see [12; pp. 67-69]): SU(n), Spin(n), Sp(n), E_{6}, E_{7}.

A connected totally geodesic submanifold M in N is said to be reflective if there exists an involutive isometry in N such that M is a component of its fixed point set. Reflective submanifolds in irreducible symmetric spaces have been discussed in several papers by D. S. P. Leung.

Theorem 3. Assume that N is simply connected. If K(p) is a totally geodesic orbit, then it is a reflective submanifold in N.

Proof. Take a totally geodesic orbit K(p) (p∈P_{0}) and a vector w ∈𝔭_{0} such that Exp(w) = p holds. By Theorem 2 the numbers 2\frac{α(iw)}{π} (α∈△) are integers, therefore by their parities the set △^{+} can be divided into two disjoint subsets which are △_{w}^{+} and △_{w}^{Z} (see Propositions 2 and 3). Using the relations (6), it can be seen that the orthogonal subspaces

𝔭_{w}^{1} = ∑_{α∈△_{w}^{+}} 𝔭_{α}, 𝔭_{w}^{2} = 𝔭_{0} + ∑_{α∈△_{w}^{Z}} 𝔭_{α}

(the upper index indicates the parity) are Lie triple systems in 𝔭 which satisfy the relations

[ [𝔭_{w}^{j}, 𝔭_{w}^{l}] 𝔭_{w}^{j}] ⊂ 𝔭_{w}^{l}, [ [𝔭_{w}^{j}, 𝔭_{w}^{l}] 𝔭_{w}^{l}] ⊂ 𝔭_{w}^{j}

for j, l ∈{1, 2}. Take the corresponding totally geodesic submanifolds P^{j} = Exp(𝔭_{w}^{j}), (j = 1, 2). By Theorem 1.1 of the paper [15] P^{1} and P^{2} are reflective submanifolds in N. Denote by τ_{1}, τ_{2} the corresponding involutive isometries.

The earlier statements of this paper imply that the normal subspace ν_{p}K(p) coincides with T_{p}P^{2}. It follows from this that K(p) is a component of the fixed point set of the involutive isometry r_{p} ∘ τ_{2} where r_{p} denotes the reflection of N on p.

If we omit the assumption that N is simply connected, then there exists a local reflection on K(p), since the submanifold Exp(ν_{q}K(p)) is totally geodesic for any q ∈K(p) (see Theorems 1 and 3 in [5]). Finally, notice that the reflection of N on Exp(\frac{w}{2} ) carries P^{1} and K(p) into each other.

The so-called curvature-adapted submanifolds have been studied in the papers [1] and [2]. The following result has already been proved for principal orbits of isotropy subgroups in symmetric spaces of the noncompact type (see [21]).

Corollary 3. All the orbits of the isometric action ρ are curvature-adapted submanifolds in N.

Proof. Take an orbit K(p) (p∈P_{0}) and a normal vector v(p)∈T_{p}P_{0}. Considering the proof of Lemma 7 and Theorem 1, we obtain that A_{v(p)} and R_{v(p)} commute on T_{p}K(p). This implies that if K(p) is a principal orbit, then it is a curvature-adapted submanifold. On the other hand, if K(p) is a singular one, then we can use the procedure applied in the proof of Theorem 2.

Remark 5. It is clear that excepting Theorems 2 and 3, analogous statements can be deduced in symmetric spaces of the noncompact type, too. Since in such a symmetric space the exponential map at a point is a diffeomorphism, we can take global normal coordinate systems. Hence, orbits of isotropy subgroups can easily be characterized. Clearly, the hyperbolic function coth has to be used in the analogue of Theorem 1. In these symmetric spaces totally geodesic foliations on isoparametric orbits have been discussed in the paper [21].

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Received October 31, 1994.