On Degenerate $3$-$(\alpha,\delta)$-Sasakian Manifolds

We propose a new method to construct degenerate $3$-$(\alpha,\delta)$-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperk\"ahler manifolds. Subsequently, we study homogeneous degenerate $3$-$(\alpha, \delta)$-Sasakian manifolds and prove that no non-trivial compact examples exist as well as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.


Introduction
Almost (3-)contact metric structures and special cases thereof are among the most wellstudied types of geometries in odd dimensions. Arguably the most famous examples are (3-)Sasakian manifolds. Agricola and Dileo have proposed 3-(α, δ)-Sasakian structures as a generalization of the latter which retain favorable properties like hypernormality or the existence of a connection with skew torsion adapted to the geometry [AD].
These classes of manifolds depend on two real parameters α, δ ∈ R, α = 0, and can be grouped into degenerate and nondegenerate type according to whether δ = 0 or δ = 0.
In this article, we focus on the degenerate case where the three contact metric structures are more independent of each other in the sense that their Reeb vector fields commute.
As will be explained in Section 3, this allows us to construct examples of such manifolds via fiber products of Boothby-Wang bundles over hyperkähler manifolds. This can be viewed as a counterpart to obtaining 3-Sasakian manifolds from the classical Konishi bundle over positive quaternionic Kähler spaces.
Agricola, Dileo and Stecker have shown that all 3-(α, δ)-Sasakian manifolds locally submerge onto a quaternionic Kähler space whose curvature depends on the sign of αδ [ADS,Theorem 2.2.1]. This result motivated them to construct plentiful examples of homogeneous non-degenerate 3-(α, δ)-Sasakian manifolds over quaternionic Kähler spaces of non-vanishing curvature [ADS,Section 3]. To some extent, this property also explains why it could be expected that there are rather few homogeneous degenerate 3-(α, δ)-Sasakian manifolds. This is because, ideally, such spaces would submerge onto homogeneous hyperkähler manifolds which are known to be completely flat, i.e. products of tori and Euclidean spaces [AK75]. Compared to the non-degenerate case, however, it is less clear if the corresponding hyperkähler space will actually be a manifold globally.
We identify two such scenarios where the base space is indeed a manifold. However, we show that there are restrictions beyond those imposed by [AK75] on the base.
Unlike in the non-degenerate case, the Reeb vector fields are now elements of the automorphism algebra of a homogeneous degenerate 3-(α, δ)-Sasakian manifold because they commute with one another. In Section 4, we show that they comprise the center of the automorphism algebra and conclude that they generate a closed subgroup of the automorphism group. Combining this with classical results due to Alekseevskii [Alek] as well as Conner and Raymond [CR], we prove that the only connected compact homogeneous degenerate 3-(α, δ)-Sasakian manifold is the trivial example of the 3-torus T 3 (Section 5). Finally, we study simply connected nilpotent Lie groups with a leftinvariant degenerate 3-(α, δ)-Sasakian structure in Section 6. Again using traditional results by Wilson [Wil] as well as Milnor [Mil], we show there is exactly one family of such spaces, the quaternionic Heisenberg groups. This parallels a result about nilpotent Sasakian Lie groups by Andrada, Fino and Vezzoni [AFV,Theorem 3.9].
To sum up, our findings paint the somewhat curious picture that there are no nontrivial compact homogeneous examples in the degenerate 3-(α, δ)-Sasakian category, de-spite there being both interesting non-compact homogeneous (quaternionic Heisenberg group) and compact inhomogeneous cases (3-Boothby-Wang bundles over compact inhomogeneous hyperkähler manifolds, see Section 3). Qualitatively, this geometry is in stark contrast to the classical 3-Sasakian one where homogeneous examples are automatically compact and there exist as many as compact simple Lie groups ( [BGM, Theorem C], see also [GRS]).

Acknowledgements:
The second named author is supported by the German Academic Scholarship Foundation. The authors thank Ilka Agricola for fruitful discussions about the subject.

Preliminaries
Let (M 2n+1 , g) be a Riemannian manifold of odd dimension. An almost contact metric structure (ξ, η, ϕ) is defined by a unit length vector field ξ ∈ X(M ), its metric dual η ∈ Ω 1 (M ) and an almost hermitian structure ϕ ∈ End(T M ) on ker η satisfying For an almost contact metric manifold (M, g, ξ, η, ϕ), we let Φ ∈ Ω 2 (M ) denote the fundamental 2-form given by Φ(X, Y ) = g(X, ϕY ). Now suppose that (M 4n+3 , g) is equipped with three almost contact metric structures 2,3 which are compatible in the sense that for any even permutation (i j k) of (1 2 3). Then, we call (M, g, ξ i , η i , ϕ i ) i=1,2,3 an almost 3-contact metric manifold. By construction, the Reeb vector fields ξ i are orthogonal and form a 3-dimensional distribution V = ξ 1 , ξ 2 , ξ 3 . We call this distribution vertical and its orthogonal complement H = i ker η i horizontal.
In case α = δ = 1, the differential condition is just dη i = 2Φ i , which corresponds to standard 3-Sasakian manifolds. We will recall only those results about 3-(α, δ)-Sasakian manifolds which we need here. For a more complete survey we refer to the introductory article [AD] as well as the third author's thesis [Ste].
Among 3-(α, δ)-Sasakian manifolds we differentiate two subclasses due to the behavior of their Reeb vector fields. .) The Reeb vector fields are Killing and for any even permutation (i j k) of (1 2 3) they satisfy the commutator relations In particular, the vertical distribution is integrable.
If δ = 0, then the Reeb vector fields span a Lie algebra isomorphic to su(2). For δ = 0 on the other hand, the vertical Lie algebra is Abelian.

A Constructive Result
Whereas plentiful examples of non-degenerate 3-(α, δ)-Sasakian structures are known , see also Example 6.1). The following construction starting with a hyperkähler manifold is an excerpt form the third author's thesis [Ste].
Let (N, g, I 1 , I 2 , I 3 ) be a hyperkähler manifold. Suppose that the fundamental two- Then we obtain a Boothby-Wang bundle [BW,Theorem 3]. In particular, these bundles have Chern class [ω i ] and the total spaces P i are equipped with Sasaki structures

Now consider the product bundle
If we denote the fiber of P i over x ∈ N by (P i ) x := π −1 i ({x}) ∼ = S 1 , then the bundle M π − → N can be seen as the fiber bundle with fiber (P 1 ) x × (P 2 ) x × (P 3 ) x over any given as ξ i generates the fiber of P i . The construction as a fiber product bundle ensures that the Reeb vector fields ξ i are linearly independent and [ξ i , ξ j ] = 0, as it ought to be for degenerate 3-(α, δ)-Sasakian manifolds.
Remark 3.2. The T 3 -bundle M also appears in Fowdar's paper [Fow] in his construction of quaternionic Kähler metrics on R × M albeit without the geometric structure, see also [Cor,Lemma 2.1].
In [For] Foreman obtains a so-called complex contact manifold by constructing a T 2bundle over N in similar fashion. His construction imposes a less restrictive assumption than hyperkähler on N . However, in this special case the complex T 2 -bundle he considers can be obtained from our construction as the quotient by one of the Reeb vector fields.
This should be considered as the analogue of the twistor space over a positive quaternionic Kähler manifold and the 3-Sasakian space above (cmp. [For2]).
Proof. Observe that (g, η i , ξ i , ϕ i ) extends the almost contact metric structure on The last identity to check is the differential condition (2). As (ξ i , η i ,φ i ) defines a Sasakian structure on P i we have g i (X,φ i Y ) = dη i (X, Y ) for vertical vectors X, Y ∈ T p i P i . This remains true for X, Y ∈ H ⊂ T p M as ϕ i is justφ i on H. If either vector of X, Y lies in V, the left hand side of (2) has to vanish. We have as η i is defined on the factor P i of the product P . In particular, it is invariant under ξ j for j = i. Finally, ξ i dη i = 0 from the Sasakian condition on P i .
We call this bundle the 3-Boothby-Wang bundle over N . As shown in [ADS, Theorem 2.2.1], locally any degenerate 3-(α, δ)-Sasakian manifold maps onto a hyperkähler space via the so-called canonical submersion. The above construction describes the converse starting from a hyperkähler manifold. By construction, the Reeb vector fields span the fiber of the bundle T 3 → M π − → N . Therefore the canonical submersion agrees globally with π mapping onto a hyperkähler manifold, namely the initial manifold N . Observe that starting with a compact hyperkähler manifold in this way one obtains a compact degenerate 3-(α, δ)-Sasakian manifold.

Remark 3.3.
In [Cor] Cortés shows that non-flat compact hyperkähler manifolds of arbitrary dimension with integral Kähler forms exist. This implies that there are compact degenerate 3-(α, δ)-Sasaki manifolds that are not quotients of a quaternionic Heisenberg group.
Example 3.4. As a toy example, we could consider N to be a single point, meaning that M is just T 3 . It turns out that this is the only connected compact degenerate 3-(α, δ)-Sasakian manifold in dimension three, which can be seen as follows: If

Homogeneous Spaces
From now on, let (M 4n+3 , g, ξ i , η i , ϕ i ) i=1,2,3 be a connected degenerate 3-(α, δ)-Sasakian Let us now assume that M is homogeneous, i.e. the identity component G := Aut 0 (M ) of its group Aut(M ) of 3-(α, δ)-Sasakian automorphisms acts transitively. The Lie algebra g is (anti-)isomorphic to the space aut(M ) of all complete Killing vector fields on M which commute with the Reeb vector fields ξ 1 , ξ 2 , ξ 3 via the map g → aut(M ), X → X, where X denotes the fundamental vector field of the left G-action.
Since the automorphism group acts transitively, the Reeb vector fields are complete. Unlike in the 3-Sasakian and more generally the non-degenerate 3-(α, δ)-Sasakian case, the Reeb vector fields commute with each other, so we have ξ 1 , ξ 2 , ξ 3 ∈ aut(M ). In fact, the corresponding elements X 1 , X 2 , X 3 ∈ g (i.e. X i = ξ i ) clearly even lie in the center Z(g) of g. We will show that g contains no other central elements using techniques very similar to those in [GRS, Section 6].
Proof. Evaluating the exterior derivative gives and the Leibniz rule for the Lie derivative implies where L X η i = 0 because G acts by 3-(α, δ)-Sasakian automorphisms.

Lemma 4.2. A vector field of the form
, then for j = 1, 2, 3 and W ∈ X(M ), we have: Since M is connected, f 1 , f 2 , f 3 have to be constant.
Proof. Let X ∈ Z(g) and choose some p ∈ M and Y ∈ g such that Y p = ϕ 1 X p (which is possible, since G acts transitively). Then, by virtue of Lemma 4.1: where v H denotes the horizontal component of a tangent vector v ∈ T M . Since α = 0 and p ∈ M was arbitrary, the vector field X has to be completely vertical and therefore of the form described in Lemma 4.2. Consequently, X = 3 i=1 a i ξ i for some a 1 , a 2 , a 3 ∈ R and X = 3 i=1 a i X i ∈ k.
Corollary 4.4. The connected subgroup K ⊂ G with Lie algebra k is closed.
Proof. Since G is connected, the Lie algebra of Z(G) is given by Z(g). Thus, Proposition 4.3 implies that K = Z 0 (G), the identity component of the center. Since Z(G) is closed in G and Z 0 (G) is closed in Z(G), it follows that K is closed in G.

Compact Homogeneous Spaces
Let us now additionally assume that M is compact. Then its isometry group Isom(M ) as well as the closed subgroups Aut(M ) and Aut 0 (M ) = G are also compact, as is K by Concretely, the hyperkähler forms are induced by dη i which are basic by (2). Since Theorem 5.2. The only connected compact homogeneous degenerate 3-(α, δ)-Sasakian manifold is T 3 .
Proof. Let L be the kernel of the G-action on the hyperkähler quotient space M/K.
Then, G/L is a connected compact Lie group which acts effectively on a closed aspherical manifold (i.e. a manifold whose universal cover is contractible). It follows from the work of Conner and Raymond that G/L is also a torus [CR,Theorem 5.6].
Since L is a normal subgroup of G, its Lie algebra l is an ideal in g. Because G is compact, we may choose an Ad(G)-invariant (and thus also ad(g)-invariant) inner product on g and consider the complementary ideal m := l ⊥ . Again by compactness of G, we have g = [g, g] ⊕ k and [g, g] = k ⊥ (independently of the choice of the inner product). Since K ⊂ L, it follows that m is an ideal inside the semisimple algebra [g, g] and is therefore itself semisimple. But on the other hand, m is isomorphic to the Lie algebra g/l of G/L, which is Abelian. Consequently, m = {0}.
It follows that G/L has to be the trivial torus, meaning that G acts trivially on M/K.
Since this action is also transitive, M/K is a singleton and M is 3-dimensional. By virtue of Example 3.4, M is isomorphic to T 3 .

Nilpotent Lie Groups
In this section, we want to investigate simply connected nilpotent Lie groups with a leftinvariant degenerate 3-(α, δ)-Sasakian structure. Let us first recall the specific example of the quaternionic Heisenberg group, which was first described as a naturally reductive space in [AFS] and later identified as a homogeneous degenerate 3-(α, δ)-Sasakian manifold in [AD].
Example 6.1. The quaternionic Heisenberg group of dimension 4n + 3 is the simply connected Lie group H determined by the following Lie algebra h: As a set, where Z is the span of the imaginary quaternions i, j, k, which we will denote by ξ 1 , ξ 2 , ξ 3 .
We define an inner product g on h by requiring that ξ 1 , ξ 2 , ξ 3 are orthonormal, g| H n ×H n is the standard Euclidean inner product on H n and g| Z×H n = 0. Let η i be the g-dual 1form of ξ i and ϕ i be the endomorphism of h such that ϕ i ξ j = ξ k and ϕ i | H n is quaternionic multiplication from the left by ξ i . Finally, the Lie bracket on h is determined by the conditions Z(h) = Z, [H n , H n ] ⊂ Z (implying that h is 2-step nilpotent) and As shown in [AFS,Subsection 2.2] and [AD,Example 2.3.2], the corresponding leftinvariant tensor fields (g, ξ i , η i , ϕ i ) i=1,2,3 constitute a homogeneous degenerate 3-(α, δ)-Sasakian structure on H with α = 1/2. As always, one can modify this structure (and in particular the value of α) by applying an H-homothetic deformation [AD,Definition 2.3.1]. In fact, the presentation of the quaternionic Heisenberg group in [AFS] and [AD] already incorporates an H-homothetic deformation with parameters a = 1, b = λ 2 − 1 and c = λ, where λ > 0.
From now on, let N 4n+3 denote a simply connected nilpotent Lie group with a leftinvariant degenerate 3-(α, δ)-Sasakian structure (g, ξ i , η i , ϕ i ) i=1,2,3 . The Lie algebra n = H ⊕ V decomposes both as vector spaces and as left-invariant distributions inside the tangent bundle.
Proof. The derivative of the left-invariant 1-form η 1 is given by for all X, Y ∈ n. Thus if X ∈ Z(n), then Condition (2) implies that for Y := ϕ 1 X: Proposition 6.3. Z(n) = V and n is 2-step nilpotent.
for all X ∈ n, so V = Z(n). Moreover, N/V is nilpotent and Ricci-flat, so a result by Milnor implies that its Lie algebra n/V = n/Z(n) is Abelian [Mil,Theorem 2.4]. Consequently, [n, n] ⊂ Z(n), so n is 2-step nilpotent.
Theorem 6.4. The only simply connected nilpotent Lie groups with a left-invariant degenerate 3-(α, δ)-Sasakian structure are the quaternionic Heisenberg groups, endowed with the structure described in Example 6.1.
Proof. Since N is simply connected, it suffices to construct a Lie algebra isomorphism ψ : n → h which preserves the structure tensors. First, let ψ map the Reeb vector fields of n to those of h. Both H ⊂ n and H n ⊂ h are left quaternionic vector spaces with compatible inner products, so they admit orthonormal bases of the form (e 1 , ϕ 1 e 1 , ϕ 2 e 1 , ϕ 3 e 1 , . . . , e n , ϕ 1 e n , ϕ 2 e n , ϕ 3 e n ). Let ψ map such a basis of H to a corresponding basis of H n .
From these definitions, it is clear that ψ is a linear isomorphism which respects the structure tensors, so it only remains to check that ψ preserves the Lie bracket. To this end, first note that by the previous proposition: ψ(Z(n)) = ψ(V) = Z = Z(h Remark 6.5. This theorem is very analogous to a result by Andrada, Fino and Vezzoni: They showed that the only simply connected nilpotent Lie groups with a leftinvariant Sasakian structure are the odd-dimensional Heisenberg groups [AFV,Theorem 3.9]. Finally, however, we would like to warn the reader about the somewhat counterintuitive fact that, at least in the Sasakian realm, there exist (non-nilpotent) Lie groups with a left-invariant structure where the Reeb vector field does not lie in the center of the Lie algebra.