Remarkable Classes of Almost 3-Contact Metric Manifolds

Abstract

We introduce a new class of almost 3-contact metric manifolds, called 3-$(0,\delta )$-Sasaki manifolds. We show fundamental geometric properties of these manifolds, analyzing analogies and differences with the known classes of 3-$(\alpha ,\delta )$-Sasaki ($\alpha \ne 0$) and 3-$\delta$-cosymplectic manifolds.

1. Introduction

An almost 3-contact metric manifold is a $(4n+3)$-dimensional differentiable manifold M endowed with three almost contact metric structures $({\phi }_{,}{\xi }_i,{\eta }_i,g)$, $i=1,2,3$, 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-$(\alpha ,\delta )$-Sasaki manifolds defined as almost 3-contact metric manifolds $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ such that

$$d{\eta }_i=2\alpha {\mathsf{\Phi }}_i+2(\alpha -\delta ){\eta }_j\wedge {\eta }_k,\alpha \in {\mathbb{R}}^{*},\delta \in \mathbb{R},$$

(1)

for every even permutation $(i,j,k)$ of $(1,2,3)$. This is a generalization of 3-Sasaki manifolds, which correspond to the values $\alpha =\delta =1$. A second class introduced in [1] is given by 3-δ-cosymplectic manifolds defined by the conditions

$$d{\eta }_i=-2\delta {\eta }_j\wedge {\eta }_k,d{\mathsf{\Phi }}_i=0,\delta \in \mathbb{R},$$

generalizing 3-cosymplectic manifolds which correspond to the value $\delta =0$.

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

$$d{\eta }_i=-2\delta {\eta }_j\wedge {\eta }_k,d{\mathsf{\Phi }}_i=-2\delta ({\eta }_j\wedge {\mathsf{\Phi }}_k-{\eta }_k\wedge {\mathsf{\Phi }}_j),\delta \in \mathbb{R}$$

(2)

for every even permutation $(i,j,k)$ of $(1,2,3)$. When $\delta =0$ we recover a 3-cosymplectic manifold. We will call these manifolds 3-$(0,\delta )$-Sasaki manifolds. The choice of name is due to the fact that for a 3-$(\alpha ,\delta )$-Sasaki manifold, Equation (1) implies

$$d{\mathsf{\Phi }}_i=2(\alpha -\delta )({\eta }_j\wedge {\mathsf{\Phi }}_k-{\eta }_k\wedge {\mathsf{\Phi }}_j),$$

(3)

so that the two equations in (2) formally correspond to (1) and (3) with $\alpha =0$, 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-$(0,\delta )$-Sasaki structures can be defined on the semidirect products $SO\left(3\right)⋉{\mathbb{R}}^{4n}$. The structure on these Lie groups was introduced in [2] as an example of canonical abelian almost 3-contact metric structure. It is also shown in [2] that the Lie group $SO\left(3\right)⋉{\mathbb{R}}^{4n}$ admits co-compact discrete subgroups, so that the corresponding compact quotients admit almost 3-contact metric structures of the same type.

One can show that for all the above three classes of manifolds, 3-$(\alpha ,\delta )$-Sasaki, 3-$\delta$-cosymplectic, and 3-$(0,\delta )$-Sasaki manifolds, the structure is hypernormal, the characteristic vector fields ${\xi }_i$, $i=1,2,3$, are Killing and they span an integrable distribution, called vertical, with totally geodesic leaves. Nevertheless, there are remarkable geometric differences between the three classes. In the 3-$(\alpha ,\delta )$-Sasaki case the 1-forms ${\eta }_i$ are all contact forms, i.e., ${\eta }_i\wedge {\left(d{\eta }_i\right)}^n\ne 0$ everywhere on M, while for the other two classes, the horizontal distribution defined by ${\eta }_i=0$, $i=1,2,3$, is integrable. Both 3-$\delta$-cosymplectic manifolds and 3-$(0,\delta )$-Sasaki manifolds are locally isometric to the Riemannian product of a 3-dimensional Lie group, tangent to the vertical distribution, and a $4n$-dimensional manifold tangent to the horizontal distribution. The Lie group is either isomorphic to $SO\left(3\right)$ or flat depending on wether $\delta \ne 0$ or $\delta =0$. Each horizontal leaf is endowed with a hyper-Kähler structure. The difference between 3-$\delta$-cosymplectic and 3-$(0,\delta )$-Sasaki manifolds lies in the projectability of the structure tensor fields ${\phi }_i$, $i=1,2,3$, with respect to the vertical foliation. They are always projectable for 3-$\delta$-cosymplectic manifolds, but not for 3-$(0,\delta )$-Sasaki manifolds with $\delta \ne 0$. In this case one can project a transverse quaternionic structure, as it happens for 3-$(\alpha ,\delta )$-Sasaki manifolds. Finally, for the three classes of manifolds, we analyze the existence of a canonical metric connection with totally skew-symmetric torsion.

2. Almost Contact and Almost 3-Contact Metric Manifolds

An almost contact manifold is a smooth manifold M of dimension $2n+1$, endowed with a structure $(\phi ,\xi ,\eta )$, where $\phi$ is a $(1,1)$-tensor field, $\xi$ a vector field, and $\eta$ a 1-form such that

$${\phi }^2=-I+\eta \otimes \xi ,\eta \left(\xi \right)=1,$$

implying that $\phi \xi =0$, $\eta \circ \phi =0$, and $\phi$ has rank $2n$. The tangent bundle of M splits as $TM=\mathcal{H}\oplus ⟨\xi ⟩$, where $\mathscr{H}$ is the $2n$-dimensional distribution defined by $\mathscr{H}$ $=\mathrm{Im}\left(\phi \right)=\mathrm{Ker}\left(\eta \right)$. The vector field $\xi$ is called the characteristic or Reeb vector field.

On the product manifold $M\times \mathbb{R}$ one can define an almost complex structure J by $J\left(X,f\frac{d}{dt}\right)=\left(\phi X-f\xi ,\eta \left(X\right)\frac{d}{dt}\right)$, where X is a vector field tangent to M, t is the coordinate of $\mathbb{R}$ and f is a ${\mathcal{C}}^{\infty }$ function on $M\times \mathbb{R}$. If J is integrable, the almost contact structure is said to be normal and this is equivalent to the vanishing of the tensor field $N_{\phi }:=[\phi ,\phi ]+d\eta \otimes \xi$, where $[\phi ,\phi ]$ is the Nijenhuis torsion of $\phi$ [3]. More precisely, for any vector fields X and Y, $N_{\phi }$ is given by

$$N_{\phi }(X,Y)=[\phi X,\phi Y]+{\phi }^2[X,Y]-\phi [\phi X,Y]-\phi [X,\phi Y]+d\eta (X,Y)\xi .$$

(4)

It is known that any almost contact manifold admits a compatible metric, that is a Riemannian metric g such that $g(\phi X,\phi Y)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right)$ for every $X,Y\in \mathfrak{X}\left(M\right)$. Then $\eta =g(·,\xi )$ and $\mathcal{H}={⟨\xi ⟩}^{\perp }$. The manifold $(M,\phi ,\xi ,\eta ,g)$ is called an almost contact metric manifold. The associated fundamental 2-form is defined by $\mathsf{\Phi }(X,Y)=g(X,\phi Y)$.

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

  $$d\eta =2\alpha \mathsf{\Phi },\alpha \in {\mathbb{R}}^{*},$$
  When $\alpha =1$, it is called a contact metric manifold; the 1-form $\eta$ is a contact form, that is $\eta \wedge {\left(d\eta \right)}^n\ne 0$ everywhere on M. An α-Sasaki manifold is a normal $\alpha$-contact metric manifold, and again such a manifold with $\alpha =1$ is called a Sasaki manifold.
• An almost cosymplectic manifold is defined as an almost contact metric manifold such that

  $$d\eta =0,d\mathsf{\Phi }=0;$$

  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 $\mathsf{\Phi }$. This class includes both $\alpha$-Sasaki and cosymplectic manifolds. The Reeb vector field of a quasi-Sasaki manifold is always Killing.

Both $\alpha$-Sasaki manifolds and cosymplectic manifolds can be characterized by means of the Levi-Civita connection ${\nabla }^g$. Indeed, one can show that an almost contact metric manifold $(M,\phi ,\xi ,\eta ,g)$ is $\alpha$-Sasaki if and only if

$$\left({\nabla }_X^g\phi \right)Y=\alpha (g(X,Y)\xi -\eta \left(X\right)Y)\forall X,Y\in \mathfrak{X}\left(M\right).$$

An almost contact metric manifold is cosymplectic if and only if ${\nabla }^g\phi =0$; 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 $({\phi }_i,{\xi }_i,{\eta }_i)$, $i=1,2,3$, satisfying the following relations,

$$\begin{array}{c}\hfill {\phi }_k={\phi }_i{\phi }_j-{\eta }_j\otimes {\xi }_i=-{\phi }_j{\phi }_i+{\eta }_i\otimes {\xi }_j,\\ \hfill {\xi }_k={\phi }_i{\xi }_j=-{\phi }_j{\xi }_i,{\eta }_k={\eta }_i\circ {\phi }_j=-{\eta }_j\circ {\phi }_i,\end{array}$$

for any even permutation $(i,j,k)$ of $(1,2,3)$ ([3]). The tangent bundle of M splits as $TM=\mathcal{H}\oplus \mathcal{V}$, where

$$\mathcal{H}:=\bigcap _{i=1}^3\mathrm{Ker}\left({\eta }_i\right),\mathcal{V}:=⟨{\xi }_1,{\xi }_2,{\xi }_3⟩.$$

In particular, $\mathscr{H}$ has rank $4n$. We call any vector belonging to the distribution $\mathscr{H}$ horizontal and any vector belonging to the distribution $\mathcal{V}$ vertical. The manifold is said to be hypernormal if each almost contact structure $({\varphi }_i,{\xi }_i,{\eta }_i)$ 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 ${\left\{({\phi }_x,{\xi }_x,{\eta }_x)\right\}}_{x\in S^2}$ of almost contact structures such that

$${\phi }_x\circ {\phi }_y-{\eta }_y\otimes {\xi }_x={\phi }_{x\times y}-(x·y)I,{\phi }_x{\xi }_y={\xi }_{x\times y},{\eta }_x\circ {\phi }_y={\eta }_{x\times y},$$

for every $x,y\in S^2$, where · and × denote the standard inner product and cross product on ${\mathbb{R}}^3$. 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 $({\phi }_i,{\xi }_i,{\eta }_i,g)$, $i=1,2,3$. For ease of notation, we will denote an almost 3-contact metric manifold by $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$, omitting $i=1,2,3$. The subbundles $\mathscr{H}$ and $\mathcal{V}$ are orthogonal with respect to g and the three Reeb vector fields ${\xi }_1,{\xi }_2,{\xi }_3$ are orthonormal. In fact, the structure group of the tangent bundle is reducible to $\mathrm{Sp}\left(n\right)\times \left\{1\right\}$ [8].

Given an almost 3-contact metric structure $({\phi }_i,{\xi }_i,{\eta }_i,g)$, an $\mathscr{H}$-homothetic deformation is defined by

$${\eta }_i^{\prime }=c{\eta }_i,{\xi }_i^{\prime }=\frac{1}{c}{\xi }_i,{\phi }_i^{\prime }={\phi }_i,g^{\prime }=ag+b\sum _{i=1}^3{\eta }_i\otimes {\eta }_i,$$

(5)

where $a,b,c$ are real numbers such that $a>0$, $c^2=a+b>0$, ensuring that $({\phi }_i^{\prime },{\xi }_i^{\prime },{\eta }_i^{\prime },g^{\prime })$ is an almost 3-contact metric structure. In particular, the fundamental 2-forms ${\mathsf{\Phi }}_i$ and ${\mathsf{\Phi }}_i^{\prime }$ associated to the structures are related by

$${\mathsf{\Phi }}_i^{\prime }=a{\mathsf{\Phi }}_i-b{\eta }_j\wedge {\eta }_k,$$

(6)

where $(i,j,k)$ is an even permutation of $(1,2,3)$.

An almost 3-contact metric manifold is called

• 3-α-Sasaki, with $\alpha \in {\mathbb{R}}^{*}$, if $({\phi }_i,{\xi }_i,{\eta }_i,g)$ is $\alpha$-Sasaki for all $i=1,2,3$, i.e., the structure is hypernormal and

  $$d{\eta }_i=2\alpha {\mathsf{\Phi }}_i,i=1,2,3;$$

  (7)

  when $\alpha =1$, it is a 3-Sasaki manifold;
• 3-cosymplectic if $({\phi }_i,{\xi }_i,{\eta }_i,g)$ is cosymplectic for all $i=1,2,3$, i.e., the structure is hypernormal and

  $$d{\eta }_i=0,d{\mathsf{\Phi }}_i=0,i=1,2,3;$$

  (8)
• 3-quasi-Sasaki manifold if each structure $({\phi }_i,{\xi }_i,{\eta }_i,g)$ is quasi-Sasaki; this class includes both 3-$\alpha$-Sasaki and 3-cosymplectic manifolds.

These classes were deeply investigated by various authors. See [5,9,10] and references therein for 3-Sasakian geometry, the papers [7,11,12] for 3-cosymplectic manifolds, and [13,14] for 3-quasi-Sasaki manifolds.

In fact, both for 3-Sasaki and 3-cosymplectic manifolds, the hypernormality is consequence of the structure Equations (7) and (8) respectively. This was proved by Kashiwada in [15] for 3-Sasaki manifolds, and in ([16], Theorem 4.13) for 3-cosymplectic manifolds.

In [1] the new classes of 3-$(\alpha ,\delta )$-Sasaki manifolds and 3-$\delta$-cosymplectic manifolds were introduced, generalizing the classes of 3-$\alpha$-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:

Lemma 1

([1]). Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be an almost 3-contact metric manifold. Then the following formula holds $\forall X,Y,Z\in \mathfrak{X}\left(M\right)$:

$$\begin{array}{c}{\displaystyle g(N_{{\phi }_i}(X,Y),Z)=}\hfill \\ =-d{\mathsf{\Phi }}_j(X,Y,{\phi }_{j}Z)+d{\mathsf{\Phi }}_j({\phi }_{i}X,{\phi }_{i}Y,{\phi }_{j}Z)+d{\mathsf{\Phi }}_k(X,{\phi }_{i}Y,{\phi }_{j}Z)+d{\mathsf{\Phi }}_k({\phi }_{i}X,Y,{\phi }_{j}Z)\hfill \\ -{\eta }_i\left(X\right)[d{\eta }_j({\phi }_{i}Y,{\phi }_{j}Z)+d{\eta }_k(Y,{\phi }_{j}Z)]+{\eta }_i\left(Y\right)[d{\eta }_j({\phi }_{i}X,{\phi }_{j}Z)+d{\eta }_k(X,{\phi }_{j}Z)]\hfill \\ +{\eta }_j\left(Z\right)[d{\eta }_j(X,Y)-d{\eta }_j({\phi }_{i}X,{\phi }_{i}Y)]-{\eta }_j\left(Z\right)[d{\eta }_k(X,{\phi }_{i}Y)+d{\eta }_k({\phi }_{i}X,Y)].\hfill \end{array}$$

(9)

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 $\delta \in {\mathcal{C}}^{\infty }\left(M\right)$ such that

$${\eta }_r\left([{\xi }_s,{\xi }_t]\right)=2\delta {ϵ}_{rst},r,s,t=1,2,3$$

where ${ϵ}_{rst}$ is the totally skew-symmetric symbol, or equivalently $d{\eta }_r({\xi }_s,{\xi }_t)=-2\delta {ϵ}_{rst}$. We call $\delta$ a Reeb commutator function, we refer to [1] for more information on this notion.

3. 3-$(\mathbf{\alpha },\mathbf{\delta })$-Sasaki Manifolds and 3-$\mathbf{\delta }$-Cosymplectic Manifolds

This section is a short review of 3-$(\alpha ,\delta )$-Sasaki manifolds and 3-$\delta$-cosymplectic manifolds. These were discussed in detail in [1,17].

Definition 1.

An almost 3-contact metric manifold $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ is called a 3-$(\alpha ,\delta )$-Sasaki manifold if it satisfies

$$d{\eta }_i=2\alpha {\mathsf{\Phi }}_i+2(\alpha -\delta ){\eta }_j\wedge {\eta }_k$$

for every even permutation $(i,j,k)$ of $(1,2,3)$, where $\alpha \ne 0$ and δ are real constants.

When $\alpha =\delta =1$, we have a 3-contact metric manifold, and hence a 3-Sasaki manifold by Kashiwada’s theorem [15]. In the following, when considering 3-$(\alpha ,\delta )$-Sasaki manifolds we will always mean $\alpha \ne 0$. As an immediate consequences of the definition one obtains the following properties:

• Each ${\xi }_i$ is an infinitesimal automorphism of the distribution $\mathscr{H}$, i.e.,

  $$d{\eta }_r(X,{\xi }_s)=0X\in \Gamma \left(\mathcal{H}\right),r,s=1,2,3;$$
• The constant $\delta$ is the Reeb commutator function;
• The differentials $d{\mathsf{\Phi }}_i$ are given by

  $$d{\mathsf{\Phi }}_i=2(\delta -\alpha )({\eta }_k\wedge {\mathsf{\Phi }}_j-{\eta }_j\wedge {\mathsf{\Phi }}_k).$$

Applying Lemma 1 one shows the following

Theorem 1

([1], Theorem 2.2.1). Any 3-$(\alpha ,\delta )$-Sasaki manifold is hypernormal.

In particular, a 3-$(\alpha ,\delta )$-Sasaki manifold with $\alpha =\delta$ is 3-$\alpha$-Sasaki. It can be also shown that the vertical distribution of any 3-$(\alpha ,\delta )$-Sasaki manifold is integrable with totally geodesic leaves and each Reeb vector field ${\xi }_i$ is Killing.

We can distinguish three main classes of 3-$(\alpha ,\delta )$-Sasaki manifolds. A 3-$(\alpha ,\delta )$-Sasaki manifold is called degenerate if $\delta =0$ and non-degenerate otherwise. Quaternionic Heisenberg groups are examples of degenerate 3-$(\alpha ,\delta )$-Sasaki manifolds (see ([1], Example 2.3.2)). Considering an $\mathscr{H}$-homothetic deformation of a 3-$(\alpha ,\delta )$-Sasaki structure, as in (5), one can verify that the obtained structure $({\phi }^{\prime },{\xi }^{\prime },{\eta }^{\prime },g^{\prime })$ is a 3-$({\alpha }^{\prime },{\delta }^{\prime })$-Sasaki with

$${\alpha }^{\prime }=\alpha \frac{c}{a},{\delta }^{\prime }=\frac{\delta }{c}.$$

In particular, $\mathscr{H}$-homothetic deformations preserve the class of degenerate manifolds. In the nondegenerate case, one sees immediately that ${\alpha }^{\prime }{\delta }^{\prime }$ has the same sign as $\alpha \delta$. This justifies the distinction between positive 3-$(\alpha ,\delta )$-Sasaki manifolds, with $\alpha \delta >0$, and negative 3-$(\alpha ,\delta )$-Sasaki manifolds, with $\alpha \delta <0$. In fact, it can be shown that a 3-$(\alpha ,\delta )$-Sasaki manifold is positive if and only if it is $\mathscr{H}$-homothetic to a 3-Sasaki manifold, and negative if and only if it is $\mathscr{H}$-homothetic to a 3-$({\alpha }^{\prime },{\delta }^{\prime })$-Sasaki manifold with ${\alpha }^{\prime }=-1$, ${\delta }^{\prime }=1$.

Examples of negative 3-$(\alpha ,\delta )$-Sasaki manifolds can be obtained in the following way. It is known that quaternionic Kähler (not hyper-Kähler) manifolds with negative scalar curvature admit a canonically associated principal $\mathrm{SO}\left(3\right)$-bundle $P\left(M\right)$ which is endowed with a negative 3-Sasaki structure [18,19]. This is a 3-structure $({\phi }_i,{\xi }_i,{\eta }_i,\tilde{g})$, $i=1,2,3$, where $({\phi }_i,{\xi }_i,{\eta }_i)$ is a normal almost 3-contact structure, and $\tilde{g}$ is a compatible semi-Riemannian metric, with signature $(3,4n)$, where $4n$ is the dimension of the base space, and $d{\eta }_i(X,Y)=2\tilde{g}(X,{\phi }_{i}Y)$. Then, one can define the Riemannian metric

$$g=-\tilde{g}+2\sum _{i=1}^3{\eta }_i\otimes {\eta }_i,$$

which is compatible with the structure $({\phi }_i,{\xi }_i,{\eta }_i)$, and satisfies $d{\eta }_i=-2{\mathsf{\Phi }}_i-4{\eta }_j\wedge {\eta }_k$, where ${\mathsf{\Phi }}_i(X,Y)=g(X,{\phi }_{i}Y)$ (see also [19]). Therefore $({\phi }_i,{\xi }_i,{\eta }_i,g)$ is a 3-$(\alpha ,\delta )$-Sasaki structure with $\alpha =-1$ and $\delta =1$.

The following Theorem regarding the transverse geometry with respect to the vertical foliation of a 3-$(\alpha ,\delta )$-Sasaki manifold is proved in [17]:

Theorem 2.

Any 3-$(\alpha ,\delta )$-Sasaki manifold M admits a locally defined Riemannian submersion $\pi :M\to N$ along its horizontal distribution $\mathcal{H}$ such that N carries a quaternionic Kähler structure given by

$$\check{{\phi }_i}={\pi }_{*}\circ {\phi }_i\circ s_{*},i=1,2,3,$$

where $s:U\to M$ is any local smooth section of the Riemannian submersion. The covariant derivatives of the almost complex structures ${\check{\phi }}_i$ are given by

$${\nabla }_X^{g_N}{\check{\phi }}_i=2\delta ({\check{\eta }}_k\left(X\right){\check{\phi }}_j-{\check{\eta }}_j\left(X\right){\check{\phi }}_k)$$

where ${\check{\eta }}_i\left(X\right)={\eta }_i\left(s_{*}X\right)\circ s$ for $i=1,2,3$. The scalar curvature of the base space N is $16n(n+2)\alpha \delta$.

The Riemannian Ricci tensor of any 3-$(\alpha ,\delta )$-Sasaki manifold is computed in [1]:

$${\mathrm{Ric}}^g=2\alpha \left(2\delta (n+2)-3\alpha \right)g+2(\alpha -\delta )\left((2n+3)\alpha -\delta \right)\sum _{i=1}^3{\eta }_i\otimes {\eta }_i.$$

(10)

In particular, a 3-$(\alpha ,\delta )$-Sasaki manifold is Riemannian Einstein if and only if $\delta =\alpha$, in which case the structure is 3-$\alpha$-Sasaki, or $\delta =(2n+3)\alpha$.

Notice that, by Theorem 2, a non-degenerate 3-$(\alpha ,\delta )$-Sasaki manifold locally fibers over a quaternionic Kähler space of positive or negative scalar curvature, according to $\alpha \delta >0$ or $\alpha \delta <0$, respectively. In [17] a systematic study of homogeneous non-degenerate 3-$(\alpha ,\delta )$-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 $\alpha \delta <0$, one can provide a general construction of homogeneous 3-$(\alpha ,\delta )$-Sasaki manifolds fibering over nonsymmetric Alekseevsky spaces.

We recall now the definition and some basic facts on 3-$\delta$-cosymplectic manifolds.

Definition 2.

A 3-$\delta$-cosymplectic manifold is an almost 3-contact metric manifold satisfying

$$d{\eta }_i=-2\delta {\eta }_j\wedge {\eta }_k,d{\mathsf{\Phi }}_i=0,$$

for some $\delta \in \mathbb{R}$ and for every even permutation $(i,j,k)$ of $(1,2,3)$.

When $\delta =0$, the fact that the forms ${\eta }_i$ and ${\mathsf{\Phi }}_i$ are all closed implies that the structure is hypernormal ([16], Theorem 4.13). In fact this immediately follows from (9). Therefore, a 3-$\delta$-cosymplectic manifold with $\delta =0$ 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:

Proposition 1

([12]). 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-$\delta$-cosymplectic manifolds with $\delta \ne 0$, 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 $\delta \ne 0$ is locally the Riemannian product of a hyper-Kähler manifold and a 3-dimensional Lie group isomorphic to $SO\left(3\right)$, with constant curvature ${\delta }^2$. Consequently, the Riemannian Ricci tensor is ${Ric}^g=2{\delta }^2{\sum }_{i=1}^3{\eta }_i\otimes {\eta }_i$.

In both cases, i.e., $\delta =0$ or $\delta \ne 0$, 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 $N\times G$, 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 $SO\left(3\right)$. If ${\xi }_1,{\xi }_2,{\xi }_3$ are generators of the Lie algebra $\mathfrak{g}$ of G, satisfying $[{\xi }_i,{\xi }_j]=2\delta {\xi }_k$, $\delta \in \mathbb{R}$, then one can define in a natural manner an almost 3-contact metric structure $({\phi }_i,{\xi }_i,{\eta }_i,g)$ on the product $N\times G$, setting

$${\phi }_i{|}_{TN}=J_i,{\phi }_i{\xi }_i=0,{\phi }_i{\xi }_j={\xi }_k,{\phi }_i{\xi }_k=-{\xi }_j,$$

$${\eta }_i{|}_{TN}=0,{\eta }_i\left({\xi }_i\right)=1,{\eta }_i\left({\xi }_j\right)={\eta }_i\left({\xi }_k\right)=0,$$

and g the product metric of h and the left invariant Riemannian metric on G with respect to which ${\xi }_1$, ${\xi }_2$, ${\xi }_3$ are an orthonormal basis of $\mathfrak{g}$.

For a comparison with the class of 3-$(0,\delta )$-Sasaki manifolds, which will be introduced in the next section, it is worth remarking that for a 3-$\delta$-cosymplectic manifold $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ the Lie derivatives of the structure tensor fields ${\phi }_i$, $i=1,2,3$ with respect to the Reeb vector fields are given by

$${\mathcal{L}}_{{\xi }_i}{\phi }_i=0,{\mathcal{L}}_{{\xi }_i}{\phi }_j=2\delta ({\eta }_i\otimes {\xi }_j-{\eta }_j\otimes {\xi }_i)=-{\mathcal{L}}_{{\xi }_j}{\phi }_i$$

(11)

for every $i,j=1,2,3$. Indeed, in a 3-$\delta$-cosymplectic manifold the Levi-Civita connection satisfies ([1], Proposition 2.1.1):

$${\nabla }_{{\xi }_i}^g{\phi }_i=0,$$

$$\left({\nabla }_{{\xi }_i}^g{\phi }_j\right)X=\delta ({\eta }_i\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_i)=-\left({\nabla }_{{\xi }_j}^g{\phi }_i\right)X,$$

$${\nabla }_X^g{\xi }_i=\delta ({\eta }_k\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_k),$$

where $(i,j,k)$ is an even permutation of $(1,2,3)$ and $X\in \mathfrak{X}\left(M\right)$. Therefore,

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{{\xi }_i}{\phi }_i\right)X& =[{\xi }_i,{\phi }_{i}X]-{\phi }_i[{\xi }_i,X]\hfill \\ \hfill & ={\nabla }_{{\xi }_i}^g\left({\phi }_{i}X\right)-{\nabla }_{{\phi }_{i}X}^g{\xi }_i-{\phi }_i\left({\nabla }_{{\xi }_i}^{g}X\right)+{\phi }_i\left({\nabla }_X^g{\xi }_i\right)\hfill \\ \hfill & =\left({\nabla }_{{\xi }_i}^g{\phi }_i\right)X-{\nabla }_{{\phi }_{i}X}^g{\xi }_i+{\phi }_i\left({\nabla }_X^g{\xi }_i\right)\hfill \\ \hfill & =-\delta ({\eta }_k\left({\phi }_{i}X\right){\xi }_j-{\eta }_j\left({\phi }_{i}X\right){\xi }_k)+\delta ({\eta }_k\left(X\right){\phi }_i{\xi }_j-{\eta }_j\left(X\right){\phi }_i{\xi }_k)=0.\hfill \end{array}$$

In the same way,

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{{\xi }_i}{\phi }_j\right)X& =\left({\nabla }_{{\xi }_i}^g{\phi }_j\right)X-{\nabla }_{{\phi }_{j}X}^g{\xi }_i+{\phi }_j\left({\nabla }_X^g{\xi }_i\right)\hfill \\ \hfill & =\delta ({\eta }_i\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_i)-\delta {\eta }_k\left({\phi }_{j}X\right){\xi }_j-\delta {\eta }_j\left(X\right){\phi }_j{\xi }_k\hfill \\ \hfill & =2\delta ({\eta }_i\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_i)=-\left({\mathcal{L}}_{{\xi }_j}{\phi }_i\right)X.\hfill \end{array}$$

4. 3-$(\mathbf{0},\mathbf{\delta })$-Sasaki Manifolds

In this section we introduce the class of 3-$(0,\delta )$-Sasaki manifolds.

Definition 3.

An almost 3-contact metric manifold $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ will be called 3-$(0,\delta )$-Sasaki manifold if

$$d{\eta }_i=-2\delta {\eta }_j\wedge {\eta }_k,d{\mathsf{\Phi }}_i=-2\delta ({\eta }_j\wedge {\mathsf{\Phi }}_k-{\eta }_k\wedge {\mathsf{\Phi }}_j)$$

(12)

for every even permutation $(i,j,k)$ of $(1,2,3)$, and for some real constant $\delta \in \mathbb{R}$.

In particular, the structure is not 3-quasi-Sasaki when $\delta \ne 0$, and we have the following basic properties for a 3-$(0,\delta )$-Sasaki manifold:

• The horizontal distribution $\mathscr{H}$ is integrable;
• Each ${\xi }_i$ is an infinitesimal automorphism of the distribution $\mathscr{H}$, i. e.

  $$d{\eta }_r(X,{\xi }_s)=0X\in \Gamma \left(\mathcal{H}\right),r,s=1,2,3;$$
• The constant $\delta$ is the Reeb commutator function.

Remark 1.

In case $\delta \ne 0$, the two equations in (12) are not completely independent. Indeed, if one assumes $d{\mathsf{\Phi }}_i=-2\gamma ({\eta }_j\wedge {\mathsf{\Phi }}_k-{\eta }_k\wedge {\mathsf{\Phi }}_j)$, $\gamma \in {\mathbb{R}}^{*}$, differentiating this equation, and combining with $d{\eta }_i=-2\delta {\eta }_j\wedge {\eta }_k$, a straightforward computation gives $\gamma =\delta$. Thus, there is no freedom for the choice of constant in the second equation.

If $({\phi }_i,{\xi }_i,{\eta }_i,g)$ is a 3-$(0,\delta )$-Sasaki structure, applying an $\mathscr{H}$-homothetic deformation as in (5), an easy computation using (6) shows that the new structure $({\phi }_i^{\prime },{\eta }_i^{\prime },{\xi }_i^{\prime },g^{\prime })$ is again 3-$(0,{\delta }^{\prime })$-Sasaki, with ${\delta }^{\prime }=\frac{\delta }{c}$.

Example 1.

Consider the abelian Lie algebra ${\mathbb{R}}^{4n}$ spanned by vectors $v_r$, $v_{n+r}$, $v_{2n+r}$, $v_{3n+r}$, $r=1,\dots ,n$, and endowed with the hypercomplex structure $\{J_1,J_2,J_3\}$ defined by

$$J_i\left(v_r\right)=v_{in+r},J_i\left(v_{in+r}\right)=-v_r,J_i\left(v_{jn+r}\right)=v_{kn+r},J_i\left(v_{kn+r}\right)=-v_{jn+r},$$

for every even permutation $(i,j,k)$ of $(1,2,3)$. Let us consider also the Lie algebra $\mathfrak{so}\left(3\right)$ spanned by ${\xi }_1,{\xi }_2,{\xi }_3$ with Lie brackets given by $[{\xi }_i,{\xi }_j]=2\delta {\xi }_k$, $\delta \ne 0$. Let ρ be the representation of $\mathfrak{so}\left(3\right)$ on ${\mathbb{R}}^{4n}$ given by

$$\rho :\mathfrak{so}\left(3\right)\to \mathfrak{gl}(4n,\mathbb{R}),\rho \left({\xi }_i\right)=\delta J_i,i=1,2,3.$$

On the Lie algebra $\mathfrak{g}=\mathfrak{so}\left(3\right){⋉}_{\rho }{\mathbb{R}}^{4n}$ on can define in a natural way an almost 3-contact metric structure $({\phi }_i,{\xi }_i,{\eta }_i,g)$, with

$${\phi }_i{|}_{{\mathbb{R}}^{4n}}=J_i,\phi \left({\xi }_i\right)=0,{\phi }_i\left({\xi }_j\right)={\xi }_k=-{\phi }_j\left({\xi }_k\right),$$

$${\eta }_i{|}_{{\mathbb{R}}^{4n}}=0,{\eta }_i\left({\xi }_i\right)=1,{\eta }_i\left({\xi }_j\right)={\eta }_i\left({\xi }_k\right)=0,$$

and where g is the inner product such that the vectors ${\xi }_i$, $v_l$, $i=1,2,3$, $l=1,\dots ,4n$ are orthonormal. In particular, the non zero brackets on $\mathfrak{g}$ are given by

$$[{\xi }_i,{\xi }_j]=2\delta {\xi }_k,[{\xi }_i,X]=\delta {\phi }_i\left(X\right),X\in {\mathbb{R}}^{4n}.$$

The representation $\rho :\mathfrak{so}\left(3\right)\to \mathfrak{gl}(4n,\mathbb{R})$ can be integrated to a representation $\tilde{\rho }:SO\left(3\right)\to GL(4n,\mathbb{R})$. Therefore, identifying ${\mathbb{R}}^{4n}$ with ${\mathbb{H}}^n$ in a natural way, the simply connected Lie group $G=SO\left(3\right){⋉}_{\tilde{\rho }}{\mathbb{H}}^n$, with Lie algebra $\mathfrak{g}$, admits a left invariant almost 3-contact metric structure $({\phi }_i,{\xi }_i,{\eta }_i,g)$. One can easily verify that this structure satisfies (12).

Remark 2.

For more details on the above example we refer to [2], where $\mathfrak{g}$ 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 $\mathfrak{g}$ 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

$${\nabla }_X{\phi }_i=2\delta ({\eta }_k\left(X\right){\phi }_j-{\eta }_j\left(X\right){\phi }_k)$$

for every vector field X and for every even permutation $(i,j,k)$ of $(1,2,3)$. The torsion of the canonical connection ∇ is $T=2\delta {\eta }_1\wedge {\eta }_2\wedge {\eta }_3$, which satisfies $\nabla T=0$.

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.

Proposition 3.

Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be a 3-$(0,\delta )$-Sasaki manifold. Then the structure is hypernormal.

Proof. 

In order to compute the tensor fields $N_{{\phi }_i}$, we apply Lemma 1. We always denote by $X,Y,Z$ horizontal vector fields and by $(i,j,k)$ an even permutation of $(1,2,3)$.

Being $d{\mathsf{\Phi }}_i(X,Y,Z)=0$, then $N_{{\phi }_i}(X,Y,Z)=0$ for every $i=1,2,3$. Furthermore, since the horizontal distribution is integrable, by the definition of the tensor field $N_{{\phi }_i}$ (see (4)), one has $N_{{\phi }_i}(X,Y,{\xi }_r)=0$ for all $r=1,2,3$. Notice that, since

$${\xi }_i⌟{\mathsf{\Phi }}_i=0,{\xi }_j⌟{\mathsf{\Phi }}_i=-{\eta }_k,{\xi }_k⌟{\mathsf{\Phi }}_i={\eta }_j,$$

from the second equation in (12), we have,

$${\xi }_i⌟d{\mathsf{\Phi }}_i=0,{\xi }_j⌟d{\mathsf{\Phi }}_i=-2\delta ({\mathsf{\Phi }}_k+{\eta }_{ij}),{\xi }_k⌟d{\mathsf{\Phi }}_i=2\delta ({\mathsf{\Phi }}_j+{\eta }_{ki}).$$

(13)

Therefore, form Lemma 1, applying (12) and (13), we have

$$\begin{array}{cc}\hfill N_{{\phi }_i}(X,{\xi }_i,Z)& =-d{\mathsf{\Phi }}_j(X,{\xi }_i,{\phi }_{j}Z)+d{\mathsf{\Phi }}_k({\phi }_{i}X,{\xi }_i,{\phi }_{j}Z)+d{\eta }_j({\phi }_{i}X,{\phi }_{j}Z)+d{\eta }_k(X,{\phi }_{j}Z)\hfill \\ \hfill & =-2\delta {\mathsf{\Phi }}_k({\phi }_{j}Z,X)-2\delta {\mathsf{\Phi }}_j({\phi }_{j}Z,{\phi }_{i}X)\hfill \\ \hfill & =2\delta {\mathsf{\Phi }}_j({\phi }_{i}X,{\phi }_{j}Z)+2\delta {\mathsf{\Phi }}_k(X,{\phi }_{j}Z)=-2\delta g({\phi }_{i}X,Z)-2\delta g(X,{\phi }_{i}Z)=0,\hfill \\ \hfill N_{{\phi }_i}(X,{\xi }_j,Z)& =d{\mathsf{\Phi }}_j({\phi }_{i}X,{\xi }_k,{\phi }_{j}Z)+d{\mathsf{\Phi }}_k({\phi }_{i}X,{\xi }_j,{\phi }_{j}Z)\hfill \\ \hfill & =-2\delta {\mathsf{\Phi }}_i({\phi }_{j}Z,{\phi }_{i}X)+2\delta {\mathsf{\Phi }}_i({\phi }_{j}Z,{\phi }_{i}X)=0,\hfill \\ \hfill N_{{\phi }_i}(X,{\xi }_k,Z)& =-d{\mathsf{\Phi }}_j(X,{\xi }_k,{\phi }_{j}Z)-d{\mathsf{\Phi }}_k(X,{\xi }_j,{\phi }_{j}Z)\hfill \\ \hfill & =2\delta {\mathsf{\Phi }}_i({\phi }_{j}Z,X)-2\delta {\mathsf{\Phi }}_i({\phi }_{j}Z,X)=0.\hfill \end{array}$$

Equations (13) implies $d{\mathsf{\Phi }}_r(X,{\xi }_s,{\xi }_t)=0$ for every $r,s,t=1,2,3$ and $X\in \Gamma \left(\mathcal{H}\right)$. Taking also into account that $d{\eta }_r(X,{\xi }_s)=0$, we deduce from (9) that

$$N_{{\phi }_r}(X,{\xi }_s,{\xi }_t)=N_{{\phi }_r}({\xi }_s,{\xi }_t,X)=0.$$

Finally, (9) implies together with $d{\eta }_r({\xi }_s,{\xi }_t)=-2\delta {ϵ}_{rst}$ that

$$N_{{\phi }_i}({\xi }_i,{\xi }_j,{\xi }_k)=N_{{\phi }_i}({\xi }_i,{\xi }_k,{\xi }_j)=N_{{\phi }_i}({\xi }_j,{\xi }_k,{\xi }_i)=0,$$

completing the proof that M is hypernormal. □

Proposition 4.

Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be a 3-$(0,\delta )$-Sasaki manifold. Then the Levi-Civita connection satisfies for all $X,Y\in \mathfrak{X}\left(M\right)$ and any cyclic permutation $(i,j,k)$ of $(1,2,3)$:

$$\begin{array}{cc}\hfill \left({\nabla }_X^g{\phi }_i\right)Y& =2\delta \left[{\eta }_k\left(X\right){\phi }_{j}Y-{\eta }_j\left(X\right){\phi }_{k}Y\right]\hfill \\ \hfill & -\delta \left[{\eta }_j\left(X\right){\eta }_j\left(Y\right)+{\eta }_k\left(X\right){\eta }_k\left(Y\right)\right]{\xi }_i+\delta {\eta }_i\left(Y\right)\left[{\eta }_j\left(X\right){\xi }_j+{\eta }_k\left(X\right){\xi }_k\right]\hfill \end{array}$$

(14)

and

$${\nabla }_X^g{\xi }_i=\delta ({\eta }_k\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_k),$$

(15)

$${\nabla }_{{\xi }_i}^g{\xi }_i=0,{\nabla }_{{\xi }_i}^g{\xi }_j=-{\nabla }_{{\xi }_j}^g{\xi }_i=\delta {\xi }_k.$$

(16)

In particular, each ${\xi }_i$ is a Killing vector field.

Proof. 

Since the structure is hypernormal, by ([3], Lemma 6.1), the Levi-Civita connection satisfies

$$\begin{array}{cc}\hfill 2g(\left({\nabla }_X^g{\phi }_i\right)Y,Z)& =d{\mathsf{\Phi }}_i(X,{\phi }_{i}Y,{\phi }_{i}Z)-d{\mathsf{\Phi }}_i(X,Y,Z)\hfill \\ \hfill & +d{\eta }_i({\phi }_{i}Y,X){\eta }_i\left(Z\right)-d{\eta }_i({\phi }_{i}Z,X){\eta }_i\left(Y\right).\hfill \end{array}$$

(17)

Further, an easy computation (see [1]) shows that for every cyclic permutation $(i,j,k)$ of $(1,2,3)$,

$$\begin{array}{cc}\hfill {\mathsf{\Phi }}_j({\phi }_{i}X,{\phi }_{i}Y)& =-{\mathsf{\Phi }}_j(X,Y)-({\eta }_k\wedge {\eta }_i)(X,Y),\hfill \\ \hfill {\mathsf{\Phi }}_k({\phi }_{i}X,{\phi }_{i}Y)& =-{\mathsf{\Phi }}_k(X,Y)-({\eta }_i\wedge {\eta }_j)(X,Y),\hfill \\ \hfill {\mathsf{\Phi }}_j({\phi }_{i}X,Y)& =-{\mathsf{\Phi }}_k(X,Y)-{\eta }_i\left(X\right){\eta }_j\left(Y\right),\hfill \\ \hfill {\mathsf{\Phi }}_k({\phi }_{i}X,Y)& ={\mathsf{\Phi }}_j(X,Y)-{\eta }_i\left(X\right){\eta }_k\left(Y\right).\hfill \end{array}$$

Then, using the second equation in (12) and the above equations, we have

$$\begin{array}{c}{\displaystyle d{\mathsf{\Phi }}_i(X,{\phi }_{i}Y,{\phi }_{i}Z)=}\hfill \\ =-2\delta [{\eta }_j\left(X\right){\mathsf{\Phi }}_k({\phi }_{i}Y,{\phi }_{i}Z)+{\eta }_j\left({\phi }_{i}Y\right){\mathsf{\Phi }}_k({\phi }_{i}Z,X)+{\eta }_j\left({\phi }_{i}Z\right){\mathsf{\Phi }}_k(X,{\phi }_{i}Y)\hfill \\ -{\eta }_k\left(X\right){\mathsf{\Phi }}_j({\phi }_{i}Y,{\phi }_{i}Z)-{\eta }_k\left({\phi }_{i}Y\right){\mathsf{\Phi }}_j({\phi }_{i}Z,X)-{\eta }_k\left({\phi }_{i}Z\right){\mathsf{\Phi }}_j(X,{\phi }_{i}Y)]\hfill \\ =-2\delta [-{\eta }_j\left(X\right){\mathsf{\Phi }}_k(Y,Z)-{\eta }_j\left(X\right)({\eta }_i\wedge {\eta }_j)(Y,Z)\hfill \\ -{\eta }_k\left(Y\right){\mathsf{\Phi }}_j(Z,X)+{\eta }_k\left(Y\right){\eta }_i\left(Z\right){\eta }_k\left(X\right)+{\eta }_k\left(Z\right){\mathsf{\Phi }}_j(Y,X)-{\eta }_k\left(Z\right){\eta }_i\left(Y\right){\eta }_k\left(X\right)\hfill \\ +{\eta }_k\left(X\right){\mathsf{\Phi }}_j(Y,Z)+{\eta }_k\left(X\right)({\eta }_k\wedge {\eta }_i)(Y,Z)\hfill \\ +{\eta }_j\left(Y\right){\mathsf{\Phi }}_k(Z,X)+{\eta }_j\left(Y\right){\eta }_i\left(Z\right){\eta }_j\left(X\right)-{\eta }_j\left(Z\right){\mathsf{\Phi }}_k(Y,X)-{\eta }_j\left(Z\right){\eta }_i\left(Y\right){\eta }_j\left(X\right)]\hfill \\ =d{\mathsf{\Phi }}_i(X,Y,Z)+4\delta [{\eta }_j\left(X\right){\mathsf{\Phi }}_k(Y,Z)-{\eta }_k\left(X\right){\mathsf{\Phi }}_j(Y,Z)]\hfill \\ +4\delta {\eta }_j\left(X\right)[{\eta }_i\left(Y\right){\eta }_j\left(Z\right)-{\eta }_j\left(Y\right){\eta }_i\left(Z\right)]\hfill \\ +4\delta {\eta }_k\left(X\right)[{\eta }_i\left(Y\right){\eta }_k\left(Z\right)-{\eta }_k\left(Y\right){\eta }_i\left(Z\right)].\hfill \end{array}$$

On the other hand, again using the first equation in (12), we obtain

$$\begin{array}{c}{\displaystyle d{\eta }_i({\phi }_{i}Y,X){\eta }_i\left(Z\right)-d{\eta }_i({\phi }_{i}Z,X){\eta }_i\left(Y\right)=}\hfill \\ =-2\delta ({\eta }_j\wedge {\eta }_k)({\phi }_{i}Y,X){\eta }_i\left(Z\right)+2\delta ({\eta }_j\wedge {\eta }_k)({\phi }_{i}Z,X){\eta }_i\left(Y\right)\hfill \\ =-2\delta {\eta }_i\left(Z\right)[-{\eta }_k\left(Y\right){\eta }_k\left(X\right)-{\eta }_j\left(X\right){\eta }_j\left(Y\right)]+2\delta {\eta }_i\left(Y\right)[-{\eta }_k\left(Z\right){\eta }_k\left(X\right)-{\eta }_j\left(X\right){\eta }_j\left(Z\right)].\hfill \end{array}$$

Inserting the above computations in (17), we conclude that

$$\begin{array}{cc}\hfill g(\left({\nabla }_X^g{\phi }_i\right)Y,Z)& =2\delta [{\eta }_k\left(X\right)g({\phi }_{j}Y,Z)-{\eta }_j\left(X\right)g({\phi }_{k}Y,Z)]\hfill \\ \hfill & -\delta {\eta }_i\left(Z\right)[{\eta }_k\left(Y\right){\eta }_k\left(X\right)+{\eta }_j\left(X\right){\eta }_j\left(Y\right)]+\delta {\eta }_i\left(Y\right)[{\eta }_k\left(Z\right){\eta }_k\left(X\right)+{\eta }_j\left(X\right){\eta }_j\left(Z\right)]\hfill \end{array}$$

which implies (14). As regards the proof (15), applying (14) for $Y={\xi }_i$, we get

$$\left({\nabla }_X^g{\phi }_i\right){\xi }_i=-\delta ({\eta }_j\left(X\right){\xi }_j+{\eta }_k\left(X\right){\xi }_k).$$

Applying ${\phi }_i$ on both hand-sides, we obtain (15). Equations (16) are immediate consequences of (15). Furthermore, we also get

$$g({\nabla }_X^g{\xi }_i,Y)=-\delta ({\eta }_j\wedge {\eta }_k)(X,Y)$$

for every $X,Y\in \mathfrak{X}\left(M\right)$. Since ${\nabla }^g{\xi }_i$ is skew-symmetric, ${\xi }_i$ is Killing. □

Corollary 1.

Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be a 3-$(0,\delta )$-Sasaki manifold. Then for every even permutation $(i,j,k)$ of $(1,2,3)$ we have

$${\mathcal{L}}_{{\xi }_i}{\phi }_i=0,{\mathcal{L}}_{{\xi }_i}{\phi }_j=-{\mathcal{L}}_{{\xi }_j}{\phi }_i=2\delta {\phi }_k.$$

(18)

Proof. 

For the first Lie derivative, notice that by (14) we have ${\nabla }_{{\xi }_i}^g{\phi }_i=0$. Then, applying also (15), for every vector field X we have

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{{\xi }_i}{\phi }_i\right)X& =\left({\nabla }_{{\xi }_i}^g{\phi }_i\right)X-{\nabla }_{{\phi }_{i}X}^g{\xi }_i+{\phi }_i\left({\nabla }_X^g{\xi }_i\right)\hfill \\ & =-\delta ({\eta }_k\left({\phi }_{i}X\right){\xi }_j-{\eta }_j\left({\phi }_{i}X\right){\xi }_k)+\delta ({\eta }_k\left(X\right){\phi }_i{\xi }_j-{\eta }_j\left(X\right){\phi }_i{\xi }_k)=0.\hfill \end{array}$$

Now, using (14) for the covariant derivative ${\nabla }^g{\phi }_j$, for every vector field Y, we have

$$\left({\nabla }_{{\xi }_i}^g{\phi }_j\right)Y=2\delta {\phi }_{k}Y-\delta \left({\eta }_i\left(Y\right){\xi }_j-{\eta }_j\left(Y\right){\xi }_i\right).$$

Therefore, applying also (15), we get

$$\begin{array}{cc}\hfill \left({\mathcal{L}}_{{\xi }_i}{\phi }_j\right)X& =\left({\nabla }_{{\xi }_i}^g{\phi }_j\right)X-{\nabla }_{{\phi }_{j}X}^g{\xi }_i+{\phi }_j\left({\nabla }_X^g{\xi }_i\right)\hfill \\ \hfill & =2\delta {\phi }_{k}X-\delta \left({\eta }_i\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_i\right)-\delta {\eta }_k\left({\phi }_{j}X\right){\xi }_j-\delta {\eta }_j\left(X\right){\phi }_j{\xi }_k\hfill \\ \hfill & =2\delta {\phi }_{k}X.\hfill \end{array}$$

Analogously, ${\mathcal{L}}_{{\xi }_j}{\phi }_i=-2\delta {\phi }_k$. □

Theorem 3.

Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be a 3-$(0,\delta )$-Sasaki manifold. 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 $SO\left(3\right)$, with constant sectional curvature ${\delta }^2$; each leaf of the horizontal distribution is endowed with a hyper-Kähler structure. Consequently, the Riemannian Ricci tensor of M is given by

$${Ric}^g=2{\delta }^2\sum _{i=1}^3{\eta }_i\otimes {\eta }_i.$$

(19)

Proof. 

We already know that the horizontal distribution $\mathscr{H}$ is integrable. From (15), for every $X,Y\in \Gamma \left(\mathcal{H}\right)$ and $i=1,2,3$, we have

$$g({\nabla }_X^{g}Y,{\xi }_i)=-g({\nabla }_X^g{\xi }_i,Y)=0,$$

so that the distribution $\mathscr{H}$ has totally geodesic leaves. Furthermore, Equation (16) implies that the vertical distribution $\mathcal{V}$ is also integrable with totally geodesic leaves. In particular $[{\xi }_i,{\xi }_j]=2\delta {\xi }_k$ for an even permutation $(i,j,k)$ of $(1,2,3)$, so that the leaves of $\mathcal{V}$ are locally isomorphic to the Lie group $SO\left(3\right)$ and have constant sectional curvature ${\delta }^2$. On each leaf of the horizontal distribution $\mathscr{H}$ one can consider the almost hyper-Hermitian structure defined by $(J_i:={\phi }_i{|}_H,g)$, which turns out to be hyper-Kähler due to (14). Consequently, M is locally the Riemannnian product of 3-dimensional sphere of curvature ${\delta }^2$ and a $4n$-dimensional manifold $M^{\prime }$, 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-$(0,\delta )$-Sasaki manifold is locally isometric to the Riemannnian product of 3-dimensional sphere and a $4n$-dimensional manifold $M^{\prime }$, 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 ${\delta }^2$ 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 $\mathcal{V}$, in both cases the Riemannian metric g is projectable, being the vector fields ${\xi }_i$, $i=1,2,3$, all Killing. In the case of 3-δ-cosymplectic manifolds, each tensor field ${\phi }_i$ is also projectable, as by (11), the Lie derivatives with respect to the Reeb vector fields satisfy $\left({\mathcal{L}}_{{\xi }_i}{\phi }_j\right)X=0$ for every $i,j=1,2,3$ and for every horizontal vector field X. In the case of 3-$(0,\delta )$-Sasaki manifolds, owing to (18), the tensor fields are not projectable. Nevertheless, taking into account the horizontal parts ${\mathsf{\Phi }}_i^{}:={\mathsf{\Phi }}_i+{\eta }_j\wedge {\eta }_k$ of the fundamental 2-forms ${\mathsf{\Phi }}_i$, one can verify that horizontal 4-form

$${\mathsf{\Phi }}_1^{\mathcal{H}}\wedge {\mathsf{\Phi }}_1^{\mathcal{H}}+{\mathsf{\Phi }}_2^{\mathcal{H}}\wedge {\mathsf{\Phi }}_2^{\mathcal{H}}+{\mathsf{\Phi }}_3^{\mathcal{H}}\wedge {\mathsf{\Phi }}_3^{\mathcal{H}}$$

is projectable and defines a transversal quaternionic structure, which turns out to be locally hyper-Kähler.

5. Connections with Totally Skew-Symmetric Torsion

In this section we will show that every 3-$(0,\delta )$-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 $T(X,Y,Z):=g\left(T\right(X,Y),Z)$ is a 3-form. The relation between ∇ and the Levi-Civita connection ${\nabla }^g$ is then given by

$${\nabla }_{X}Y={\nabla }_X^{g}Y+\frac{1}{2}T(X,Y).$$

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 $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ is called canonical if the following conditions are satisfied:

(i)

each $N_{{\phi }_i}$ is totally skew-symmetric on $\mathcal{H}$,

(ii)

each ${\xi }_i$ is a Killing vector field,

(iii)

for any $X,Y,Z\in \Gamma \left(\mathcal{H}\right)$ and any $i,j=1,2,3$,

$$N_{{\phi }_i}(X,Y,Z)-d{\mathsf{\Phi }}_i({\phi }_{i}X,{\phi }_{i}Y,{\phi }_{i}Z)=N_{{\phi }_j}(X,Y,Z)-d{\mathsf{\Phi }}_j({\phi }_{j}X,{\phi }_{j}Y,{\phi }_{j}Z),$$

(iv)

M admits a Reeb Killing function $\beta \in C^{\infty }\left(M\right)$, that is the tensor fields $A_{ij}$ defined on $\mathcal{H}$ by

$$A_{ij}(X,Y):=g(\left({\mathcal{L}}_{{\xi }_j}{\phi }_i\right)X,Y)+d{\eta }_j(X,{\phi }_{i}Y)+d{\eta }_j({\phi }_{i}X,Y),$$

satisfy

$$A_{ii}(X,Y)=0,A_{ij}(X,Y)=-A_{ji}(X,Y)=\beta {\mathsf{\Phi }}_k(X,Y),$$

for every $X,Y\in \Gamma \left(\mathcal{H}\right)$ and every even permutation $(i,j,k)$ of $(1,2,3)$.

Here $N_{{\phi }_i}$ also denotes the $(0,3)$-tensor field defined by $N_{{\phi }_i}(X,Y,Z):=g(N_{{\phi }_i}(X,Y),Z)$ and we say that $N_{{\phi }_i}$ is totally skew-symmetric on $\mathcal{H}$ if the $(0,3)$-tensor is a 3-form on $\mathcal{H}$.

Theorem 4

([1]). An almost 3-contact metric manifold $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ is canonical, with Reeb Killing function β, if and only if it admits a metric connection ∇ with skew torsion such that

$${\nabla }_X{\phi }_i=\beta ({\eta }_k\left(X\right){\phi }_j-{\eta }_j\left(X\right){\phi }_k)$$

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

$$\begin{array}{cc}\hfill T(X,Y,Z)& =N_{{\phi }_i}(X,Y,Z)-d{\mathsf{\Phi }}_i({\phi }_{i}X,{\phi }_{i}Y,{\phi }_{i}Z),\hfill \\ \hfill T(X,Y,{\xi }_i)& =d{\eta }_i(X,Y),\hfill \\ \hfill T(X,{\xi }_i,{\xi }_j)& =-g([{\xi }_i,{\xi }_j],X),\hfill \\ \hfill T({\xi }_1,{\xi }_2,{\xi }_3)& =2(\beta -\delta ),\hfill \end{array}$$

for every $X,Y,Z\in \Gamma \left(\mathcal{H}\right)$, 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

$${\nabla }_X{\xi }_i=\beta ({\eta }_k\left(X\right){\xi }_j-{\eta }_j\left(X\right){\xi }_k),{\nabla }_X{\eta }_i=\beta ({\eta }_k\left(X\right){\eta }_j-{\eta }_j\left(X\right){\eta }_k)$$

(20)

for every vector field X on M. Therefore, when $\beta =0$ the canonical connection parallelizes all the structure tensor fields, in which case we call the almost 3-contact metric manifold parallel.

Both 3-$(\alpha ,\delta )$-Sasaki manifolds and 3-$\delta$-cosymplectic manifolds turn out to be canonical. In particular,

Theorem 5

([1]). Every 3-$(\alpha ,\delta )$-Sasaki manifold is a canonical almost 3-contact metric manifold, with constant Reeb Killing function $\beta =2(\delta -2\alpha )$. The torsion T of the canonical connection ∇ is given by

$$T=\sum _{i=1}^3{\eta }_i\wedge d{\eta }_i+8(\delta -\alpha ){\eta }_{123}=2\alpha \sum _{i=1}^3{\eta }_i\wedge {\mathsf{\Phi }}_i+2(\delta -4\alpha ){\eta }_{123}$$

and satisfies $\nabla T=0$.

We denote by ${\eta }_{123}$ the 3-form ${\eta }_1\wedge {\eta }_2\wedge {\eta }_3$. From the above theorem, it follows that any 3-$(\alpha ,\delta )$-Sasaki manifold is a parallel canonical manifold if and only if $\delta =2\alpha$, in which case the 3-$(\alpha ,\delta )$-Sasaki structure is positive ($\alpha \delta >0$).

Regarding 3-$\delta$-cosymplectic manifolds, we have:

Proposition 5

([1]). Any 3-δ-cosymplectic manifold is a parallel canonical almost 3-contact metric manifold. The torsion of the canonical connection is given by

$$T=-2\delta {\eta }_{123}.$$

For the class of 3-$(0,\delta )$-Sasaki manifolds, we have the following

Proposition 6.

Every 3-$(0,\delta )$-Sasaki manifold is a canonical almost 3-contact metric manifold, with constant Reeb Killing function $\beta =2\delta$. The torsion T of the canonical connection ∇ is given by

$$T=2\delta {\eta }_{123},$$

which satisfies $\nabla T=0$.

Proof. 

Let $(M,{\phi }_i,{\xi }_i,{\eta }_i,g)$ be a 3-$(0,\delta )$-Sasaki manifold. We showed that the structure is hypernormal and the Reeb vector fields are Killing. Furthermore, by the second equation in (12), $d{\mathsf{\Phi }}_i(X,Y,Z)=0$ for every $X,Y,Z\in \Gamma \left(\mathcal{H}\right)$. 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 $X,Y\in \Gamma \left(\mathcal{H}\right)$ we have

$$A_{ii}(X,Y)=0,A_{ij}(X,Y)=-A_{ji}(X,Y)=2\delta {\mathsf{\Phi }}_k(X,Y).$$

Hence, the structure is canonical with Reeb commutator function $\beta =2\delta$. 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 $T({\xi }_1,{\xi }_2,{\xi }_3)=2\delta$, so that $T=2\delta {\eta }_{123}$. Although the structure is not parallel when $\delta \ne 0$, the torsion satisfies $\nabla T=0$, as by (20), the 3-form ${\eta }_{123}$ 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).