FOCAL SETS, MOMENT MAPS AND INDUCED HOPF BUNDLES IN THE QUATERNIONIC PROJECTIVE SPACE

LIVIU ORNEA AND PAOLO PICCINNI

Abstract. We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space ℍP^{n}. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex structure on the Stiefel manifolds V_{2}(ℂ^{n+1}) and Ṽ_{4}(ℝ^{n+1}), the one on the former manifold being non compatible with its known hypercomplex structure.

1. Introduction

A motivation for the present work has been to understand some aspects of the Riemannian geometry of the focal set F oc_{ℍP^{n}}ℂP^{n}, i.e. of the set of points in the quaternionic projective space ℍP^{n} that are critical values of the normal exponential map with respect to the totally geodesic submanifold ℂP^{n} ⊂ ℍP^{n}. Our starting point has been the Habilitationsschrift by J. Berndt [7], [8], where it is proven, in particular, that F oc_{ℍP^{n}}ℂP^{n} fibers in circles over Gr_{2}(ℂ^{n+1}), the Grassmannian of 2-planes in ℂ^{n+1}. In the simplest cases n = 1, 2 these focal points distribute on spheres respectively S^{1}, S^{5} and n = 3 seems thus to be the first significant case. Indeed, since F oc_{ℍP^{3}}ℂP^{3} is a circle bundle over the Klein quadric Gr_{2}(ℂ^{4}) ≅ ℂQ^{4} ⊂ ℂP^{5}, both the classical projective geometry of ℂQ^{4} (described for example in [36], pp. 26-35) and its two quaternion Kähler structures [26] can be related to the total space F oc_{ℍP^{3}}ℂP^{3}. More generally, the twofold Kähler-Einstein and quaternion Kähler properties of the Grassmannian Gr_{2}(ℂ^{n+1}) suggested us a link between the focal set F oc_{ℍP^{n}}ℂP^{n} and the Sasakian-Einstein geometry of the induced Hopf bundles considered in our previous work [30].

1991 Mathematics Subject Classification. 53C15, 53C25, 53C55.

Key words and phrases. Quaternion Kähler manifold, Sasakian structure, 3-Sasakian structure, complex structure, focal set, Riemannian submersion, moment map, induced Hopf bundle.

This paper begins by giving an explicit identification of the focal set F oc_{ℍP^{n}}ℂP^{n} with the zero level set of the moment map μ associated to the diagonal U(1) - action on ℍP^{n}. This is the simplest isometric action giving examples of reductions in quaternion Kähler geometry, and it is well known that the reduced manifold is precisely the quaternion Kähler Wolf space Gr_{2}(ℂ^{n+1}) [19], [20]. According to the notation in use, we write ℍP^{n}///U(1) = Gr_{2}(ℂ^{n+1}) to indicate this reduction procedure, that has been proven to be the unique one with respect to isometric circle actions and positive quaternion Kähler manifolds [3], [4]. Next, the diagonal Sp(1)-action on ℍP^{n} produces as reduction the quaternion Kähler manifold ℍP^{n}///Sp(1) = G̃r_{4}(ℝ^{n+1}), the Grassmannian of oriented 4-planes in ℝ^{n+1}. The zero set of the corresponding moment map ν : ℍP^{n} → ℍ can be also related to focal sets in ℍP^{n}. (For these identifications cf. theorems 3.1, 3.2).

On the other hand, the zero level sets μ^{−1}(0) and ν^{−1}(0) of these moment maps μ and ν can be identified with the total spaces of some induced Hopf S^{1}-bundles. This enables to define on them respectively a Sasakian-Einstein and a 3-Sasakian metric, and hence to write a diagram where these zero sets μ^{−1}(0) and ν^{−1}(0) are base spaces of S^{3}-bundles, now induced by the Hopf fibration S^{4n+3} → ℍP^{n}. The corresponding total spaces are the Stiefel manifolds respectively V_{2}(ℂ^{n+1}) and Ṽ_{4}(ℝ^{n+1}). The 3-Sasakian structure of the fibers S^{3} and the Sasakian structure of the base spaces μ^{−1}(0) and ν^{−1}(0) allow to construct a complex structure on V_{2}(ℂ^{n+1}) and on Ṽ_{4}(ℝ^{n+1}). On the former Stiefel manifold this complex structure is non-compatible with its standard hypercomplex structure and we believe with any of the hypercomplex structures that can be constructed via the 3-Sasakian reduction: [2], [14], [23]. We don’t know previous references concerning these complex structures, whose construction is however very much in the spirit of Calabi-Eckmann manifolds, exibiting the Stiefel manifolds as fibrations in complex hermitian products of 3-spheres over quaternion Kähler Wolf spaces.

Acknowledgements. This paper was written while the first named author was visiting the University of Rome ”La Sapienza” and the ”Max-Planck-Institut für Mathematik” in Bonn. He thanks both institutions for financial support and their staff for hospitality.

Both authors thank Victor Vuletescu for some helpful discussions.

2. Preliminaries

We collect here some definitions and basic facts that will be used throughout the paper.

Let (M, g) be a Riemannian manifold and let N be an isometrically immersed submanifold. The critical values of the restriction exp_{T^{⊥}N} of the exponential map of M to the normal bundle of N are called focal points of N and the set of all focal points, here denoted with F oc_{M}N, is called the focal set of N (cf. for example [17] p. 23, [18], p. 227,

[27] p. 283). Although the focal set of a submanifold may not be a submanifold (for smooth plane curves and for regular surfaces in ℝ^{3} the focal sets are respectively the evolutes and the surfaces of the centers, and both of them can have singular points, cf. [18], pp. 237-238 and p. 232), so happens for several examples, including the ones considered in the present paper. Indeed, it seems that not many focal sets of Riemannian submanifolds have been explicitely determined, but we can mention totally geodesic spheres in spheres [31], and hypersurfaces in space forms [15], [16].

The focal set we are mainly interested in is F oc_{ℍP^{n}}ℂP^{n} with respect to the totally geodesic ℂP^{n} in the quaternionic projective space ℍP^{n}. This focal set appears in J. Berndt’s work [7], [8], there denoted with Q^{n}, and studied in connection with both the complex Kähler and the quaternion Kähler structure of Gr_{2}(ℂ^{n+1}), the Grassmannian of complex two-planes in ℂ^{n+1}.

Two geometric structures which appear naturally in our context are the Sasakian and the 3-Sasakian ones. We briefly recall the definitions, sending the reader to the survey [11] for further informations.

A Sasakian manifold is a (2n + 1)-dimensional Riemannian manifold (S, g) equipped with a unitary Killing vector field ξ such that the field of endomorphisms φ := ∇ξ satisfy the differential equation

∇φ = Id ⊗ η − g ⊗ ξ,

where ∇ is the Levi-Civita connection of g and η is the dual 1-form of ξ. The datum of a Sasakian structure on the manifold S is easily seen to be equivalent to the requirement that the cône metric dr^{2} + r^{2}g, on ℝ_{+} × S have holonomy contained in U(n + 1). Note that η is a contact form on S, hence ξ is its Reeb field. In the simplest example, the canonical sphere S^{2n+1}, the Killing vector field is ξ = −JU, J being the standard complex structure of ℂ^{n+1} and U the unit outward normal to the sphere.

More generally, look at the induced Hopf S^{1}-bundle π : V → M, over a smooth submanifold of ℂP^{N}. Its total space V carries a Sasakian structure (V, g, ξ) induced from the one of (S^{2N+1}, can). If the Fubini-Study metric of ℂP^{N} induces an Einstein metric h with Einstein constant α on M, it can be seen (cf. [30], lemma 1) that the Ricci tensor of the metric g has the form

Ric(g) = λg + μη ⊗ η

with λ = α−2, μ = dim V +1−α. This is known as the η-Einstein property in Sasakian geometry and, following S. Tanno [35], an η-Einstein Sasakian metric can be deformed to a Sasakian-Einstein one by setting

g′ = Ag + A(A − 1)η ⊗ η, (2.1)

ξ′ = A^{−1}ξ, with A = \frac{λ+2}{dim V +1}. Since π is a Riemannian submersion with respect to the metrics h and g, to preserve the Riemannian submersion with g′, the scaled metric h′ = Ah has to be considered on the basis M.

As Sasakian geometry may be viewed as the odd-dimensional counterpart of Kähler geometry, the odd-dimensional correspondent of hyperKähler manifolds are the 3-Sasakian ones (cf. [11]). Precisely, a (4n + 3)-dimensional Riemannian manifold (S, g) is 3-Sasakian if it is endowed with three mutually orthogonal unit Killing vector fields ξ_{1}, ξ_{2}, ξ_{3}, each one defining a Sasakian structure and satisfying the conditions

[ξ_{1}, ξ_{2}] = 2ξ_{3}, [ξ_{2}, ξ_{3}] = 2ξ_{1}, [ξ_{3}, ξ_{1}] = 2ξ_{2}.

As above, an equivalent definition requires that the cône metric dr^{2} + r^{2}g on ℝ_{+} ×S be hyperkähler, i. e. its holonomy group be contained in Sp(n + 1). 3-Sasakian manifolds are necessarily Einstein with positive scalar curvature and their Einstein constant is 4n + 2. Given a positive quaternion Kähler manifold P, one constructs its Kähler-Einstein twistor space Z_{P} and then an S^{1} bundle over it whose Chern class is, up to torsion, the one of an induced Hopf bundle. The total space S thus obtained is an SO(3)-principal bundle over P with 3-Sasakian structure. Moreover, all three fibrations involved are Riemannian submersions.

We recall now two basic moment maps of quaternion Kähler geometry. Let [h_{0} : h_{1} : ... : h_{n}] be the homogeneous coordinates on ℍP^{n} and, for each a = 0, 1, ..., n, we shall write the complex and real components of h_{a} as follows:

h_{a} = z_{a} + w_{a}j = α_{a} + β_{a}i + γ_{a}j + δ_{a}k (2.2) where z_{a} = α_{a} +β_{a}i, and w_{a} = γ_{a} +δ_{a}i. The first moment map, induced by the diagonal action of U(1) on ℍP^{n}, is, at the hyperKähler level of the bundle ℍ^{n+1}\{0} → ℍP^{n}, given by:

μ : ℍ^{n+1}\\{0\} → ℍ, μ = ∑_{a=0}^{n} \overline{h}_{a}ih_{a}. (2.3)

On ℍP^{n}, one has to look at the corresponding moment map μ_{qK} as a section of S^{2}H, the rank 3 vector bundle of the compatible almost complex structures. In fact, by using the metric, μ_{qK} appears as a 2-form solution of

dμ_{qK} = i_{X}Ω,

where Ω is the Kähler 4-form and X the Killing vector field generating the U(1)-action [34], [1]. However, since the zero level sets μ^{−1}(0) ⊂ ℍ^{n+1} and μ_{qK}^{−1}(0) ⊂ ℍP^{n} correspond each other in the bundle projection ℍ^{n+1}\{0} → ℍP^{n}, we shall refer to the definition μ = ∑_{a=0}^{n} \overline{h}_{a}ih_{a}, whose zero set makes sense also when the h_{a} are the homogeneous coordinates of ℍP^{n} [11]. The reduced manifold μ^{−1}(0)/U(1) = μ_{qK}^{−1}(0)/U(1) turns out to be the quaternion Kähler Wolf space \frac{SU(n+1)}{S(U(n−1)×U(2))} ≅ Gr_{2}(ℂ^{n+1}) [19], [20].

The second moment map to be considered, generated by the action of Sp(1), is (again at the hyperKähler level):

ν : ℍ^{n+1}\\{0\} → ℍ^{3}, ν = (∑_{a=0}^{n} \overline{h}_{a}ih_{a}, ∑_{a=0}^{n} \overline{h}_{a}jh_{a}, ∑_{a=0}^{n} \overline{h}_{a}kh_{a}), (2.4)

and its corresponding quaternion Kähler moment map ν_{qK} can be viewed as a triple of 2-forms associated to the frame of Killing vector fields defining the Sp(1)-action. The corresponding reduced manifold ν^{−1}(0)/Sp(1) = ν_{qK}^{−1}(0)/Sp(1) is now the Wolf space \frac{SO(n+1)}{SO(n−3)×SO(4)} ≅

G̃r_{4}(ℝ^{n+1}), the Grassmannian of oriented four-planes of ℝ^{n+1}.

3. Statement of results

Theorem 3.1. (i) The focal set F oc_{ℍP^{n}}ℂP^{n} coincides with μ^{−1}(0), and it is isometric to the total space of the induced Hopf S^{1}-bundle through the Plücker embedding Gr_{2}(ℂ^{n+1}) ↪ ℂP^{N}.

(ii) The metric g_{1}, induced on μ^{−1}(0) by the Plücker embedding, allows to define on μ^{−1}(0) a Sasakian Einstein metric g.

(iii) The Stiefel manifold V_{2}(ℂ^{n+1}) of orthonormal 2-frames in ℂ^{n+1}, diffeomorphic to the total space of the induced Hopf S^{3}-bundle through the embedding μ^{−1}(0) ⊂ ℍP^{n}, admis an (integrable) complex structure J, non compatible with its standard hypercomplex structure.

We have a similar statement regarding the moment map ν induced by the action of Sp(1) on ℍP^{n}. We need to consider the following mutually congruent, totally geodesic embeddings of ℂP^{n} in ℍP^{n}.

\begin{matrix}ℂP_{i}^{n} = & \{h ∈ ℍP^{n} ; γ_{a} = δ_{a} = 0, a = 1, ..., n\}, \\ ℂP_{j}^{n} = & \{h ∈ ℍP^{n} ; β_{a} = δ_{a} = 0, a = 1, ..., n\}, \\ ℂP_{k}^{n} = & \{h ∈ ℍP^{n} ; β_{a} = γ_{a} = 0, a = 1, ..., n\}, \end{matrix} (3.1)

and of course ℂP_{i}^{n} is the standard ℂP^{n} appearing in theorem 3.1.

Theorem 3.2. (i) The zero level set ν^{−1}(0) coincides with the intersection M = F oc_{ℍP^{n}}ℂP_{i}^{n} ∩ F oc_{ℍP^{n}}ℂP_{j}^{n} ∩ F oc_{ℍP^{n}}ℂP_{k}^{n} and it is isometric to the total space of the induced Hopf S^{1}-bundle over the Fano manifold Z_{G̃r_{4}(ℝ^{n+1})}, through the embeddings Z_{G̃r_{4}(ℝ^{n+1})} ↪ Gr_{2}(ℂ^{n+1}) ↪ ℂP^{N}, the first of which is defined by looking at Z_{G̃r_{4}(ℝ^{n+1})} as the space of totally isotropic two-planes in ℂ^{n+1}. This isometry allows the construction of a Sasakian Einstein metric on ν^{−1}(0), and consequently its identification with the homogeneous 3-Sasakian manifold SO(n + 1)/(SO(n − 3) × Sp(1)).

(ii) The Stiefel manifold Ṽ_{4}(ℝ^{n+1}) admits an (integrable) complex structure, projecting to the complex structure of Z_{G̃r_{4}(ℝ^{n+1})}.

Here Z_{G̃r_{4}(ℝ^{n+1})} is the twistor space of the quaternion Kähler Wolf

space G̃r_{4}(ℝ^{n+1}), given by the Sp(1) reduction on ℍP^{n}. Z_{G̃r_{4}(ℝ^{n+1})} is known to be a complex submanifold of the Kähler-Einstein Grassmannian Gr_{2}(ℂ^{n+1}), cf. for example [24] p. 14 or [22] p. 702.

Statements 3.1, 3.2 give, in particular, some fibrations that can be collected in a larger diagram as follows. Here V_{k}(ℂ^{n+1}), Ṽ_{k}(ℝ^{n+1}) denote the Stiefel manifolds of k-frames in ℂ^{n+1} and of oriented k-frames in ℝ^{n+1}, respectively.

Corollary 3.1. There is a commutative diagram

Ṽ_{4}(ℝ^{n+1}) ↪ V_{2}(ℂ^{n+1}) ↪ S^{4n+3}
↓S^{3}  ↓S^{3}  ↓S^{3}
ν^{−1}(0) ↪ μ^{−1}(0) ↪ ℍP^{n}  S^{2N−1}
↓S^{1}  ↓S^{1}  ↙
Z_{G̃r_{4}(ℝ^{n+1})} ↪ Gr_{2}(ℂ^{n+1}) ↪ ℂP^{N}
↓S^{2}

G̃r_{4}(ℝ^{n+1})

of principal S^{1} and S^{3}-bundles and Riemannian submersions. The zero level sets μ^{−1}(0) and ν^{−1}(0) are thus total spaces of induced Hopf S^{1}- bundles and as such are minimal submanifolds in the sphere S^{2N−1}.

4. The proofs

Proof of theorem 3.1 (i) and (ii). The first observation is that:

F oc_{ℍP^{1}}ℂP^{1} is the level set of the moment map associated to the standard U(1)-action on ℍP^{1}.

In the homogeneous coordinates [h_{0}, h_{1}] of ℍP^{1}, and by using notations from formula (2.1), ℂP^{1} is given by:

w_{0} = 0, w_{1} = 0 or γ_{0} = δ_{0} = γ_{1} = δ_{1} = 0. (4.1)

The pair ℂP^{1} ⊂ ℍP^{1} can be identified with S^{2} ⊂ S^{4}, and the focal set of a totally geodesic sphere S^{p} in S^{n} is the unit S^{n−p−1} in the (n − p)-dimensional orthogonal complement of the ℝ^{p+1} containing S^{p} ([31], p. 286). This identification can be made explicit using the coordinate h = α + βi + γj + δk = h_{0}h_{1}^{−1} in the affine line h_{1} ≠ 0, whose corresponding real coordinates are:

\begin{aligned}α & = \frac{α_{0}α_{1} + β_{0}β_{1} + γ_{0}γ_{1} + δ_{0}δ_{1}}{r^{2}} , β = \frac{α_{1}β_{0} − α_{0}β_{1} − γ_{0}δ_{1} + γ_{1}δ_{0}}{r^{2}} , \\ γ & = \frac{α_{1}γ_{0} − α_{0}γ_{1} − δ_{0}β_{1} + β_{0}δ_{1}}{r^{2}} , δ = \frac{α_{1}δ_{0} − α_{0}δ_{1} − γ_{1}β_{0} + γ_{0}δ_{1}}{r^{2}} . \end{aligned}

where r^{2} = α_{1}^{2} + β_{1}^{2} + γ_{1}^{2} + δ_{1}^{2}. If (x_{1}, ..., x_{5}) are the coordinates on ℝ^{5}, the (inverse) stereographic projection ℝ^{4} → S^{4} reads:

(α, β, γ, δ) ↦ (x_{1}, ..., x_{5}) = (\frac{2α}{r^{2} + 1}, \frac{2β}{r^{2} + 1}, \frac{2γ}{r^{2} + 1}, \frac{2δ}{r^{2} + 1}, \frac{r^{2} − 1}{r^{2} + 1}). (4.2)

Thus, if x_{3} = x_{4} = 0 are the equations of a totally geodesic S^{2} in S^{4}, the focal set of S^{2} in S^{4} is the S^{1} of equations x_{1} = x_{2} = x_{5} = 0, corresponding to:

α = β = 0, γ^{2} + δ^{2} = 1. (4.3)

On the other hand, the moment map μ on ℍP^{1} can be written in the non-homogeneous coordinate h = h_{0}h_{1}^{−1} as μ = \overline{h}ih + i. Its level set μ^{−1}(0) is thus:

α_{1}^{2} + β_{1}^{2} − γ_{1}^{2} − δ_{1}^{2} + 1 = 0, αδ − βγ = 0, αγ + βγ = 0, (4.4)

and the systems (4.3) and (4.4) are equivalent.

We now treat the case n > 1. From the definition of the focal set, we only need to look at geodesics normal to ℂP^{n}. These are normal to all the complex projective lines ℂL^{1} ⊂ ℂP^{n}, thus their focal points with respect to ℂP^{n} are also focal points for all the lines ℂL^{1} contained in ℂP^{n}. Any such line ℂL^{1} in ℍP^{n} belongs to a unique quaternionic projective line ℍL^{1} and the latter is totally geodesic in ℍP^{n}. It follows that any geodesic that is normal to a ℂL^{1} and tangent to the corresponding ℍL^{1} at a given point, remains tangent to ℍL^{1} for its entire length. Hence, if we show that through any point x of ℂP^{n} and for any v ∈ T_{x}^{⊥}ℂP^{n} there exists a projective quaternionic line ℍL^{1} containing v, we may deduce:

F oc_{ℍP^{n}}ℂP^{n} is the union of all the focal sets F oc_{ℍL^{1}}ℂL^{1}.

This can be seen through the diagram:

ℂ^{n+1}\{0} ↪ ℍ^{n+1}\{0}
↓  ↓
ℂP^{n} ↪ ℍP^{n}.

by looking at the complex planes L_{2}^{ℂ} in ℂ^{n+1} containig the fibre ℂ^{∗} and the corresponding hypercomplex 2-planes L_{2}^{ℍ} in ℍ^{n+1} containig the fibre ℍ^{∗}. Observe that for any vector \overrightarrow{v} ∈ ℍ^{n+1}, normal to the standard embedded ℂ^{n+1}, there exist a L_{2}^{ℂ} and a L_{2}^{ℍ} with span_{ℝ}\{L_{2}^{ℂ}, \overrightarrow{v}\} ⊂ L_{2}^{ℍ}: indeed, if ℍ^{n+1} = span_{ℍ}\{\overrightarrow{e_{0}}, ..., \overrightarrow{e_{n}}\} and ℂ^{n+1} = span_{ℂ}\{\overrightarrow{e_{0}}, ..., \overrightarrow{e_{n}}\}, we have:

\overrightarrow{v} ⊥ ℂ^{n+1} if and only if \overrightarrow{v} = ∑_{a=0}^{n} λ_{a}j\overrightarrow{e_{a}} + μ_{a}k\overrightarrow{e_{a}}.

Hence, if

\overrightarrow{w} = −j\overrightarrow{v} = ∑_{a=0}^{n} λ_{a}\overrightarrow{e_{a}} − μ_{a}i\overrightarrow{e_{a}},

we obtain

L_{2}^{ℂ} = span_{ℂ}\{\overrightarrow{e_{0}}, \overrightarrow{w}\}

and

L_{2}^{ℍ} = span_{ℝ}\{\overrightarrow{e_{0}}, i\overrightarrow{e_{0}}, j\overrightarrow{e_{0}}, k\overrightarrow{e_{0}}, \overrightarrow{w}, i\overrightarrow{w}, j\overrightarrow{w}, k\overrightarrow{w}\}

satisfy the condition.

The identification of F oc_{ℍP^{n}}ℂP^{n} with μ^{−1}(0) is then completed by the following:

The subset μ^{−1}(0) ⊂ ℍP^{n} is the union of the zero sets of the moment maps μ_{1} associated to the standard U(1)-action on all the projective quaternionic lines ℍL^{1} ⊂ ℍP^{n}.

Let {p_{0}, p_{1}, ..., p_{n}} be the canonical frame of ℍP^{n} with unit point u and let [h_{0} : ... : h_{n}] be the homogeneous coordinates in this frame. Accordingly, the moment map reads μ(h) = ∑_{a=0}^{n} \overline{h}_{a}ih_{a}. Fix a q ∈ μ^{−1}(0), i.e. ∑_{a=0}^{n} \overline{q}_{a}iq_{a} = 0 and note that q cannot be real; however, we may suppose q_{0} ≠ 0.

Let ℍL^{1} ≅ ℍP^{1} be the quaternionic projective line through q and p_{1}. To compare the intersection μ^{−1}(0) ∩ ℍL^{1} with the zero level set of the moment map in ℍL^{1} ≅ ℍP^{1}, we change the frame in ℍP^{n} as follows. We want new homogeneous coordinates [k_{0} : k_{1} : ... : k_{n}] such that q = [1 : j : 0 : ... : 0] and the coordinates of p_{1},..., p_{n} remain unchanged. With the new unit point v = [q_{0} : 1 : −1 : −1 : ... : −1], the matrix of the change of coordinates is

A = (\begin{matrix}q_{0} & & 0 & & 0 & & … & & & 0 \\ q_{1} − j & & 1 & & 0 & & … & & & 0 \\ q_{2} & & 0 & & −1 & & … & & & 0 \\ . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . & . \\ q_{n} & & 0 & & 0 & & … & & & −1 \end{matrix})

Thus, ^{t}[h_{0}, ..., h_{n}] = A^{t}[k_{0}, ..., k_{n}] and the moment map is

μ = ∑_{a=0}^{n} \overline{h_{a}}ih_{a} = \overline{k_{0}}(∑_{a=0}^{n} \overline{q_{a}}iq_{a})k_{0} − \overline{k_{0}}jijk_{0} − ∑_{b=1}^{n} \overline{k_{b}}ik_{b} = − ∑_{b=0}^{n} \overline{k_{b}}ik_{b}.

It follows that μ^{−1}(0) is invariant under these projective changes of coordinates. As the coordinates on ℍL^{1} are [k_{0} : k_{1}], its moment map is μ_{1} = −\overline{k_{0}}ik_{0}−\overline{k_{1}}ik_{1} with the same level set μ_{1}^{−1}(0) described for ℍP^{1}. This proves the inclusion μ^{−1}(0)|_{ℍL^{1}} ⊂ μ_{1}. As the converse inclusion is clear, for any projective line ℍL^{1} in ℍP^{n}, the proof of the identification F oc_{ℍP^{n}}ℂP^{n} = μ^{−1}(0) is complete.

Next, we prove that F oc_{ℍP^{n}}ℂP^{n} is isometric to the total space of the induced Hopf bundle over the Grassmannian Gr_{2}(ℂ^{n+1}). In fact, since F oc_{ℍP^{n}}ℂP^{n} is simply connected ([7], p. 17), a diffeomorphism with the induced Hopf bundle is a consequence of the following observation:

Let π : P → Gr_{2}(ℂ^{n+1}) be a principal circle bundle with simply connected P. Then M is diffeomorphic with the total space of the induced Hopf bundle of S^{2N−1} → ℂP^{N}, N = \binom{n+1}{2} − 1, through the Plücker embedding Gr_{2}(ℂ^{n+1}) ↪ ℂP^{N}.

In fact, the principal S^{1} -bundles over the base B are classified by their Chern class in H^{2}(B, ℤ). Since H^{2}(Gr_{2}(ℂ^{n+1}), ℤ) ≅ ℤ is generated by the class of the Kähler form, one can denote by P_{a} the S^{1}-bundle over Gr_{2}(ℂ^{n+1}) associated to a ∈ ℤ. Observe that P_{1} (resp. P_{−1}) is the circle bundle associated to the canonical line bundle O_{Gr_{2}(ℂ^{n+1})}(1) (resp. O_{Gr_{2}(ℂ^{n+1})}(−1), dual of the former). But dual complex line bundles are diffeomorphic as real vector bundles. Hence P_{1} is diffeomorphic with P_{−1}.

Let us now show that the if P_{a} is simply connected, then a = ±1. From π_{1}(P) = 0, we have H^{1}(P, ℤ) = 0 and H^{2}(P, ℤ) torsion free. The Gysin sequence of π:

\begin{aligned}0 → H^{1}(P, ℤ) & → H^{0}(Gr_{2}(ℂ^{n+1}), ℤ) → H^{2}(Gr_{2}(ℂ^{n+1}), ℤ) \\ & → H^{2}(P, ℤ) → 0 \end{aligned}

reduces thus to:

0 ⟶ ℤ ⟶^{∪c_{1}} ℤ ⟶ H^{2}(P, ℤ) ⟶ 0,

where c_{1} is the Chern class of π. Hence, c_{1} ≠ ±1 implies H^{2}(P, ℤ) ≅ ℤ_{n} for some n ≥ 2. Consequently, c_{1} is, up to sign, the Chern class of the induced Hopf bundle.

As regards the identification of the metric g_{1} inherited by F oc_{ℍP^{n}}(ℂP^{n}) from (S^{2N−1}, can) as total space of the induced Hopf bundle and the metric g induced on F oc_{ℍP^{n}}(ℂP^{n}) from ℍP^{n}, we note that both (F oc_{ℍP^{n}}(ℂP^{n}), g_{1}) and (F oc_{ℍP^{n}}(ℂP^{n}), g) are Riemannian submersions with geodesic fibres S^{1} over Gr_{2}(ℂ^{n+1}) (cf. [30] for g_{1} and [7] for g). Thus theorem 9.59 in [6], p. 249, can be used to conclude that g = g_{1}. This completes the proof of (i) and (ii) in theorem 3.1.

Remark 4.1. An alternative way of recognizing that F oc_{ℍP^{n}}ℂP^{n} = μ^{−1}(0) is to look at the standard isometric action of SU(n+1) on ℍP^{n} and to see that the two subset are obtained as homogeneous spaces of SU(n + 1) with the same isotropy groups. The homogeneous space is in fact \frac{SU(n+1)}{SU(2)×SU(n−1)}, showed in [7], p. 17 (and there denoted by Q^{n}) to be a singular orbit of the action of SU(n + 1). A similar description of μ^{−1}(0) as homogeneous space is in [1], p. 65, in relation with the problem of studying local compatible complex structures in ℍP^{n}. Cfr. also [33], p. 171, where \frac{SU(n+1)}{SU(2)×SU(n−1)} is called the ”Grassmannian of oriented two-planes” of ℂ^{n+1}.

Remark 4.2. We proved in [30] that the metric g_{1} is Sasakian and η-Einstein, and this is in accordance with formulas following prop. 9 in [7]. As for the extrinsic geometry of F oc_{ℍP^{n}}(ℂP^{n}), J. Berndt proves (cor. 1 in [7]) that the immersion in ℍP^{n} is minimal. It is interesting to observe that the same holds for the immersion (F oc_{ℍP^{n}}ℂP^{n}, g_{1}) in (S^{2N−1}, can), cf. [30].

Proof of Theorem 3.2 (i). The identification of ν^{−1}(0) with the intersection M is an immediate consequence of the definition of ν and of the first statement of theorem 3.1. To see that ν^{−1}(0) is isometric to the above mentioned induced Hopf bundle, we notice that the last observation in the proof of theorem 3.1 (i) and (ii) still holds for principal circle bundles with simply connected P over any complex algebraic projective submanifold B of ℂP^{n} with H^{2}(B, ℤ) ≅ ℤ. This is the case for our Z, hence it remains to prove that ν^{−1}(0) is simply connected. To this purpose we note that ν^{−1}(0) can be looked at as homogeneous space \frac{SO(n+1)}{SO(n−3)×Sp(1)} through the transitive action of SO(n + 1) on ν^{−1}(0). This last action comes, in fact, from the natural action of SO(n+1) on ℍP^{n} and J. Berndt’s observation that SU(n+1) acts transitively on F oc_{ℍP^{n}}(ℂP^{n}). His argument can be triplicated to produce transitive actions of groups SU_{i}(n+1), SU_{j}(n+1), SU_{k}(n+1) (the first one is the standard SU(n + 1) in Sp(n + 1), the other two are similarly defined by interchanging the rôles of the unit quaternions i, j, k) on the zero level sets μ_{i}^{−1}(0), μ_{j}^{−1}(0), μ_{k}^{−1}(0). The isotropy subgroup of the action on ν^{−1}(0) at the point [1 : i : j : k : 0... : 0] is then Sp(1) × SO(n − 3). The homotopy sequence associated to the homogeneous manifold \frac{SO(n+1)}{Sp(1)×SO(n−3)} thus shows that ν^{−1}(0) is simply connected, so that it is diffeomorphic with the total space of the induced Hopf bundle, now over the Fano manifold Z_{G̃r_{4}(ℝ^{n+1})}. As an induced S^{1}-Hopf bundle, ν^{−1}(0) has a Sasakian η - Einstein structure. This can be deformed to a Sasakian-Einstein metric, which is still a Riemannian submersion after rescaling the standard Kähler-Einstein metric of Z_{G̃r_{4}(ℝ^{n+1})} [30]. On the other hand, the composition of the fiberings

ν^{−1}(0) →^{S^{1}} Z_{G̃r_{4}(ℝ^{n+1})} →^{S^{2}} G̃r_{4}(ℝ^{n+1})

is an SO(3)-bundle which endows ν^{−1}(0) with a 3-Sasakian structure via the inversion theorem 4.6 of [10].

Proof of Theorem 3.1 (iii) and Theorem 3.2 (ii). In both cases we use the same idea: we construct a complex structure on the total space of a Riemannian submersion with fibre S^{3} and with a Sasakian structure on its base. As both Stiefel manifolds under discussion are induced Hopf S^{3}-bundles (as easily recognized by looking at them as homogeneous spaces), we may endow them naturally with metrics making these S^{3}-bundles Riemannian submersions. By naturality, these are the usual homogeneous metrics on the considered Stiefel manifolds. Now the proof follows from:

Proposition 4.1. Let π : P → B be a totally geodesic Riemannian submersion with fibre S^{3}. Suppose the base B has a Sasakian structure (φ, ξ, η, g^{B}). Then P admits a complex structure with respect to which g^{P} is Hermitian, but not Kählerian.

Proof. For any Y ∈ X (B) we let X^{∗} be its horizontal lift on P. Let ξ_{1}, ξ_{2}, ξ_{3} be the unit Killing vector fields which give the usual 3-Sasakian structure of S^{3} and η_{1}, η_{2}, η_{3} their dual w.r.t. the canonical metric of S^{3}. ξ_{i} may be viewed as vector fields on P. Let η̂_{i} be their dual forms w.r.t. the metric g^{P}. Restricted to any fibre, η̂_{i} coincide with η_{i}. The usual splitting of T P into vertical and horizontal parts may be refined to

T P = span{ξ_{1}, ξ_{2}, ξ_{3}} ⊕ span{ξ^{∗}} ⊕ H′

where H′ represents the horizontal vector fields orthogonal to ξ^{∗}. Define an almost complex structure J on P as:

• Jξ_{1} = ξ_{2}, Jξ_{2} = −ξ_{1}.

• Jξ_{3} = ξ^{∗}, Jξ^{∗} = −ξ_{3}

• JX^{∗} = (ϕX)^{∗} for any X ∈ X (B) orthogonal to ξ. To show J is integrable, we compute its Nijenhuis tensor field:

[J, J](A, B) = [A, B]+J[JA, B]+J[A, JB]−[JA, JB], A, B ∈ X (P).

We shall need the following

Lemma 4.1. [ξ_{i}, Y^{∗}] = 0 for any Y ∈ X (B) and i = 1, 2, 3.

Proof. It is enough to show that [ξ_{i}, Y^{∗}] is both horizontal and vertical. Indeed, on one hand:

dη̂_{i}(ξ_{j}, Y^{∗}) = g(φ_{i}ξ_{j}, Y^{∗}) = g(ϵ_{ijk}ξ_{k}, Y^{∗}) = 0

and also

dη̂_{i}(ξ_{j}, Y^{∗}) = −η̂_{i}([ξ_{i}, Y^{∗}]).

Hence [η_{i}, Y^{∗}] is horizontal. And on the other hand [ξ_{i}, Y^{∗}] is vertical (as the bracket of any vertical field with a projectable one, cf. [6] p.240). The conclusion follows.

We analyse separately the different possible positions of A, B, noting that, due to the tensorial character of [J, J], when dealing with horizontal vector fields it is enough to work with basic ones.

1) Clearly [J, J](ξ_{1}, ξ_{2}) = 0.

2) [J, J](ξ_{a}, ξ_{3}) = 0 for a = 1, 2 because (computation for a = 1, similar for a = 2):

[J, J](ξ_{1}, ξ_{3}) = [ξ_{1}, ξ_{3}] + J[Jξ_{1}, ξ_{3}] + J[ξ_{1}, Jξ_{3}] − [Jξ_{1}, Jξ_{3}] =

= −2ξ_{2} + J[ξ_{2}, ξ_{3}] + J[ξ_{1}, ξ^{∗}] − [ξ_{2}, ξ^{∗}] = = −2ξ_{2} + J(2ξ_{1}) = = 0

where in the middle equality the last two brackets are zero according to the lemma.

3) [J, J](ξ_{a}, ξ^{∗}) = 0 because (computation for a = 1, similar for a = 2):

[J, J](ξ_{1}, ξ^{∗}) = [ξ_{1}, ξ^{∗}] + J[Jξ_{1}, ξ^{∗}] + J[ξ_{1}, Jξ^{∗}] − [Jξ_{1}, Jξ^{∗}] =
= J[ξ_{2}, ξ^{∗}] + J[ξ_{1}, −ξ_{3}] − [ξ_{2}, −ξ_{3}] =
= J(2ξ_{2}) + 2ξ_{1} = 0

(note that some of the brackets are zero by the lemma).

4) [J, J](ξ_{a}, ξ^{∗}) = 0 because :

\begin{aligned}[J, J](ξ_{3}, ξ^{∗}) & = [ξ_{3}, ξ^{∗}] + J[Jξ_{3}, ξ^{∗}] + J[ξ_{3}, Jξ^{∗}] − [Jξ_{3}, Jξ^{∗}] = \\ & = [ξ_{3}, ξ^{∗}] + J[ξ_{2}, ξ^{∗}] + J[ξ^{∗}, ξ^{∗}] + J[ξ_{3}, −ξ_{3}] − [ξ^{∗}, −ξ_{3}] = 0, \end{aligned}

using again the lemma to cancel the first, second and third terms. For the last four cases, X ∈ X (B) will be orthogonal to ξ.

5) We compute [J, J](ξ^{∗}, X^{∗}). Here we have to observe that ξ and ξ^{∗} (resp. X and X^{∗}) being π-related, one has [ξ^{∗}, X^{∗}] = [ξ, X]^{∗}. Then:

[J, J](ξ^{∗}, X^{∗}) = [ξ, X]^{∗} + J[Jξ^{∗}, X^{∗}] + J[ξ^{∗}, JX^{∗}] − [Jξ^{∗}, X^{∗}] =

= [ξ, X]^{∗} + J[−ξ_{3}, X^{∗}] + J[ξ^{∗}, (φX)^{∗}] − [−ξ_{3}, X^{∗}] =

= [ξ, X]^{∗} + J[ξ^{∗}, (φX)^{∗}] =
= ([ξ, X] + φ[ξ, φX])^{∗}.

where again in the middle equality two terms are zero by the lemma But, on B, we have:

[ξ, X] = ∇_{ξ}X − ∇_{X}ξ = ∇_{ξ}X − φX;
[ξ, φX] = ∇_{ξ}(φX) − ∇_{(}φX)ξ =
= (∇_{ξ}φ)X + φ∇_{ξ}X − φ^{2}X =
= η(X)ξ − g^{B}(ξ, X)ξ + φ∇_{ξ}X + X − η(X)ξ =
= X + φ∇_{ξ}X − η(X)ξ;
φ[ξ, φX] = φX − ∇_{ξ}X + η(∇_{ξ}X)ξ.

Finally, by g^{B}(ξ, X) = 0, taking the derivative along ξ, using ∇_{ξ}ξ = φξ = 0, we get η(∇_{ξ}X) = g(ξ, ∇_{ξ}X) = 0. Then

φ[ξ, φX] = φX − ∇_{ξ}X,

and the proof of this case is complete.

6) [J, J](ξ_{a}, X^{∗}) = 0, a = 1, 2 taking into account the Lemma and the fact that JX^{∗} = (φX)^{∗}. Next:

7) For [J, J](ξ_{3}, X^{∗}) we compute:

[J, J](ξ_{3}, X^{∗}) = [ξ_{3}, X^{∗}] + J[ξ^{∗}, X^{∗}] + J[ξ_{3}, (φX)^{∗}] − [ξ^{∗}, (φX)^{∗}] =
= J[ξ, X]^{∗} − [ξ, (φX)^{∗}] = as above
= (φ[ξ, X] − X − φ∇_{ξ}X + η(X)ξ)^{∗} = 0.

Finally, for any X, Y ⊥ ξ,

[J, J](X^{∗}, Y^{∗}) = [φ, φ](X, Y)^{∗} = 0.

due to the normality property of the Sasakian structure of B. The compatibility of J with g^{P} is immediate. The Kähler form of (P, g^{P} , J) is

ω = dπ^{∗}η + π^{∗}η ∧ η_{3} − dη_{3},

that shows that dω ≠ 0.

Remark 4.3. A similar computation proves that L_{ξ^{∗}}J = L_{ξ_{3}}J = 0, thus ξ^{∗} and ξ_{3} are infinitesimal automorphisms of the constructed complex structure. Moreover, ξ^{∗} and ξ_{3} are also Killing fields with respect to g^{P}.

Note that the complex structure obtained in this way projects onto the complex structure underlying the Kähler structure of Gr_{2}(ℂ^{n+1}). This latter is not compatible with the quaternion Kähler structure of this Grassmannian. But it is precisely this quaternion Kähler structure which is lifted to a 3-Sasakian structure and then, by means of an appropriate circle bundle, to a hypercomplex structure on V_{2}(ℂ^{n+1}), cf. [2], [14]. Hence our complex structure is not compatible with the standard hypercomplex one of V_{2}(ℂ^{n+1}).

This completes the proof of the theorem.

Proof of Corollary 3.1. It remains only to show that the total space of an induced S^{1} Hopf bundle is minimal in (S^{2N−1}, can). In fact, more generally, in a commutative diagram

\overline{N} ⟶^{\overline{i}} \overline{M}
π_{N} ↓  ↓^{π_{M}}
N ⟶^{i} M

of Riemannian submersions π_{N}, π_{M} with totally geodesic fibres, and immersions i (resp. \overline{i}), one has that N is minimal in M if and only if \overline{N} is minimal in \overline{M}. This follows by a direct computation of the mean curvature vector fields of i and \overline{i} using Gauss’ formula of a submanifold and formula (9.25 a) in [6].

5. Further Observations

As mentioned in the introduction, the zero level set μ^{−1}(0) ⊂ ℍP^{2} is diffeomorphic to a sphere S^{5} and the projection to the reduced manifold can be identified with the Hopf fibration S^{5} → ℂP^{2}. Going to the next case n = 3, the projection μ^{−1}(0) → Gr_{2}(ℂ^{4}), now an induced Hopf fibration, can be described by looking at the families of submanifolds in Gr_{2}(ℂ^{4}) that are either Kähler-Einstein or quaternion Kähler. The Grassmannian Gr_{2}(ℂ^{4}) admits in fact, besides a natural complex Kähler structure, two distinct quaternion Kähler structures in connection with the isomorphism of vector bundles T Gr_{2}(ℂ^{4}) ≅ V ⊗ V^{⊥} and to the (almost) hypercomplex structure on the tautological vector bundle V or on its orthogonal complement V^{⊥} (cf. [26]). The families of submanifolds we want to look at on Gr_{2}(ℂ^{4}) are described as follows (cf. [7], [8], [25], [32]).

There are two families F, F′ of complex projective planes CP^{2}, CP^{2′}, a family F′′ of products ℂP^{1} × ℂP^{1}, and a family F′′′ of spheres S^{4}; they are given by:

F ={planes contained in a 3-space of ℂ^{4}},
F′ ={planes through a line of ℂ^{4}},
F′′ ={planes that are invariant for a hypercomplex
structure J of ℂ^{4}},
F′′′ ={planes given by pairs of lines in two fixed orthogonal
planes of ℂ^{4}}.

All these 4-dimensional submanifolds of Gr_{2}(ℂ^{4}) have nice intersection properties, some of which can also be formulated in the framework of projective geometry of the lines in the 3-dimensional space ℂP^{3}, the context where the Klein quadric ℂQ^{4} ⊂ ℂP^{5}, isometric to Gr_{2}(ℂ^{4}), was first introduced. Instead of listing these intersection properties on Gr_{2}(ℂ^{4}) (cf. [25], pp. 508-512, for some of them, and the remaining ones can be similarly deduced), we formulate the corresponding properties for the families of 5-dimensional submanifolds obtained at the Sasakian Einstein level as induced Hopf bundles over the members of families F,..., F′′′.

Proposition 5.1. The 9-dimensional Sasakian Einstein focal set F oc_{ℍP^{3}}ℂP^{3} ≅ μ^{−1}(0) ⊂ ℍP^{3} contains the following families of 5dimensional submanifolds, each of which fibers in circles over a complex or a quaternionic submanifold of Gr_{2}(ℂ^{4}). There are two families E, E′ of Sasakian 5-spheres S^{5}, S′^{5}, a family E′′ of Sasakian S^{3} × S^{2} and a family E′′′ of products S^{4} × S^{1}. The induced metrics on members of the families E, E′,E′′ are Sasakian η-Einstein, and can be modified to Sasakian Einstein metrics through formula (2.1). The intersection properties of these 5-dimentional submanifolds are the following.

(a) Pairs of 5-spheres in the same family intersect in a circle, and pairs of 5-spheres in different families either do not intersect or intersect in a S^{3}.

(b) Pairs of submanifolds of type S^{3} × S^{2} intersect either in a S^{3} or in a pair of disjoint circles. A submanifold of type S^{3} × S^{2} intersects a 5-sphere in a S^{3}.

(c) Any submanifold of type S^{4} ×S^{1} intersects a 5-sphere in a circle, intersects a S^{3} × S^{2} in two disjoint circles, and any pair submanifolds of type S^{4} × S^{1} intersect in two disjoint circles.

This kind of geometry of submanifolds, now described for the level set μ^{−1}(0) ⊂ ℍP^{3}, can be formulated for all the zero level sets of moment maps appearing in the diagram of corollary 3.1. There is such a level set for any odd dimension. For example, ν^{−1}(0) ⊂ ℍP^{4} is diffeomorphic to a sphere S^{7}, yielding as reduced manifold ℍP^{1} and fibering in circles over its twistor space ℂP^{3}, Kähler submanifold of the Grassmannian Gr_{2}(ℂ^{5}). Thus Sasakian (η)-Einstein submanifolds of type S^{5} and S^{3}×S^{2} can be determined in S^{7} (cf. [30]), and intersection properties like in prop. 5.1 are obtained.

The 11-dimensional example is the level set ν^{−1}(0) ⊂ ℍP^{5}, identified in theorem 3.2 with an intersection of three focal sets in ℍP^{5}. This 11dimensional manifold is diffeomorphic to the (unique) 3-Sasakian homogeneous manifold projecting in SO(3) over the Wolf space G̃r_{4}(ℝ^{6}), and this latter manifold is isometric to Gr_{2}(ℂ^{4}). Thus again the geometry of 3-Sasakian and of Sasakian (η)-Einstein submanifolds ν^{−1}(0) ⊂ ℍP^{4} is obtained from the same starting point of proposition 5.1.

Besides this kind of geometry of submanifolds, the Sasakian-Einstein level sets μ^{−1}(0) seem to share with (some of) the 3-Sasakian level sets ν^{−1}(0) a common expression of their Poincaré polynomials. We have in fact:

Proposition 5.2. The Poincaré polynomial of μ^{−1}(0) ⊂ ℍP^{n} is given by:

P oin_{μ^{−1}(0)}(t) = ∑_{i=0}^{[ \frac{n−1}{2} ]}(t^{4i} + t^{4n−3−4i})

This is obtained by the Gysin sequence of the S^{1}-bundle μ^{−1}(0) → Gr_{2}(ℂ^{n+1}), where the connecting homomorphism

H^{p}(Gr_{2}(ℂ^{n+1})) → H^{p+2}(Gr_{2}(ℂ^{n+1}))

is given by the wedge product with the Kähler form of the Grassmannian. Since this wedge product is injective up to the middle real dimension p + 2 = 2n − 2 (cf. [21], lemma 3.1, for the similar 3-Sasakian situation), the Gysin sequence reduces to a series of short exact sequences up to

H^{2n−4}(Gr_{2}(ℂ^{n+1})) → H^{2n−2}(Gr_{2}(ℂ^{n+1})) → H^{2n−2}(μ^{−1}(0)).

This allows to compute the Betti numbers of (μ^{−1}(0)) by differences of consecutive even Betti numbers in Gr_{2}(ℂ^{n+1}). The Poincaré polynomial of Gr_{2}(ℂ^{n+1}) is well known (for example [9], p.292) and by writing it as:

P oin_{Gr_{2}(ℂ^{n+1})} = (1 + t^{2} + ... + t^{2n−2})(1 + t^{4} + t^{4m−4}), (for n + 1 = 2m)

P oin_{Gr_{2}(ℂ^{n+1})} = (1 + t^{4} + ... + t^{4m−4})(1 + t^{2} + t^{2n}), (for n + 1 = 2m + 1)

the conclusion is easily obtained.

The following table contains the Poincaré polynomial of μ^{−1}(0) ⊂ ℍP^{n} for low values of n

\begin{matrix}n = 3 & 1 + t^{4} + t^{5} + t^{9} \\ n = 4 & 1 + t^{4} + t^{9} + t^{13} \\ n = 5 & 1 + t^{4} + t^{8} + t^{9} + t^{13} + t^{17} \\ n = 6 & 1 + t^{4} + t^{8} + t^{13} + t^{17} + t^{21} \\ n = 7 & 1 + t^{4} + t^{8} + t^{12} + t^{13} + t^{17} + t^{21} + t^{25} \\ n = 8 & 1 + t^{4} + t^{8} + t^{12} + t^{17} + t^{21} + t^{25} + t^{29} \end{matrix}

The Poincaré polynomial of μ^{−1}(0) can be compared with that of ν^{−1}(0), computed as for the homogeneous 3-Sasakian manifold SO(n+ 1)/(SO(n − 3) × Sp(1)). The latter has two different expressions, according to whether n + 1 is even or odd (see [21] or [11], p. 28). For odd values of n + 1 = 2k + 3 this expression is:

P oin_{ν^{−1}(0)}(t) = ∑_{i=0}^{k−1}(t^{4i} + t^{8k−1−4i}),

and taking account of the dimensions, this is the same formula given in proposition 5.2 for P oin_{μ^{−1}(0)}(t).

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University of Bucharest, Faculty of Mathematics, 14 Academiei str., 70109 Bucharest, Romania

  E-mail address: lornea@geo.math.unibuc.ro

Università degli Studi di Roma ”La Sapienza”, Piazzale Aldo Moro 2, I-00185 Roma, Italia

  E-mail address: piccinni@mat.uniroma1.it