Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 43 (2002), No. 2, 521-536.

On Homogeneous Hypersurfaces in Complex Grassmannians^{∗}

Alicia N. García  Eduardo G. Hulett  Cristián U. Sánchez

Fa.M.A.F. - CIEM, Universidad Nacional de Córdoba Ciudad Universitaria, 5000 Córdoba, Argentina e-mail: agarcia@mate.uncor.edu e-mail: hulett@mate.uncor.edu e-mail: csanchez@mate.uncor.edu

Abstract. In the present article we consider a class of real hypersurfaces of the Grassmann manifold of k-planes in ℂ^{n}, G_{k}(ℂ^{n}), for k > 2. Namely the family of tubes around G_{k}(ℂ^{m}) with m < n and around the quaternionic Grassmann manifold of k/2-quaternionic planes in ℍ^{n/2}, G_{k/2}(ℍ^{n/2}), when k and n are even. We determine which of those tubes are homogeneous and for them we find the spectral decomposition of the shape operator. As a consequence we show that they are Hopf hypersurfaces. MSC 2000: 53C30, 53C35, 53C42 Keywords: complex Grassmannians, real hypersurfaces, tubes, shape operator, Kaehler structure

1. Introduction

The study of real hypersurfaces in complex projective spaces has a long and interesting history. A nice survey can be found in [11]. A particular subclass is that of the homogeneous hypersurfaces which at the present time seems to be very well understood. In this respect the reader may get well acquainted with this topic by reading the article [8] and references therein.

The study of real hypersurfaces (homogeneous or not) in ℂℙ^{n} suggests immediately the problem of understanding real hypersurfaces in other compact hermitian symmetric spaces.

∗Partially supported by CONICET and SECYT-UNCba, Argentina 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2002 Heldermann Verlag

A very nice step in this direction is the paper [2] where the authors, using deep knowledge of the Grassmannians obtained in [1], study real hypersurfaces in the complex two-plane Grassmannian obtaining an important result about hypersurfaces in G_{2}(ℂ^{m+2}) (cf [2], Th.1, p.2). Those authors take particular advantage from the fact that G_{2}(ℂ^{m+2}) is the only compact, Kaehler, quaternionic Kaehler manifold with positive scalar curvature. They use then the Kaehler structure J and the quaternionic Kaehler structure J to determine all the real hypersurfaces M in G_{2}(ℂ^{m+2}) for which J(⊥M) and J (⊥M) are invariant by the shape operator of M in G_{2}(ℂ^{m+2}). If one takes complete hypersurfaces it turns out that the only solutions are tubes around a totally geodesic G_{2}(ℂ^{m+1}) and tubes around a totally geodesic ℍℙ^{n} in G_{2}(ℂ^{2n+2}) which are real homogeneous hypersurfaces.

The present paper grew out of our attempt to study the more general situation of other complex Grassmannians as ground manifolds. As soon as one takes other Grassmannians, the quaternionic Kaehler condition is lost and the road, which is not easy in [2] gets rougher.

The goal of the present article is to initiate the study of real hypersurfaces in G_{k}(ℂ^{n}) for k > 2 and we think that our results may be found interesting.

We study a certain class of real hypersurfaces of the Grassmann manifold of k-planes in ℂ^{n}, G_{k}(ℂ^{n}) for k > 2. This class consists of tubes around G_{k}(ℂ^{m}) with m < n and those around the quaternionic Grassmann manifold of k/2-quaternionic planes in ℍ^{n/2}, G_{k/2}(ℍ^{n/2}), when k and n are even.

Our objective is to determine which tubes of the considered families are homogeneous and for them to find the spectral decomposition of the shape operator. According to J. Berndt (private communication) the classification of homogeneous hypersurfaces in complex Grassmanians seems to be obtainable from the results of A. Kollross [10] concerning hyperpolar actions on irreducible simply connected symmetric spaces of compact type. Our methods however are based on the root structure associated to the Grassmann manifold G_{k}(ℂ^{m}).

In Section 2 we study the natural action on the tubes around G_{k}(ℂ^{m}) for m < n and around G_{k/2}(ℍ^{n/2}) when k and n are even and identify those for which the isotropy group of the center acts transitively on the zero centered spheres in the normal space. These are precisely the homogeneous tubes centered at G_{k}(ℂ^{n−1}) and G_{q}(ℍ^{q+1}) for n = 2q + 2 and k = 2q (Theorem 1).

Section 3 deals with the computation of the spectrum of the Jacobi operator R_{Z} = R(., Z)Z in the direction of Z (for suitable Z) where R is the curvature of the Riemannian connection on G_{k}(ℂ^{n}). This is a nontrivial task and we indicate some of the required calculations. The results are summarized in Proposition 1. Using these results and via some delicate computations, we finally obtain Theorem 2 which together Theorem 1 gives a complete classification of the homogeneous tubes of the considered families.

Section 4 contains our main result, Theorem 3, which gives the spectral decomposition of the shape operator of the family of tubes around G_{k}(ℂ^{n−1}) showing that they are Hopf hypersurfaces of G_{k}(ℂ^{n}).

2. Some homogeneous tubes in G_{k}(ℂ^{n})

The objective of this section is to study two families of homogeneous tubes in the Grassmannian manifold G_{k}(ℂ^{n}) and describe their homogeneous structure.

The complex and quaternionic Grassmann manifolds are defined by

G_{k}(ℂ^{s}) = \frac{SU(s)}{S(U(k) × U(s − k))} and G_{q}(ℍ^{p}) = \frac{Sp(p)}{Sp(q) × Sp(p − q)}

with k < s and q < p. Their real dimensions are 2k(s − k) and 4q(p − q) respectively.

For each of the above spaces we denote by o the class in the quotient of the identity element of the group.

Let us consider the natural embeddings

j_{1} : G_{k}(ℂ^{m}) ↪ G_{k}(ℂ^{n}) for m < n
j_{2} : G_{q}(ℍ^{p}) ↪ G_{2q}(ℂ^{2p}) for q < p

given by

j_{1}(π(A)) = π(Ã) with Ã = [ \begin{matrix}A & 0 \\ 0 & I_{n−m} \end{matrix} ], A ∈ SU(m)
j_{2}(π(A)) = π(J_{p,q}AJ_{p,q}^{−1}), A ∈ Sp(p)

where π denotes the corresponding projections map onto the quotient spaces,

J_{p,q} = [ \begin{matrix}I_{q} & 0 & 0 & 0 \\ 0 & 0 & I_{q} & 0 \\ 0 & I_{p−q} & 0 & 0 \\ 0 & 0 & 0 & I_{p−q} \end{matrix} ] ∈ U(2p)  (1)

and Sp(q) × Sp(p − q) is included into Sp(p) via

([ \begin{matrix}A_{1} & B_{1} \\ C_{1} & D_{1} \end{matrix} ] , [ \begin{matrix}A_{2} & B_{2} \\ C_{2} & D_{2} \end{matrix} ]) ↪^{j} [ \begin{matrix}A_{1} & 0 & B_{1} & 0 \\ 0 & A_{2} & 0 & B_{2} \\ C_{1} & 0 & D_{1} & 0 \\ 0 & C_{2} & 0 & D_{2} \end{matrix} ] .  (2)

On the ambient space G_{k}(ℂ^{n}) we consider the SU(n)-invariant metric induced by the opposite of the Killing form of 𝔰𝔲(n). This metric determines invariant metrics on G_{k}(ℂ^{m}) and G_{q}(ℍ^{p}) making the above defined j_{1} and j_{2}, isometric imbeddings.

It is well known that the tubes centered at G_{k}(ℂ^{m}) and G_{q}(ℍ^{p}) are globally defined for sufficiently small radii (see for instance [7]).

Note that SU(n) acts transitively on G_{k}(ℂ^{n}) by isometries and for m < n the action of SU(m) on G_{k}(ℂ^{m}) is the restriction of the former one. Moreover SU(m) acts on each one of the tubes around G_{k}(ℂ^{m}) by

g.(exp_{p} rX) = exp_{g.p} rg_{∗p}X  (3)

where p ∈ G_{k}(ℂ^{m}) and X ∈ (T_{p}G_{k}(ℂ^{m}))^{⊥} with ∥X∥ = 1.

This observation is also true when we replace SU(m) by Sp(p) and G_{k}(ℂ^{m}) by G_{q}(ℍ^{p}), for n = 2p and k = 2q.

Let us now study the natural action of S(U(k)×U(m−k)) (the isotropy group of SU(m) at the point o ∈ G_{k}(ℂ^{m})) on the normal space (T_{o}G_{k}(ℂ^{m}))^{⊥}.

Since S(U(k) × U(m − k)) ⊂ S(U(k) × U(n − k)) via

g = [ \begin{matrix}A & 0 \\ 0 & B \end{matrix} ] ↪ g = [ \begin{matrix}A & 0 & 0 \\ 0 & B & 0 \\ 0 & 0 & I_{n−m} \end{matrix} ]

the elements of T_{o}G_{k}(ℂ^{m}) are included in T_{o}G_{k}(ℂ^{n}) in the form

Y = [ \begin{matrix}0 & Y_{1} & 0 \\ −Y_{1}^{∗} & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ] , Y_{1} ∈ M_{k×(m−k)}(ℂ)  (4)

and the elements of (T_{o}G_{k}(ℂ^{m}))^{⊥} are of the form

Z = [ \begin{matrix}0 & 0 & Z_{1} \\ 0 & 0 & 0 \\ −Z_{1}^{∗} & 0 & 0 \end{matrix} ] , Z_{1} ∈ M_{k×(n−m)}(ℂ).  (5)

Keeping this notation, we see that if g ∈ S(U(k) × U(m − k)) and Z ∈ (T_{o}G_{k}(ℂ^{m}))^{⊥}, then

g_{∗o}Z = gZg^{−1} = [ \begin{matrix}0 & 0 & AZ_{1} \\ 0 & 0 & 0 \\ −(AZ_{1})^{∗} & 0 & 0 \end{matrix} ] .  (6)

Hence the action of S(U(k) × U(m − k)) on (T_{o}G_{k}(ℂ^{m}))^{⊥} is nothing but the action of the group U(k) on M_{k×(n−m)}(ℂ) by matrix multiplication. Then we obtain the following Lemma.

Lemma 1. The natural action of S(U(k) × U(m − k)) on the unit sphere of (T_{o}G_{k}(ℂ^{m}))^{⊥} is transitive if and only if m = n − 1. □

In order to study the natural action of Sp(q) × Sp(p − q) (the isotropy group of Sp(p) at the point o ∈ G_{q}(ℍ^{p})) on the normal space (T_{o}G_{q}(ℍ^{p}))^{⊥} ⊂ T_{o}G_{2q}(ℂ^{2p}) we split each matrix A in Sp(p) into blocks as follows:

J_{p,q}AJ_{p,q}^{−1} = J_{p,q} [ \begin{matrix}A_{11} & A_{12} & A_{13} & A_{14} \\ A_{21} & A_{22} & A_{23} & A_{24} \\ A_{31} & A_{32} & A_{33} & A_{34} \\ A_{41} & A_{42} & A_{43} & A_{44} \end{matrix} ] J_{p,q}^{−1} =

= [ \begin{matrix}A_{11} & A_{13} & A_{12} & A_{14} \\ A_{31} & A_{33} & A_{32} & A_{34} \\ A_{21} & A_{23} & A_{22} & A_{24} \\ A_{41} & A_{43} & A_{42} & A_{44} \end{matrix} ] .

(A_{11}, A_{33} are matrices of order q × q and A_{22}, A_{44} are of order (p − q) × (p − q)).

This allows us to include the subgroup Sp(q)×Sp(p − q) of Sp(p) into S(U(2q)×U(2(p − q)))

via (g_{1}, g_{2}) ↪ [ \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} ] and at the same time to view the elements of T_{o}G_{q}(ℍ^{p}) and

(T_{o}G_{q}(ℍ^{p})^{⊥} included into T_{o}G_{2q}(ℂ^{2p}) respectively as

[ \begin{matrix}0 & 0 & X & Y \\ 0 & 0 & −\overline{Y} & \overline{X} \\ −X^{∗} & Y^{t} & 0 & 0 \\ −Y^{∗} & −X^{t} & 0 & 0 \end{matrix} ] and [ \begin{matrix}0 & 0 & X & Y \\ 0 & 0 & \overline{Y} & −\overline{X} \\ −X^{∗} & −Y^{t} & 0 & 0 \\ −Y^{∗} & X^{t} & 0 & 0 \end{matrix} ]  (7)

where X, Y ∈ M_{q×(p−q)}(ℂ).

For g = [ \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} ] ∈ Sp(q) × Sp(p − q) and Z = [ \begin{matrix}0 & Z_{1} \\ −Z_{1}^{∗} & 0 \end{matrix} ] ∈ T_{o}G_{2q}(ℂ^{2p}) it is clear

that

g_{∗o}Z = gZg^{−1} = [ \begin{matrix}0 & g_{1}Z_{1}g_{2}^{∗} \\ −g_{2}Z_{1}^{∗}g_{1}^{∗} & 0 \end{matrix} ] .  (8)

Since the matrices g_{j} are of the form g_{j} = [ \begin{matrix}A_{j} & B_{j} \\ −\overline{B}_{j} & \overline{A}_{j} \end{matrix} ], it is easy to see that gZg^{−1} ∈

(T_{o}G_{q}(ℍ^{p}))^{⊥}, ∀Z ∈ (T_{o}G_{q}(ℍ^{p}))^{⊥} and therefore Sp(q) × Sp(p − q) acts on the unit sphere of (T_{o}G_{q}(ℍ^{p}))^{⊥}.

In ([3], p. 301, 10.94) a complete classification of Lie groups acting transitively on spheres is given. According to this information Sp(q) × Sp(p − q) (2 ≤ q < p) acts transitively on the sphere S^{l−1} only when p − q = 1, and in this case l = 4q.

Moreover, for p − q = 1 the action given in (8) is transitive on the unit sphere of (T_{o}G_{q}(ℍ^{p}))^{⊥}. We have then established the following result.

Lemma 2. The natural action of Sp(q) × Sp(p − q) on the unit sphere of (T_{o}G_{q}(ℍ^{p}))^{⊥} is transitive if and only if p = q + 1. □

For any submanifold M of G_{k}(ℂ^{n}) we denote by (M)_{r} the tube of radius r > 0 around M.

Set p_{0} = exp_{o} rZ where Z ∈ (T_{o}G_{k}(ℂ^{n−1}))^{⊥} is such that Z_{1} in the expression (5) is a suitable multiple of the first vector of the canonical basis of ℂ^{k}. From (3) and (6) we may conclude that the isotropy group of the action of SU(n − 1) on the tube (G_{k}(ℂ^{n−1}))_{r} at p_{0} is given by

S({1} × U(k − 1) × U(n − 1 − k)) ⊂ SU(n − 1).

Set now q_{0} = exp_{o} rZ where Z ∈ (T_{o}G_{q}(ℍ^{q+1}))^{⊥} corresponds in the representation (7), to X = 0 and Y a suitable multiple of the first vector of the canonical basis of ℂ^{q}. Let us denote by H the isotropy group of the natural action of Sp(q + 1) on the tube (G_{q}(ℍ^{q+1}))_{r} at q_{0}.

From (3) and (8) we conclude

g ∈ H ⟺ g = [ \begin{matrix}g_{1} & 0 \\ 0 & g_{2} \end{matrix} ] ∈ Sp(q) × Sp(1) and gZg^{−1} = Z.

Further, if g_{1} = [ \begin{matrix}A & B \\ −\overline{B} & \overline{A} \end{matrix} ] and g_{2} = [ \begin{matrix}a & b \\ −\overline{b} & \overline{a} \end{matrix} ] , performing the computation indicated in

(8) we obtain

\{ \begin{matrix}\overline{a}B_{1} + \overline{b}A_{1} = 0 \\ −bB_{1} + aA_{1} = e_{1} \end{matrix}

where e_{1} is the first vector of the canonical basis of ℂ^{q} and A_{1}, B_{1} denote the first columns of A, B respectively. This implies that A_{1} = \overline{a}e_{1} and B_{1} = −\overline{b}e_{1} and so it is easy to conclude that

H = Sp(1) × Sp(q − 1)

which is embedded into Sp(q + 1) via the map

(P, Q) ↪ [ \begin{matrix}j(uP u^{−1}, Q) & 0 \\ 0 & P \end{matrix} ] ,

where u = [ \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ] and j is the inclusion given in (2).

Lemmas 1, 2 and the preceeding discusion may be summarized in the following theorem.

Theorem 1. For small enough radii r > 0 the tubes (M)_{r} in G_{k}(ℂ^{n}) around M = G_{k}(ℂ^{n−1}) and around M = G_{q}(ℍ^{q+1}) for n = 2q + 2 and k = 2q, are homogeneous real hypersurfaces of G_{k}(ℂ^{n}). Moreover these tubes are of the form

(G_{k}(ℂ^{n−1}))_{r} = SU(n − 1)/S({1} × U(k − 1) × U(n − 1 − k))

(G_{q}(ℍ^{q+1}))_{r} = Sp(q + 1)/Sp(1) × Sp(q − 1). □

3. The spectrum of R_{Z}

If R is the curvature tensor of the Riemannian manifold N, the Jacobi operator in the Zdirection for Z ∈ T N, is defined by R_{Z} := R(., Z)Z.

The goal of the present section is to compute the spectral decomposition of the Jacobi operator R_{Z} for suitable Z ∈ T_{o}N in order to give a necessary condition for the tubes under consideration to be homogeneous.

It is well known and easy to see that for a Riemannian symmetric space N = G/K the Jacobi operator R_{Z}, for Z ∈ T_{o}N, is given by

R_{Z} = −(adZ)^{2}

where ad is the adjoint representation of the Lie algebra of G.

Let N = G_{k}(ℂ^{n}). We need to introduce some Lie algebraic ingredients. In the complex simple Lie algebra 𝔰𝔩(n, ℂ) we take the Cartan subalgebra 𝔥 consisting of diagonal matrices with zero trace and let Δ be the root system of 𝔰𝔩(n, ℂ) relative to 𝔥. We may write

𝔰𝔩(n, ℂ) = 𝔥 ⊕ ∑_{γ∈Δ^{+}}(𝔤_{γ} + 𝔤_{−γ})

where Δ^{+} indicates the set of positive roots with respect to the usual order. Let π = {α_{1}, …, α_{n−1}} ⊂ Δ^{+} be the standard system of simple roots. We also take in 𝔰𝔩(n, ℂ) a Weyl basis {X_{γ} : γ ∈ Δ} ∪ {H_{β} : β ∈ π} (see [9] p. 176). The following set of vectors provides a basis of the compact real form 𝔰𝔲(n)

\begin{matrix}U_{γ} = \frac{1}{\sqrt{2}}(X_{γ} − X_{−γ}) & γ ∈ Δ^{+}, \\ U_{−γ} = \frac{i}{\sqrt{2}}(X_{γ} + X_{−γ}) & γ ∈ Δ^{+}, \\ iH_{β} & β ∈ π. \end{matrix}  (9)

We shall also denote by 𝔥_{u} the real vector space generated by {iH_{β} : β ∈ π} and set 𝔪_{γ} = ℝU_{γ} ⊕ ℝU_{−γ}. Then we have

𝔰𝔲(n) = 𝔥_{u} ⊕ ∑_{γ∈Δ^{+}} 𝔪_{γ}

For m < n we include 𝔰𝔩(m, ℂ) into 𝔰𝔩(n, ℂ) in a natural way and hence we can write

𝔰𝔩(m, ℂ) = 𝔥_{1} ⊕ ∑_{γ∈Δ_{1}}(𝔤_{γ} + 𝔤_{−γ})

where 𝔥_{1} = \{[ \begin{matrix}A & 0 \\ 0 & 0 \end{matrix} ] : A ∈ 𝔰𝔩(m, ℂ)\} and Δ_{1} = \{∑ l_{j}α_{j} ∈ Δ^{+} : l_{j} = 0, m ≤ j ≤ n − 1\}.

Analogously we have

𝔰𝔲(m) = 𝔥_{1u} ⊕ ∑_{γ∈Δ_{1}} 𝔪_{γ}

Let 𝔨 and 𝔨_{1} ⊂ 𝔨 be the Lie algebras of S(U(k)×U(n−k)) and S(U(k)×U(m−k)) respectively. Then by [[5], 4.1] we may write

𝔰𝔲(n) = 𝔨 ⊕ ∑_{γ∈Γ} 𝔪_{γ} where  Γ = {∑ l_{j}α_{j} ∈ Δ^{+} : l_{k} = 1}
𝔰𝔲(m) = 𝔨_{1} ⊕ ∑_{γ∈Γ_{1}} 𝔪_{γ} where Γ_{1} = {γ ∈ Γ : l_{j} = 0, m ≤ j ≤ n − 1}.

As usual we identify

T_{o}G_{k}(ℂ^{n}) ≡ ∑_{γ∈Γ} 𝔪_{γ} and T_{o}G_{k}(ℂ^{m}) ≡ ∑_{γ∈Γ_{1}} 𝔪_{γ}  (10)

and hence

(T_{o}G_{k}(ℂ^{m}))^{⊥} = ∑_{γ∈Γ−Γ_{1}} 𝔪_{γ}.  (11)

The following notation will be useful for the roots of 𝔰𝔩(n, ℂ)

ε_{i,j} = ∑_{l=i}^{j} α_{l}  (12)

however we shall denote the maximal root by μ or ε_{1,n−1} indistinctly.

When n = 2p and k = 2q, for

1 ≤ i ≤ q and 2q ≤ j ≤ p + q − 1

we define the following tangent vectors of T_{o}G_{2q}(ℂ^{2p})

\begin{matrix}X_{i,j}^{+} = \frac{1}{\sqrt{2}}(U_{ε_{i,j}} + U_{ε_{q+i,j+p−q}}) & V_{i,j}^{+} = \frac{1}{\sqrt{2}}(U_{ε_{i,j}} − U_{ε_{q+i,j+p−q}}) \\ X_{i,j}^{−} = \frac{1}{\sqrt{2}}(U_{−ε_{i,j}} − U_{−ε_{q+i,j+p−q}}) & V_{i,j}^{−} = \frac{1}{\sqrt{2}}(U_{−ε_{i,j}} + U_{−ε_{q+i,j+p−q}}) \\ Y_{i,j}^{+} = \frac{1}{\sqrt{2}}(U_{ε_{i,j+p−q}} − U_{ε_{q+i,j}}) & W_{i,j}^{+} = \frac{1}{\sqrt{2}}(U_{ε_{i,j+p−q}} + U_{ε_{q+i,j}}) \\ Y_{i,j}^{−} = \frac{1}{\sqrt{2}}(U_{−ε_{i,j+p−q}} + U_{−ε_{q+i,j}}) & W_{i,j}^{−} = \frac{1}{\sqrt{2}}(U_{−ε_{i,j+p−q}} − U_{−ε_{q+i,j}}) \end{matrix}  (13)

It is easy to see that

\begin{matrix}T_{o}G_{q}(ℍ^{p}) ≡ span_{ℝ}\{X_{i,j}^{±}, Y_{i,j}^{±}\} \\ (T_{o}G_{q}(ℍ^{p}))^{⊥} ≡ span_{ℝ}\{V_{i,j}^{±}, W_{i,j}^{±}\}. \end{matrix}  (14)

Our next task is to find the spectral decomposition of the Jacobi operator R_{Z} for Z = U_{μ} = U_{ε_{1,n−1}} and Z = W_{1,p+q−1}^{+} respectively. To that end we introduce some extra notation.

Let ε and ρ be elements of Δ^{+} such that ε − ρ ∈ Δ. We define

sg(ε − ρ) = \{ \begin{matrix}1 & if ε − ρ ∈ Δ^{+} \\ −1 & if ρ − ε ∈ Δ^{+} \end{matrix}

and

|ε − ρ| = \{ \begin{matrix}ε − ρ & if ε − ρ ∈ Δ^{+} \\ ρ − ε & if ρ − ε ∈ Δ^{+}. \end{matrix}

Then for ε ≠ ρ we get the following formulae (see [5] p. 223)

\begin{matrix}[U_{ε}, U_{−ε}] = iH_{ε} \\ [U_{ε}, U_{ρ}] = \frac{1}{\sqrt{2}}\{N_{ε,ρ}U_{ε+ρ} + sg(ρ − ε)N_{ε,−ρ}U_{|ε−ρ|}\} \\ [U_{ε}, U_{−ρ}] = \frac{1}{\sqrt{2}}\{N_{ε,ρ}U_{−(ε+ρ)} + N_{ε,−ρ}U_{−|ε−ρ|}\}. \end{matrix}  (15)

Here we understand that if ε ± ρ are not roots then the terms N_{ε,−ρ}U_{|ε−ρ|}, N_{ε,ρ}U_{ε+ρ}, etc., vanish. Therefore, if γ ∈ Δ^{+} we have

[U_{μ}, U_{γ}] = \{ \begin{matrix}−\frac{1}{\sqrt{2}}N_{μ,−γ}U_{μ−γ} & if μ − γ ∈ Δ \\ 0 & if μ − γ ∉ Δ. \end{matrix}

Using again the formulae (15) we obtain

[U_{μ}, [U_{μ}, U_{γ}]] = \{ \begin{matrix}\frac{1}{2}N_{μ,−γ}N_{μ,γ−μ}U_{γ} & if μ − γ ∈ Δ \\ 0 & if μ − γ ∉ Δ. \end{matrix}

Due to μ − γ + (γ − μ) = 0 we have N_{μ,−γ} = −N_{μ,γ−μ}. Further, since for the algebra 𝔰𝔩(n, ℂ) (of type 𝔞_{n−1}) the constants N_{α,β} are given by

N_{α,β} = \{ \begin{matrix}±\frac{1}{\sqrt{2n}} & if α + β ∈ Δ \\ 0 & if α + β ∉ Δ \end{matrix}

we may write

[U_{μ}, [U_{μ}, U_{γ}]] = \{ \begin{matrix}−\frac{1}{4n}U_{γ} & if μ − γ ∈ Δ \\ 0 & if μ − γ ∉ Δ \end{matrix}

and so, for γ ∈ Δ^{+} we obtain

R_{U_{μ}}(U_{γ}) = \{ \begin{matrix}\frac{1}{4n}U_{γ} & if μ − γ ∈ Δ \\ 0 & if μ − γ ∉ Δ. \end{matrix}

In an analogous way, for γ ∈ Δ^{+} and γ ≠ μ we have

R_{U_{μ}}(U_{−γ}) = \{ \begin{matrix}\frac{1}{4n}U_{−γ} & if μ − γ ∈ Δ \\ 0 & if μ − γ ∉ Δ. \end{matrix}

Since

[U_{μ}, H_{μ}] = −\frac{1}{\sqrt{2}}\{μ(H_{μ})X_{μ} − (−μ(H_{μ}))X_{−μ}\} = iμ(H_{μ})U_{−μ}  (16)

we get

[U_{μ}, [U_{μ}, U_{−μ}]] = [U_{μ}, iH_{μ}] = −μ(H_{μ})U_{−μ}.

Let E_{ij} denote a square matrix with entry 1 where the i-th row and the j-th column meet, all other entries being 0. Then H_{μ} = \frac{1}{2n}(E_{11} − E_{nn}) and consequently

μ(H_{μ}) = B(H_{μ}, H_{μ}) = 2ntr(H_{μ}^{2}) = \frac{1}{n}.

Thus we deduce that R_{U_{μ}}(U_{−μ}) = \frac{1}{n}U_{−μ}. For γ ∈ Δ^{+}, the above calculations may be resumed into

R_{U_{μ}}(X) = \{ \begin{matrix}\frac{1}{4n}X & if X = U_{±γ} and μ − γ ∈ Δ \\ 0 & if X = U_{μ} or U_{±γ} and 0 ≠ μ − γ ∉ Δ \\ \frac{1}{n}X & if X = U_{−μ}. \end{matrix}  (17)

Let now n = 2p and k = 2q. For Z = W_{1,p+q−1}^{+} by proceeding as in the previous situation, with some more effort, we obtain the following formulae.

R_{Z}(X) = \{ \begin{matrix}\frac{1}{8n}X & \{ \begin{matrix}if X is one of the following X_{s,t}^{±}, Y_{s,t}^{±}, V_{s,t}^{±}, W_{s,t}^{±} with \\ (s, t) = (1, j), 2q ≤ j ≤ p + q − 2 or \\ (s, t) = (i, p + q − 1), 2 ≤ i ≤ q \end{matrix} \\ \frac{1}{2n}X & if X = V_{1,p+q−1}^{±} or X = Y_{1,p+q−1}^{−} or X = W_{1,p+q−1}^{−} \\ 0 & for the remaining vectors of (13) \end{matrix}  (18)

The above calculations may be summarized in the following proposition.

Proposition 1. For each one of the tangent vectors Z ∈ T_{o}G_{k}(ℂ^{n}) given below, with the same notation of (9) and (13), we list the eigenvalues c of R_{Z}, the corresponding eigenspace V_{c} and their dimensions. i) Z = U_{μ} = U_{ε_{1,n−1}}

c  V_{c}  dimV_{c}
1/4n  ∑ 𝔪_{γ}, for γ ∈ Δ^{+}, μ − γ ∈ Δ  2(n − 2)
1/n  ℝU_{−μ}  1
0 ℝU_{μ} ⊕ ∑ 𝔪_{γ}, for γ ∈ Δ^{+}, 0 ≠ μ − γ ∉ Δ 2(k − 1)(n − 1 − k) + 1

ii) n = 2p, k = 2q, Z = W_{1,p+q−1}^{+}

\begin{matrix}c & V_{c} & dimV_{c} \\ 1/8n & ∑_{(s,t)∈I_{1}} 𝔪_{s,t} & 8(p − 2) \\ 1/2n & span\{V_{1,p+q−1}^{±}, Y_{1,p+q−1}^{−}, W_{1,p+q−1}^{−}\} & 4 \\ 0 & ∑_{(s,t)∈I−I_{1}} 𝔪_{s,t} ⊕ 𝔫 & 8(q − 1)(p − q − 1) + 4 \end{matrix}

where 𝔪_{s,t} = span{X_{s,t}^{±}, Y_{s,t}^{±}, V_{s,t}^{±}, W_{s,t}^{±}},

𝔫 = span{X_{1,p+q−1}^{±}, Y_{1,p+q−1}^{+}, W_{1,p+q−1}^{+}},

I = {(i, j) : 1 ≤ i ≤ q and 2q ≤ j ≤ p + q − 2} and

I_{1} = {(1, j) : 2q ≤ j ≤ q + p − 2} ∪ {(i, q + p − 1) : 2 ≤ i ≤ q}. □

Remark 1. Note that from (10), (14) and the eigenspaces decomposition in the preceeding proposition it follows that T_{o}G_{k}(ℂ^{m}) and (T_{o}G_{k}(ℂ^{m}))^{⊥} are invariant by R_{U_{μ}} and the spaces T_{o}G_{q}(ℍ^{p}) and (T_{o}G_{q}(ℍ^{p}))^{⊥} are invariant by R_{W_{1,p+q−1}^{+}}.

Our next immediate objective is to decide which of the considered tubes are homogeneous. To that end we need to perform the following calculations.

Set Z = aZ_{1} + bZ_{2} + cZ_{3} where Z_{i} ∈ T_{o}G_{k}(ℂ^{n}) and a, b, c ∈ ℝ. By bilinearity of the Lie bracket and Jacobi´ s identity we have for γ ∈ Δ

\begin{aligned}[Z, [Z, U_{γ}]] & = & a^{2} [Z_{1}, [Z_{1}, U_{γ}]] + b^{2} [Z_{2}, [Z_{2}, U_{γ}]] + c^{2} [Z_{3}, [Z_{3}, U_{γ}]] + \\ & & +2ab [Z_{1}, [Z_{2}, U_{γ}]] + 2ac [Z_{1}, [Z_{3}, U_{γ}]] + 2bc [Z_{2}, [Z_{3}, U_{γ}]] + \\ & & +ab [U_{γ}, [Z_{1}, Z_{2}]] + ac [U_{γ}, [Z_{1}, Z_{3}]] + bc [U_{γ}, [Z_{2}, Z_{3}]] \end{aligned}  (19)

Let now be k < n − 2.

Let us take now Z_{1} = U_{ε_{1,n−2}}, Z_{2} = U_{ε_{1,n−1}}, Z_{3} = U_{ε_{2,n−2}} and pick the coefficients a, b, c ∈ ℝ so that a^{2} + b^{2} + c^{2} = 1 (hence ∥Z∥ = 1).

By (15) and computing the corresponding coefficients N_{α,β} we have that

\begin{aligned}[Z_{1}, Z_{2}] & = & −\frac{1}{2\sqrt{n}}U_{ε_{n−1,n−1}} \\ [Z_{1}, Z_{3}] & = & −\frac{1}{2\sqrt{n}}U_{ε_{1,1}} \\ [Z_{2}, Z_{3}] & = & 0 \end{aligned}  (20)

It is also easy to obtain that

\begin{matrix}[Z_{1}, U_{ε_{1,k}} ] = \frac{1}{2\sqrt{n}}U_{ε_{k+1,n−2}} & [Z_{2}, U_{ε_{1,k}} ] = \frac{1}{2\sqrt{n}}U_{ε_{k+1,n−1}} \\ [Z_{3}, U_{ε_{1,k}} ] = 0 & [Z_{1}, [Z_{1}, U_{ε_{1,k}} ]] = −\frac{1}{4n}U_{ε_{1,k}} \\ [Z_{1}, [Z_{2}, U_{ε_{1,k}} ]] = 0 & [Z_{2}, [Z_{2}, U_{ε_{1,k}} ]] = −\frac{1}{4n}U_{ε_{1,k}} \\ [U_{ε_{2,k}} , [Z_{1}, Z_{2}]] = 0 & [U_{ε_{2,k}} , [Z_{1}, Z_{3}]] = −\frac{1}{4n}U_{ε_{2,k}} \end{matrix}

and

\begin{matrix}[Z_{1}, U_{ε_{2,k}} ] = 0 & [Z_{2}, U_{ε_{2,k}} ] = 0 \\ [Z_{3}, U_{ε_{2,k}} ] = \frac{1}{2\sqrt{n}}U_{ε_{k+1,n−2}} & [Z_{1}, [Z_{3}, U_{ε_{2,k}} ]] = −\frac{1}{4n}U_{ε_{1,k}} \\ [Z_{2}, [Z_{3}, U_{ε_{2,k}} ]] = 0 & [Z_{3}, [Z_{3}, U_{ε_{2,k}} ]] = −\frac{1}{4n}U_{ε_{2,k}} \\ [U_{ε_{2,k}} , [Z_{1}, Z_{2}]] = 0 & [U_{ε_{2,k}} , [Z_{1}, Z_{3}]] = \frac{1}{4n}U_{ε_{1,k}} . \end{matrix}

Using these relations and (20) in the expresion (19) we obtain

[Z, [Z, U_{ε_{1,k}} ]] = −\frac{1}{4n}\{(a^{2} + b^{2})U_{ε_{1,k}} + acU_{ε_{2,k}} \}

and

[Z, [Z, U_{ε_{2,k}} ]] = −\frac{1}{4n}\{acU_{ε_{1,k}} + c^{2}U_{ε_{2,k}} \}.

Choosing a = \frac{1}{2} \sqrt{\frac{3}{2}}, b = \frac{1}{2\sqrt{2}} and c = \frac{1}{\sqrt{2}} we set

Z_{0} = \frac{1}{2}\sqrt{\frac{3}{2}}U_{ε_{1,n−2}} + \frac{1}{2\sqrt{2}}U_{ε_{1,n−1}} + \frac{1}{\sqrt{2}}U_{ε_{2,n−2}}  (21)

and then

[Z_{0}, [Z_{0}, U_{ε_{1,k}} + U_{ε_{2,k}} ]] = −\frac{2 + \sqrt{3}}{16n}(U_{ε_{1,k}} + U_{ε_{2,k}}).  (22)

Let now n = 2p, k = 2q and q ≤ p − 2. Using the formula (19) and performing a straightforward but very tedious calculation, for Z_{1} = U_{ε_{1,2p−2}} + U_{ε_{q+1,p+q−2}}, Z_{2} = U_{ε_{1,2p−1}} + U_{ε_{q+1,p+q−1}}, Z_{3} = U_{ε_{2,2p−2}} + U_{ε_{q+2,p+q−2}}, W = aZ_{1} + bZ_{2} + bZ_{3} with a^{2} + 2b^{2} = \frac{1}{2} and Y = U_{ε_{2,p+q−2}} + U_{ε_{q+2,2p−2}} − U_{ε_{1,p+q−1}} − U_{ε_{q+1,2p−1}}, we obtain that

[W, [W, Y]] = −\frac{a^{2}}{4n}Y.  (23)

The preceeding discussion gives us a necessary condition under which the tubes around G_{k}(ℂ^{m}) and G_{q}(ℍ^{p}) are homogeneous with the natural action of a subgroup of the isometry group of G_{k}(ℂ^{n}).

Remark 2. It is known that if a subgroup of the isometry group of G_{k}(ℂ^{n}) acts transitively on the tube (M)_{r} then (R_{Z_{1}})_{p_{1}} and (R_{Z_{2}})_{p_{2}} have the same eigenvalues for p_{i} ∈ (M)_{r} and Z_{i} ∈ (T_{p_{i}}(M)_{r})^{⊥}, ∥Z_{i}∥ = 1, i = 1, 2.

Theorem 2. If the tube (M)_{r} in G_{k}(ℂ^{n}) around M is homogeneous then i) m = n − 1 if M = G_{k}(ℂ^{m});

ii) p = q + 1 if M = G_{q}(ℍ^{p}) , k = 2q and n = 2p.

Proof. i) Let M = G_{k}(ℂ^{m}). By (11), if n − m ≥ 2 then U_{μ} and Z_{0} given by (21) are normal vectors of M at o. We consider now γ and η, the radial geodesics in G_{k}(ℂ^{n}) starting from o with velocities U_{μ} and Z_{0} respectively.

Since G_{k}(ℂ^{n}) is a Riemannian symmetric space, the Jacobi operators Rγ̇ and Rη̇ have constant eigenvalues along γ and η respectively. Thus, by Proposition 1 the spectrum of R_{γ̇(r)} is \{0, \frac{1}{4n}, \frac{1}{n}\} and by formula (22), \frac{2+\sqrt{3}}{16n} belongs to the spectrum of R_{η̇(r)}. This shows that both spectrums are different and hence, by the Remark 2 the tube (M)_{r} is not homogeneous. ii) Let now M = G_{q}(ℍ^{p}). By (14), if p − q ≥ 2, then W_{1,p+q−1}^{+} and W (see above (23)) are normal vectors of M at o.

By an analogous argument and using the Proposition 1 and formula (23) with 0 < a^{2} < \frac{1}{2}, we may conclude that the tube (M)_{r} is not homogeneous. □

Theorems 1 and 2 classify completely the homogeneous tubes centered at G_{k}(ℂ^{m}) for m < n and centered at G_{q}(ℍ^{p}) when k = 2q and n = 2p. In the last case, the tubes are homogeneous only when the ambient space is G_{2q}(ℂ^{2q+2}) ≡ G_{2}(ℂ^{2q+2}). This case has received much attention by J. Berndt and Y. Suh ([2]) who obtained very interesting results as we mention in Section 1. For this reason, from now on we shall only consider tubes in G_{k}(ℂ^{n}) around G_{k}(ℂ^{n−1}) for k > 2.

4. G_{k}(ℂ^{n−1})-centered tubes in G_{k}(ℂ^{n})

In this paragraph we shall make use of the spectral decomposition of the Jacobi operator R_{Z} obtained in Section 3, to determine the focal set of G_{k}(ℂ^{n−1}) in G_{k}(ℂ^{n}) and to obtain some information about the geometry of the family of G_{k}(ℂ^{n−1})-centered tubes.

Let M be a submanifold of a complete Riemannian manifold N and let Z ∈ T_{p}M^{⊥} be a unit normal vector at some point p ∈ M. Let γ_{Z} be the geodesic in N with γ_{Z}(0) = p and γ̇_{Z}(0) = Z. We recall that the point γ_{Z}(t_{0}) with t_{0} > 0 is said to be a focal point of M along γ_{Z} if the differential of the normal exponential map of M is singular at t_{0}Z. Equivalently γ_{Z}(t_{0}) with t_{0} > 0 is a focal point of M along γ_{Z} if there exists a Jacobi vector field J(t) along γ_{Z} satisfying

\begin{matrix}(i) & J(0) ∈ T_{p}M \\ (ii) & J′(0) + A_{Z}(J(0)) ∈ T_{p}M^{⊥} \\ (iii) & J(t_{0}) = 0 \end{matrix}  (24)

where A_{Z} is the shape operator of M in the direction of Z (see for instance [6]).

If there are focal points along γ_{Z}, one defines

t_{Z} := min{t_{0} > 0 : γ_{Z}(t_{0}) is a focal point of M along γ_{Z} }

and call γ_{Z}(t_{Z}) the first focal point of M along γ_{Z}. By the focal set of M we mean the set F M consisting of first focal points of M along all the geodesics γ_{Z} departing from M with Z normal unit vectors to M.

From now on M = G_{k}(ℂ^{n−1}) ⊂ G_{k}(ℂ^{n}) = N.

Note that from (11) we know that U_{μ} ∈ (T_{o}M)^{⊥} and since M is totally geodesic in N condition (ii) of (24) becomes J′(0) ∈ (T_{o}M)^{⊥}.

Let d = 2k(n−k) = dimN and d_{1} = 2k(n−1−k) = dimM. Then Proposition 1 provides an orthonormal basis of eigenvectors of R_{U_{μ}},{E_{1}, …, E_{d}} such that

{E_{1}, …, E_{d_{1}}} ⊂ T_{o}M
{E_{d_{1}+1}, …, E_{d}} ⊂ (T_{o}M)^{⊥}

Reordering we may assume that E_{d−1} = U_{μ}, E_{d} = U_{−μ}, the 2(n − 1 − k) first vectors E_{1}, …, E_{2(n−1−k)} and the 2(k − 1) vectors E_{d_{1}+1}, …, E_{d−2} are eigenvectors corresponding to the eigenvalue \frac{1}{4n}.

If E_{k}(t) denotes the parallel transport of E_{k} along γ_{U_{μ}}(t) then it is not difficult to see that any Jacobi vector field along γ_{U_{μ}}(t) satisfying (24) at p = o and for some t_{0} > 0 is given by

J(t) = ∑_{i=1}^{2(n−1−k)} p_{i} cos(\frac{t}{2\sqrt{n}})E_{i}(t) + q sin(\frac{t}{\sqrt{n}})E_{d}(t)  (25)

or

J(t) = ∑_{i=d_{1}+1}^{d−2} p_{i} sin(\frac{t}{2\sqrt{n}})E_{i}(t) + q sin(\frac{t}{\sqrt{n}})E_{d}(t)  (26)

where q and p_{i} are arbitrary real constants.

The Jacobi vector fields given by (25) vanish on t_{0} = \sqrt{n}(2l +1)π, l ∈ ℤ while those given by (26) vanish on t_{0} = \sqrt{n}2lπ, l ∈ ℤ. Hence the first focal point of M along γ_{U_{μ}} is precisely γ_{U_{μ}}(\sqrt{n}π) i.e. corresponds to t_{U_{μ}} = \sqrt{n}π.

If J(t) is a Jacobi vector field given by (25), for any g ∈ SU(n), g_{∗}J(t) is a Jacobi vector field

along the geodesic γ_{g_{∗}U_{μ}}(t) = g.γ_{U_{μ}}(t). Further if g ∈ SU(n − 1) ⊂ SU(n) via g ↪ [ \begin{matrix}g & 0 \\ 0 & 1 \end{matrix} ] ,

g_{∗}J(t) satisfies (24) at p = g.o ∈ M for t_{0} = \sqrt{n}(2l + 1)π, l ∈ ℤ.

Since SU(n−1) acts transitively on M by isometries of N and the natural action of S(U(k)× U(n − 1 − k)) on the unit sphere of (T_{o}M)^{⊥} is transitive (see Lemma 1), we may conclude:

i) The focal set F M of M consists of those points in G_{k}(ℂ^{n}) which are at distance \sqrt{n}π from M.

ii) The natural action of SU(n − 1) on the focal set F M given by (3) is transitive.

It is known that the geodesic γ_{U_{μ}}(t) is given by γ_{U_{μ}}(t) = e^{tU_{μ}}.o where

e^{tU_{μ}} = [ \begin{matrix}cos(\frac{t}{2\sqrt{n}}) & 0 & sin(\frac{t}{2\sqrt{n}}) \\ 0 & I_{n−2} & 0 \\ − sin(\frac{t}{2\sqrt{n}}) & 0 & cos(\frac{t}{2\sqrt{n}}) \end{matrix} ] .

Then the focal point on γ_{U_{μ}} turns out to be γ_{U_{μ}}(\sqrt{n}π) = g_{0}.o, where

g_{0} = [ \begin{matrix}0 & 0 & 1 \\ 0 & I_{n−2} & 0 \\ −1 & 0 & 0 \end{matrix} ] .

Thus by a straightforward calculation we may conclude that:

h ∈ SU(n − 1) ⊂ SU(n) belongs to the isotropy subgroup of SU(n − 1) at γ_{U_{μ}}(\sqrt{n}π) if and only if g_{0}^{−1}hg_{0} ∈ H = S({1} × U(k − 1) × U(n − k)).

Hence the isotropy subgroup of the point γ_{U_{μ}}(π\sqrt{n}) is g_{0}Hg_{0}^{−1} and therefore

F M = SU(n − 1)/g_{0}Hg_{0}^{−1} ≃ i(SU(n − 1))/H

where i : SU(n − 1) ↪ SU(n) is the inclusion g ↪ [ \begin{matrix}1 & 0 \\ 0 & g \end{matrix} ] . It turns out that F M is the

Grassmann manifold

F M ≃ SU(n − 1)/S(U(k − 1) × U(n − k)) = G_{k−1}(ℂ^{n−1}).

Now we compute the shape operator of the tube (M)_{r}. Denote by J_{X} the Jacobi vector field along the geodesic γ(t) = γ_{U_{μ}}(t) with initial conditions J_{X}(0) = 0 and J′_{X}(0) = X if X ∈ (T_{o}M)^{⊥} and with J_{X}(0) = X and J′_{X}(0) = 0 if X ∈ T_{o}M.

Using the results of Proposition 1 we obtain

J_{X}(t) = \sqrt{n} sin(\frac{t}{\sqrt{n}})X(t) if X ∈ V_{\frac{1}{n}}
J_{X}(t) = 2\sqrt{n} sin(\frac{t}{2\sqrt{n}})X(t) if X ∈ V_{\frac{1}{4n}} ∩ (T_{o}M)^{⊥}
J_{X}(t) = cos(\frac{t}{2\sqrt{n}})X(t)  if X ∈ V_{\frac{1}{4n}} ∩ T_{o}M
J_{X}(t) = X(t)  if X ∈ V_{0} ∩ T_{o}M
J_{X}(t) = tγ̇(t)  if X = γ̇(0)

(27)

where X(t) is the parallel transport of X along γ(t).

For 0 < r < π\sqrt{n} let A^{r} be the shape operator of the tube (M)_{r} with respect to the “outward” normal unit field.

It is well known that for the Jacobi vector fields listed in (27) with X ≠ U_{μ} = γ̇(0) the following formula holds

A_{γ̇(r)}^{r}J_{X}(r) = −J′_{X}(r)

(see for example [4] p. 152).

From this formula, Proposition 1 and (27) we obtain that the eigenvalues of A_{γ̇(r)}^{r} are

−\frac{1}{\sqrt{n}} cot(\frac{r}{\sqrt{n}}), −\frac{1}{2\sqrt{n}} cot(\frac{r}{2\sqrt{n}}), \frac{1}{2\sqrt{n}} tan(\frac{r}{2\sqrt{n}}), 0  (28)

and its respective multiplicities are

1, 2(k − 1), 2(n − 1 − k), 2(k − 1)(n − 1 − k).

Since SU(n − 1) acts on (M)_{r} transitively by isometries of N, A^{r} has the same eigenvalues at every point of (M)_{r} and hence (M)_{r} is a real isoparametric hypersurface whose principal curvatures are given in (28).

On the other hand, it is known that the natural complex Kaehler structure 𝕁 of N is given by

𝕁_{o} = Ad(h)|_{T_{o}N} with h = [ \begin{matrix}−I_{k} & 0 \\ 0 & iI_{n−k} \end{matrix} ] .

Keeping the notation of the matrix E_{ij} given below (16), we may write U_{μ} = \frac{1}{2\sqrt{n}}(E_{1n} − E_{n1}) and U_{−μ} = \frac{i}{2\sqrt{n}}(E_{1n} + E_{n1}), then 𝕁_{o}U_{μ} = U_{−μ}. By homogeneity of N, the parallel translated vectors of U_{μ} and U_{−μ} satisfy 𝕁_{γ(r)}E_{d−1}(r) = E_{d}(r) and consequently A_{γ̇(r)}^{r}𝕁_{γ(r)}E_{d−1}(r) = \frac{1}{n}E_{d}(r).

Hence, if Z is the “outward” unit normal field of (M)_{r} then 𝕁Z is an eigenvector of the shape operator A^{r} with eigenvalue \frac{1}{n}. This says that (M)_{r} is a Hopf hypersurface of N ([11] p. 244).

We summarize the discussion of the present section in the following result which gives a generalization of Theorem 18 in [1].

Theorem 3. Let M = G_{k}(ℂ^{n−1}) ⊂ G_{k}(ℂ^{n}) = N. The focal set F M of M in N is the Grassmannian manifold G_{k−1}(ℂ^{n−1}) and consists precisely of all points in N at distance \sqrt{n}π from M. Any tube (M)_{r} of radius 0 < r < \sqrt{n}π around M is an embedded isoparametric real hypersurface of N with four (resp. three for r = \frac{\sqrt{n}}{2}π) distinct principal curvatures. The principal curvatures k_{i} of (M)_{r} and its respective multiplicities m(k_{i}), with respect to the outward unit normal field Z, are

k_{i}  m(k_{i})
−\frac{1}{\sqrt{n}} cot(\frac{r}{\sqrt{n}})  1
−\frac{1}{2\sqrt{n}} cot(\frac{r}{2\sqrt{n}})  2(k − 1)
\frac{1}{2\sqrt{n}} tan(\frac{r}{2\sqrt{n}})  2(n − 1 − k)
0  2(k − 1)(n − 1 − k)

and the corresponding eigenspaces of A^{r} at γ_{U_{μ}}(r) are the parallel translated, along the radial geodesic γ_{U_{μ}},of the following subspaces of T_{o}N

T (k_{1}) = ℝU_{−μ}
T (k_{2}) = ∑_{j=2}^{k} 𝔪_{ε_{j,n−1}}

T (k_{3}) = ∑_{j=k}^{n−2} 𝔪_{ε_{1,j}}
T (k_{4}) = ∑ 𝔪_{ε_{i,j}}with 2 ⩽ i ⩽ k, k ⩽ j ⩽ n − 2

where 𝔪_{ε_{i,j}} = ℝU_{ε_{i,j}} ⊕ ℝU_{−ε_{i,j}} and ε_{i,j} is defined in (12). Furthermore (M)_{r} is a Hopf hypersurface of N. □

References

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Received June 21, 2001