Remarkable Classes of Almost 3-Contact Metric Manifolds

We introduce a new class of almost 3-contact metric manifolds, called 3-(0, δ)-Sasaki manifolds. We show fundamental geometric properties of these manifolds, analyzing analogies and differences with the known classes of 3-(α, δ)-Sasaki (α 6= 0) and 3-δ-cosymplectic manifolds.


Introduction
An almost 3-contact metric manifold is a (4n + 3)-dimensional differentiable manifold M endowed with three almost contact metric structures (ϕ , ξ i , η i , g), i = 1, 2, 3, sharing the same Riemannian metric g and satisfying suitable compatibility conditions, equivalent to the existence of a sphere of almost contact metric structures. In the recent paper [1], new classes of almost 3-contact metric manifolds were introduced and studied. The first remarkable class is given by 3-(α, δ)-Sasaki manifolds defined as almost 3-contact metric manifolds (M, ϕ i , ξ i , η i , g) such that for every even permutation (i, j, k) of (1, 2, 3). This is a generalization of 3-Sasaki manifolds, which correspond to the values α = δ = 1. A second class introduced in [1] is given by 3-δ-cosymplectic manifolds defined by the conditions generalizing 3-cosymplectic manifolds which correspond to the value δ = 0.
In the present paper we will introduce a third class of almost 3-contact metric manifolds, which is in fact a second (and alternative) generalization of 3-cosymplectic manifolds. We will consider almost 3-contact metric manifolds whose structure tensor fields satisfy for every even permutation (i, j, k) of (1, 2, 3). When δ = 0 we recover a 3-cosymplectic manifold. We will call these manifolds 3-(0, δ)-Sasaki manifolds. The choice of name is due to the fact that for a 3-(α, δ)-Sasaki manifold, Equation (1) implies so that the two equations in (2) formally correspond to (1) and (3) with α = 0, although in this case the second equation is no more a consequence of the first one. In fact the two conditions in (2) are not completely independent (see Remark 1). Examples of 3-(0, δ)-Sasaki structures can be defined on the semidirect products SO(3) R 4n . The structure on these Lie groups was introduced in [2] as an example of canonical abelian almost 3-contact implying that ϕξ = 0, η • ϕ = 0, and ϕ has rank 2n. The tangent bundle of M splits as TM = H ⊕ ξ , where H is the 2n-dimensional distribution defined by H = Im(ϕ) = Ker(η). The vector field ξ is called the characteristic or Reeb vector field. On the product manifold M × R one can define an almost complex structure J by J X, f d dt = ϕX − f ξ, η(X) d dt , where X is a vector field tangent to M, t is the coordinate of R and f is a C ∞ function on M × R. If J is integrable, the almost contact structure is said to be normal and this is equivalent to the vanishing of the tensor field N ϕ := [ϕ, ϕ] + dη ⊗ ξ, where [ϕ, ϕ] is the Nijenhuis torsion of ϕ [3]. More precisely, for any vector fields X and Y, N ϕ is given by It is known that any almost contact manifold admits a compatible metric, that is a Riemannian metric g such that g(ϕX, ϕY) = g(X, Y) − η(X)η(Y) for every X, Y ∈ X(M). Then η = g(·, ξ) and H = ξ ⊥ . The manifold (M, ϕ, ξ, η, g) is called an almost contact metric manifold. The associated fundamental 2-form is defined by Φ(X, Y) = g(X, ϕY).
We recall some remarkable classes of almost contact metric manifolds.
• An α-contact metric manifold is defined as an almost contact metric manifold such that When α = 1, it is called a contact metric manifold; the 1-form η is a contact form, that is η ∧ (dη) n = 0 everywhere on M. An α-Sasaki manifold is a normal α-contact metric manifold, and again such a manifold with α = 1 is called a Sasaki manifold.

•
An almost cosymplectic manifold is defined as an almost contact metric manifold such that dη = 0, dΦ = 0; if furhermore the structure is normal, M is called a cosymplectic manifold. It is worth remarking that some authors call these manifolds almost coKähler and coKähler, respectively ( [4]). • A quasi-Sasaki manifold is a normal almost contact metric manifold with closed 2-form Φ. This class includes both α-Sasaki and cosymplectic manifolds. The Reeb vector field of a quasi-Sasaki manifold is always Killing.
Both α-Sasaki manifolds and cosymplectic manifolds can be characterized by means of the Levi-Civita connection ∇ g . Indeed, one can show that an almost contact metric An almost contact metric manifold is cosymplectic if and only if ∇ g ϕ = 0; further, this is equivalent to requiring the manifold to be locally isometric to the Riemannian product of a real line (tangent to the Reeb vector field) and a Kähler manifold.
For a comprehensive introduction to almost contact metric manifolds we refer to [3]. For Sasaki geometry, we also recommend the monograph [5]; the survey [4] covers fundamental properties and recent results on cosymplectic geometry.
An almost 3-contact manifold is a differentiable manifold M of dimension 4n + 3 endowed with three almost contact structures (ϕ i , ξ i , η i ), i = 1, 2, 3, satisfying the following relations, for any even permutation (i, j, k) of (1, 2, 3) ( [3]). The tangent bundle of M splits as In particular, H has rank 4n. We call any vector belonging to the distribution H horizontal and any vector belonging to the distribution V vertical. The manifold is said to be hypernormal if each almost contact structure (φ i , ξ i , η i ) is normal. In [6] it was proved that if two of the almost contact structures are normal, then so is the third.
The existence of an almost 3-contact structure is equivalent to the existence of a sphere {(ϕ x , ξ x , η x )} x∈S 2 of almost contact structures such that for every x, y ∈ S 2 , where · and × denote the standard inner product and cross product on R 3 . In fact, if the structure is hypernormal, then every structure in the sphere is normal ( [7]). Any almost 3-contact manifold admits a Riemannian metric g which is compatible with each of the three structures. Then M is said to be an almost 3-contact metric manifold with structure (ϕ i , ξ i , η i , g), i = 1, 2, 3. For ease of notation, we will denote an almost 3-contact metric manifold by (M, ϕ i , ξ i , η i , g), omitting i = 1, 2, 3. The subbundles H and V are orthogonal with respect to g and the three Reeb vector fields ξ 1 , ξ 2 , ξ 3 are orthonormal. In fact, the structure group of the tangent bundle is reducible to Sp(n) × {1} [8].
Given an almost 3-contact metric structure (ϕ i , ξ i , η i , g), an H-homothetic deformation is defined by where a, b, c are real numbers such that a > 0, c 2 = a + b > 0, ensuring that (ϕ i , ξ i , η i , g ) is an almost 3-contact metric structure. In particular, the fundamental 2-forms Φ i and Φ i associated to the structures are related by where (i, j, k) is an even permutation of (1, 2, 3). An almost 3-contact metric manifold is called is α-Sasaki for all i = 1, 2, 3 , i.e. the structure is hypernormal and when α = 1, it is a 3-Sasaki manifold; is cosymplectic for all i = 1, 2, 3, i.e. the structure is hypernormal and is quasi-Sasaki; this class includes both 3-α-Sasaki and 3-cosymplectic manifolds.
In [1] the new classes of 3-(α, δ)-Sasaki manifolds and 3-δ-cosymplectic manifolds were introduced, generalizing the classes of 3-α-Sasaki and 3-cosymplectic manifolds, respectively. We will review the definitions and the basic properties of these manifolds in the next section. For both these two classes the hypernormality is a consequence of the defining structure equations for the manifolds, thus generalizing the analogous results for 3-Sasaki and 3-cosymplectic manifolds. This is obtained by using the following Lemma: be an almost 3-contact metric manifold. Then the following formula holds ∀X, Y, Z ∈ X(M): In the following we will be concerned with various classes of almost 3-contact metric manifolds where the three Reeb vector fields are all Killing. In this case one can show that there exists a function δ ∈ C ∞ (M) such that where rst is the totally skew-symmetric symbol, or equivalently dη r (ξ s , ξ t ) = −2δ rst . We call δ a Reeb commutator function, we refer to [1] for more information on this notion.
When α = δ = 1, we have a 3-contact metric manifold, and hence a 3-Sasaki manifold by Kashiwada's theorem [15]. In the following, when considering 3-(α, δ)-Sasaki manifolds we will always mean α = 0. As an immediate consequences of the definition one obtains the following properties:

1.
Each ξ i is an infinitesimal automorphism of the distribution H, i. e.
The constant δ is the Reeb commutator function; 3.
The differentials dΦ i are given by .
Applying Lemma 1 one shows the following In particular, a 3-(α, δ)-Sasaki manifold with α = δ is 3-α-Sasaki. It can be also shown that the vertical distribution of any 3-(α, δ)-Sasaki manifold is integrable with totally geodesic leaves and each Reeb vector field ξ i is Killing.
The following Theorem regarding the transverse geometry with respect to the vertical foliation of a 3-(α, δ)-Sasaki manifold is proved in [17]: Theorem 2. Any 3-(α, δ)-Sasaki manifold M admits a locally defined Riemannian submersion π : M → N along its horizontal distribution H such that N carries a quaternionic Kähler structure given byφ where s : U → M is any local smooth section of the Riemannian submersion. The covariant derivatives of the almost complex structuresφ i are given by The scalar curvature of the base space N is 16n(n + 2)αδ.
Notice that, by Theorem 2, a non-degenerate 3-(α, δ)-Sasaki manifold locally fibers over a quaternionic Kähler space of positive or negative scalar curvature, according to αδ > 0 or αδ < 0, respectively. In [17] a systematic study of homogeneous non-degenerate 3-(α, δ)-Sasaki manifolds has been carried out, obtaining a complete classification in the positive case, where the base space of the homogeneous fibration turns out to be a symmetric Wolf space. In the case αδ < 0, one can provide a general construction of homogeneous 3-(α, δ)-Sasaki manifolds fibering over nonsymmetric Alekseevsky spaces.
We recall now the definition and some basic facts on 3-δ-cosymplectic manifolds.
When δ = 0, the fact that the forms η i and Φ i are all closed implies that the structure is hypernormal ( [16], Theorem 4.13). In fact this immediately follows from (9). Therefore, a 3-δ-cosymplectic manifold with δ = 0 is 3-cosymplectic. In particular, it is 3-quasi-Sasaki and the Reeb vector fields are all Killing. The local structure of these manifolds is described by the following: . Any 3-cosymplectic manifold of dimension 4n + 3 is locally the Riemannian product of a hyper-Kähler manifold of dimension 4n and a 3-dimensional flat abelian Lie group.
As a consequence, since every hyper-Kähler manifold is Ricci flat, even the Riemannian Ricci tensor of any 3-cosymplectic manifold vanishes.
As regards 3-δ-cosymplectic manifolds with δ = 0, even in this case one can show that the structure is hypernormal, the Reeb vector fields are Killing, and the manifold locally decomposes as a Riemannian product [1]. In particular, Proposition 2. Any 3-δ-cosymplectic manifold with δ = 0 is locally the Riemannian product of a hyper-Kähler manifold and a 3-dimensional Lie group isomorphic to SO(3), with constant curvature δ 2 . Consequently, the Riemannian Ricci tensor is In both cases, i.e., δ = 0 or δ = 0, the hyper-Kähler manifold is tangent to the horizontal distribution, while the 3-dimensional Lie group is tangent to the vertical distribution. In fact, examples of these manifolds can be obtained taking Riemannian products N × G, where (N, J i , h), i = 1, 2, 3, is a hyper-Kähler manifold, and G is a 3-dimensional Lie group, which is either abelian, or isomorphic to SO(3). If ξ 1 , ξ 2 , ξ 3 are generators of the Lie algebra g of G, satisfying [ξ i , ξ j ] = 2δξ k , δ ∈ R, then one can define in a natural manner an almost 3-contact metric structure (ϕ i , ξ i , η i , g) on the product N × G, setting and g the product metric of h and the left invariant Riemannian metric on G with respect to which ξ 1 , ξ 2 , ξ 3 are an orthonormal basis of g. For a comparison with the class of 3-(0, δ)-Sasaki manifolds, which will be introduced in the next section, it is worth remarking that for a 3-δ-cosymplectic manifold (M, ϕ i , ξ i , η i , g) the Lie derivatives of the structure tensor fields ϕ i , i = 1, 2, 3 with respect to the Reeb vector fields are given by for every i, j = 1, 2, 3. Indeed, in a 3-δ-cosymplectic manifold the Levi-Civita connection satisfies ([1], Proposition 2.1. is an even permutation of (1, 2, 3) and X ∈ X(M). Therefore, In the same way,
The horizontal distribution H is integrable;

3.
The constant δ is the Reeb commutator function.

Remark 1.
In case δ = 0, the two equations in (12) are not completely independent. Indeed, if one assumes dΦ i = −2γ(η j ∧ Φ k − η k ∧ Φ j ), γ ∈ R * , differentiating this equation, and combining with dη i = −2δη j ∧ η k , a straightforward computation gives γ = δ. Thus, there is no freedom for the choice of constant in the second equation.

Example 1.
Consider the abelian Lie algebra R 4n spanned by vectors v r , v n+r , v 2n+r , v 3n+r , r = 1, . . . , n, and endowed with the hypercomplex structure {J 1 , J 2 , J 3 } defined by for every even permutation (i, j, k) of (1, 2, 3). Let us consider also the Lie algebra so(3) spanned by ξ 1 , ξ 2 , ξ 3 with Lie brackets given by [ξ i , ξ j ] = 2δξ k , δ = 0. Let ρ be the representation of so(3) on R 4n given by On the Lie algebra g = so(3) ρ R 4n on can define in a natural way an almost 3-contact metric structure (ϕ i , ξ i , η i , g), with and where g is the inner product such that the vectors ξ i , v l , i = 1, 2, 3, l = 1, . . . , 4n are orthonormal. In particular, the non zero brackets on g are given by The representation ρ : so(3) → gl(4n, R) can be integrated to a representationρ : SO(3) → GL(4n, R). Therefore, identifying R 4n with H n in a natural way, the simply connected Lie group G = SO(3) ρ H n , with Lie algebra g, admits a left invariant almost 3-contact metric structure (ϕ i , ξ i , η i , g). One can easily verify that this structure satisfies (12).

Remark 2.
For more details on the above example we refer to [2], where g is described as a remarkable example of a Lie algebra endowed with an abelian almost 3-contact metric structure. In fact, the structure defined on g belongs to the class of canonical abelian structures, so that the Lie group G admits a unique metric connection with totally skew symmetric torsion ∇ such that for every vector field X and for every even permutation (i, j, k) of (1, 2, 3). The torsion of the canonical connection ∇ is T = 2δη 1 ∧ η 2 ∧ η 3 , which satisfies ∇T = 0.
It is also shown in [2] that the Lie group G admits co-compact discrete subgroups, so that the corresponding compact quotients admit almost 3-contact metric structures of the same type. Proof. In order to compute the tensor fields N ϕ i , we apply Lemma 1. We always denote by X, Y, Z horizontal vector fields and by (i, j, k) an even permutation of (1, 2, 3).
Proof. Since the structure is hypernormal, by ([3], Lemma 6.1), the Levi-Civita connection satisfies Further, an easy computation (see [1]) shows that for every cyclic permutation (i, j, k) of (1, 2, 3), Then, using the second equation in (12) and the above equations, we have On the other hand, again using the first equation in (12), we obtain Inserting the above computations in (17), we conclude that which implies (14). As regards the proof (15), applying (14) for Y = ξ i , we get Applying ϕ i on both hand-sides, we obtain (15). Equations (16) are immediate consequences of (15). Furthermore, we also get for every X, Y ∈ X(M). Since ∇ g ξ i is skew-symmetric, ξ i is Killing.
be a 3-(0, δ)-Sasaki manifold. Then for every even permutation (i, j, k) of (1, 2, 3) we have Proof. For the first Lie derivative, notice that by (14) we have ∇ g ξ i ϕ i = 0. Then, applying also (15), for every vector field X we have Now, using (14) for the covariant derivative ∇ g ϕ j , for every vector field Y, we have Therefore, applying also (15), we get Then both the horizontal and the vertical distribution are integrable with totally geodesic leaves. Each leaf of the vertical distribution is locally isomorphic to the Lie group SO(3), with constant sectional curvature δ 2 ; each leaf of the horizontal distribution is endowed with a hyper-Kähler structure. Consequently, the Riemannian Ricci tensor of M is given by Proof. We already know that the horizontal distribution H is integrable. From (15), for every X, Y ∈ Γ(H) and i = 1, 2, 3, we have so that the distribution H has totally geodesic leaves. Furthermore, Equation (16) implies that the vertical distribution V is also integrable with totally geodesic leaves. In particular [ξ i , ξ j ] = 2δξ k for an even permutation (i, j, k) of (1, 2, 3), so that the leaves of V are locally isomorphic to the Lie group SO(3) and have constant sectional curvature δ 2 . On each leaf of the horizontal distribution H one can consider the almost hyper-Hermitian structure defined by (J i := ϕ i | H , g), which turns out to be hyper-Kähler due to (14). Consequently, M is locally the Riemannnian product of 3-dimensional sphere of curvature δ 2 and a 4ndimensional manifold M , which is endowed with a hyper-Kähler structure. Since any hyper-Kähler manifold is Ricci flat, we obtain that the Riemannian Ricci tensor of M is given by (19).

Remark 3.
From Theorem 3 it follows that any 3-(0, δ)-Sasaki manifold is locally isometric to the Riemannnian product of 3-dimensional sphere and a 4n-dimensional manifold M , which is endowed with a hyper-Kähler structure. We recall that 3-δ-cosymplectic manifolds are also locally isometric to the Riemannian product of a 3-dimensional sphere of constant curvature δ 2 and a hyper-Kähler manifold. Nevertheless, there is a difference between the two geometries. Looking at the transverse geometry of the foliation defined by the vertical distribution V, in both cases the Riemannian metric g is projectable, being the vector fields ξ i , i = 1, 2, 3, all Killing. In the case of 3-δ-cosymplectic manifolds, each tensor field ϕ i is also projectable, as by (11), the Lie derivatives with respect to the Reeb vector fields satisfy (L ξ i ϕ j )X = 0 for every i, j = 1, 2, 3 and for every horizontal vector field X. In the case of 3-(0, δ)-Sasaki manifolds, owing to (18), the tensor fields are not projectable. Nevertheless, taking into account the horizontal parts Φ H i := Φ i + η j ∧ η k of the fundamental 2-forms Φ i , one can verify that horizontal 4-form is projectable and defines a transversal quaternionic structure, which turns out to be locally hyper-Kähler.

Connections with Totally Skew-Symmetric Torsion
In this section we will show that every 3-(0, δ)-Sasaki manifold is canonical in the sense of the definition given in [1], thus admitting a special metric connection with totally skew-symmetric torsion, called canonical. Recall that a metric connection ∇ with torsion T on a Riemannian manifold (M, g) is said to have totally skew-symmetric torsion, or skew torsion for short, if the (0, 3)-tensor field T defined by T(X, Y, Z) := g(T(X, Y), Z) is a 3-form. The relation between ∇ and the Levi-Civita connection ∇ g is then given by For more details we refer to [20]. We recall now the definition and the characterization of canonical almost 3-contact metric manifolds.

Definition 4 ([1]
). An almost 3-contact metric manifold (M, ϕ i , ξ i , η i , g) is called canonical if the following conditions are satisfied: (i) each N ϕ i is totally skew-symmetric on H, (ii) each ξ i is a Killing vector field, (iii) for any X, Y, Z ∈ Γ(H) and any i, j = 1, 2, 3, (iv) M admits a Reeb Killing function β ∈ C ∞ (M), that is the tensor fields A ij defined on H by for every X, Y ∈ Γ(H) and every even permutation (i, j, k) of (1, 2, 3).
Here N ϕ i also denotes the (0, 3)-tensor field defined by N ϕ i (X, Y, Z) := g(N ϕ i (X, Y), Z) and we say that N ϕ i is totally skew-symmetric on H if the (0, 3)-tensor is a 3-form on H.

Theorem 4 ([1]
). An almost 3-contact metric manifold (M, ϕ i , ξ i , η i , g) is canonical, with Reeb Killing function β, if and only if it admits a metric connection ∇ with skew torsion such that for every vector field X on M and for every even permutation (i, j, k) of (1, 2, 3). If such a connection ∇ exists, it is unique and its torsion is given by for every X, Y, Z ∈ Γ(H), and i, j = 1, 2, 3, and where δ is the Reeb commutator function.
The connection ∇ is called the canonical connection of M, and also satisfies ∇ X ξ i = β(η k (X)ξ j − η j (X)ξ k ), ∇ X η i = β(η k (X)η j − η j (X)η k ) (20) for every vector field X on M. Therefore, when β = 0 the canonical connection parallelizes all the structure tensor fields, in which case we call the almost 3-contact metric manifold parallel.
Proof. Let (M, ϕ i , ξ i , η i , g) be a 3-(0, δ)-Sasaki manifold. We showed that the structure is hypernormal and the Reeb vector fields are Killing. Furthermore, by the second equation in (12), dΦ i (X, Y, Z) = 0 for every X, Y, Z ∈ Γ(H). Therefore, conditions (i), (ii) and (iii) in Definition 4 are easily verified. As regards condition (iv), applying the first equation in (4) and Corollary 1, for every X, Y ∈ Γ(H) we have Hence, the structure is canonical with Reeb commutator function β = 2δ. Now, by Theorem 4, taking also into account the fact that the vertical distribution is integrable, the only non-vanishing term of the canonical connection is T(ξ 1 , ξ 2 , ξ 3 ) = 2δ, so that T = 2δ η 123 . Although the structure is not parallel when δ = 0, the torsion satisfies ∇T = 0, as by (20), the 3-form η 123 is parallel with respect to ∇.
The above result generalizes the result obtained in [2] for the Lie group described in Example 1 (see also Remark 2).