On nearly Sasakian and nearly cosymplectic manifolds

Abstract

We prove that every nearly Sasakian manifold of dimension greater than five is Sasakian. This provides a new criterion for an almost contact metric manifold to be Sasakian. Moreover, we classify nearly cosymplectic manifolds of dimension greater than five.

1 Introduction

One of the most successful attempts to relax the definition of a Kähler manifold is provided by the notion of a nearly Kähler manifold. Namely, nearly Kähler manifolds are defined as almost Hermitian manifolds (M, J, g) such that the covariant derivative of the almost complex structure with respect to the Levi-Civita connection is skew-symmetric, that is

$$\begin{aligned} (\nabla _{X}J)X = 0, \end{aligned}$$

for every vector field X on M. A remarkable classification of nearly Kähler manifolds was obtained by Nagy in [14]. This result reveals how 6-dimensional nearly Kähler manifolds play a central role, appearing as one of the possible factors in the de Rham decomposition of a complete simply connected strict nearly Kähler manifold.

Notice that in the defining condition of a nearly Kähler manifold, only the symmetric part of $\nabla J$ vanishes, in contrast to the Kähler case where $\nabla J=0$. Nearly Sasakian and nearly cosymplectic manifolds were defined in the same spirit starting from Sasakian and coKähler (sometimes also called cosymplectic) manifolds, respectively.

A smooth manifold M endowed with an almost contact metric structure $(\phi ,\xi ,\eta ,g)$ is said to be nearly Sasakian if

$$\begin{aligned} (\nabla _X\phi )X=g(X,X)\xi -\eta (X)X, \end{aligned}$$

(1)

for every vector field X on M. Similarly, the condition for M to be nearly cosymplectic is given by

$$\begin{aligned} (\nabla _X\phi )X=0, \end{aligned}$$

(2)

for every vector field X on M.

The notion of a nearly Sasakian manifold was introduced by Blair and his collaborators in [4], while nearly cosymplectic manifolds were studied by Blair and Showers in [1, 3]. In the subsequent literature on the topic, quite important were the papers of Olszak [15, 16] for nearly Sasakian manifolds and those of Endo [9, 10] on nearly cosymplectic manifolds. Later on, these two classes have played a role in the Chinea-Gonzalez’s classification of almost contact metric manifolds ([8]). They also appeared in the study of harmonic almost contact structures (cf. [11, 17]). In [13], Loubeau and Vergara-Diaz proved that a nearly cosymplectic structure, once identified with a section of a twistor bundle, always defines a harmonic map.

Recently, a systematic study of nearly Sasakian and nearly cosymplectic manifolds was carried forward in [7]. In that paper, the authors proved that any nearly Sasakian manifold is a contact manifold. In the five-dimensional case, they showed that any nearly Sasakian manifold admits a nearly hypo SU(2)-structure that can be deformed to give a Sasaki–Einstein structure. Moreover, they proved that any nearly Sasakian manifold of dimension 5 has an associated nearly cosymplectic structure, thereby showing the close relation between these two notions. For five-dimensional nearly cosymplectic manifolds, they proved that any such manifold is Einstein with positive scalar curvature. It is also worth remarking that (1-parameter families of) examples of both nearly Sasakian and nearly cosymplectic structures are provided by every 5-dimensional manifold endowed with a Sasaki-Einstein SU(2)-structure.

While Sasakian manifolds are characterized by the equality

$$\begin{aligned} (\nabla _{X}\phi )Y=g(X,Y)\xi -\eta (Y)X, \end{aligned}$$

the defining condition (1) of a nearly Sasakian manifold gives a constraint only on the symmetric part of $\nabla \phi$. In this paper we show that, surprisingly, in dimension higher than five, condition (1) is enough for the manifold to be Sasakian.

Concerning nearly cosymplectic manifolds, we prove that a nearly cosymplectic non-coKähler manifold M of dimension $2n+1>5$ is locally isometric to one of the following Riemannian products:

$$\begin{aligned} \mathbb {R}\times N^{2n}, \qquad M^5\times N^{2n-4}, \end{aligned}$$

where $N^{2n}$ is a nearly Kähler non-Kähler manifold, $N^{2n-4}$ is a nearly Kähler manifold, and $M^5$ is a nearly cosymplectic non-coKähler manifold. If one makes the further assumption that the manifold is complete and simply connected, then the isometry becomes global.

2 Definitions and known results

An almost contact metric manifold is a differentiable manifold M of odd dimension $2n+1$, endowed with a structure $(\phi , \xi , \eta , g)$, given by a tensor field $\phi$ of type (1, 1), a vector field $\xi$, a 1-form $\eta$ and a Riemannian metric g satisfying

$$\begin{aligned} \phi ^2={}-I+\eta \otimes \xi ,\quad \eta (\xi )=1,\quad g(\phi X,\phi Y)=g(X,Y)-\eta (X)\eta (Y) \end{aligned}$$

for all vector fields X, Y on M (see [2, 5] for further details). From the definition it follows that $\phi \xi =0$ and $\eta \circ \phi =0$. Moreover, $\phi$ is skew-symmetric with respect to g, so that the bilinear form $\Phi :=g(-,\phi -)$ defines a 2-form on M, called fundamental 2-form. An almost contact metric manifold such that $\text {d}\eta =2\Phi$ is called a contact metric manifold. In this case $\eta$ is a contact form, i.e., $\eta \wedge (\text {d}\eta )^n\ne 0$ everywhere on M.

A Sasakian manifold is defined as a contact metric manifold such that the tensor field $N_{\phi }:=[\phi ,\phi ]+\text {d}\eta \otimes \xi$ vanishes identically. It is well known that an almost contact metric manifold is Sasakian if and only if the Levi-Civita connection satisfies:

$$\begin{aligned} (\nabla _{X}\phi )Y=g(X,Y)\xi -\eta (Y)X. \end{aligned}$$

(3)

A nearly Sasakian manifold is an almost contact metric manifold $(M,\phi , \xi ,\eta ,g)$ such that

$$\begin{aligned} (\nabla _X\phi )Y+(\nabla _Y\phi )X=2g(X,Y)\xi -\eta (X)Y-\eta (Y)X \end{aligned}$$

(4)

for all vector fields X, Y on M, or, equivalently, (1) is satisfied.

We recall some basic facts about nearly Sasakian manifolds. We refer to [4, 7, 15, 16] for the details.

In any nearly Sasakian manifold $(M,\phi , \xi ,\eta ,g)$, the characteristic vector field $\xi$ is Killing and the Levi-Civita connection satisfies $\nabla _\xi \xi =0$ and $\nabla _\xi \eta =0$. One can define a tensor field h of type (1, 1) by putting

$$\begin{aligned} \nabla _X\xi =-\phi X+hX. \end{aligned}$$

(5)

The operator h is skew-symmetric and anticommutes with $\phi$. It satisfies $h\xi =0$, $\eta \circ h=0$ and

$$\begin{aligned} \nabla _\xi h=\nabla _\xi \phi =\phi h=\frac{1}{3}\mathcal {L}_\xi \phi , \end{aligned}$$

where $\mathcal {L}_\xi$ denotes the Lie derivative with respect to $\xi$. The vanishing of h provides a necessary and sufficient condition for a nearly Sasakian manifold to be Sasakian ([16]). In [15] the following formulas are proved:

$$\begin{aligned} g((\nabla _X\phi )Y, hZ)= & {} \eta (Y)g(h^2X,\phi Z)-\eta (X)g(h^2Y,\phi Z)+\eta (Y)g(hX,Z), \end{aligned}$$

(6)

$$\begin{aligned} (\nabla _Xh^2)Y= & {} \eta (Y)(\phi -h)h^2X+g((\phi -h)h^2X,Y)\xi , \end{aligned}$$

(7)

$$\begin{aligned} R(\xi ,X)Y= & {} (\nabla _X\phi )Y-(\nabla _Xh)Y=g(X-h^2X,Y)\xi -\eta (Y)(X-h^2X), \end{aligned}$$

(8)

where R is the Riemannian curvature of g.

A central role in the study of nearly Sasakian geometry is played by the symmetric operator $h^2$. We recall the fundamental result due to Olszak [15]:

Theorem 2.1

If a nearly Sasakian non-Sasakian manifold $(M,\phi ,\xi ,\eta ,g)$ satisfies the condition

$$\begin{aligned} h^2 = \lambda (I-\eta \otimes \xi ) \end{aligned}$$

for some real number $\lambda$, then $\dim (M)=5$.

In [16] Olszak also proved that any 5-dimensional nearly Sasakian non-Sasakian manifold is Einstein with scalar curvature ${>}20$. In [7] it is proved that the eigenvalues of $h^2$ are constant. Being h skew-symmetric, the nonvanishing eigenvalues of $h^2$ are negative, so that the spectrum of $h^2$ is of type

$$\begin{aligned} \text {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}, \end{aligned}$$

$\lambda _i\ne 0$ and $\lambda _i\ne \lambda _j$ for $i\ne j$. Further, if X is an eigenvector of $h^2$ with eigenvalue $-\lambda _i^2$, then X, $\phi X$, hX, $h\phi X$ are orthogonal eigenvectors of $h^2$ with eigenvalue $-\lambda _i^2$. Hence the minimum dimension for a nearly Sasakian non-Sasakian manifold is 5. In the following we denote by $[\xi ]$ the 1-dimensional distribution generated by $\xi$, and by ${\mathcal D}(0)$ and ${\mathcal D}(-\lambda _i^2)$ the distributions of the eigenvectors 0 and $-\lambda _i^2$, respectively. We shall also denote by $\overline{\mathcal {D}}$ the distribution $\left[ \xi \right] \oplus \mathcal {D}(-\lambda _{1}^2)\oplus \cdots \oplus \mathcal {D}(-\lambda _{r}^2)$, and by ${\mathcal D}_0$ the distribution orthogonal to $\overline{\mathcal D}$, so that $\mathcal D(0)=[\xi ]\oplus {\mathcal {D}}_0$.

We will use the following results, proved in [7], concerning nearly Sasakian manifolds of dimension ${\ge }5$.

Theorem 2.2

Let M be a nearly Sasakian manifold with structure $(\phi ,\xi ,\eta ,g)$ and let $\mathrm {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}$ be the spectrum of $h^2$. Then the distributions $\mathcal D(0)$ and $[\xi ]\oplus \mathcal D(-\lambda _i^2)$ are integrable with totally geodesic leaves. In particular,

• (a) the eigenvalue 0 has multiplicity $2p+1$, $p\ge 0$. If $p>0$, the leaves of $\mathcal D(0)$ are $(2p+1)$-dimensional Sasakian manifolds;

• (b) each negative eigenvalue $-\lambda _i^2$ has multiplicity 4 and the leaves of the distribution $[\xi ]\oplus {\mathcal D}(-\lambda _i^2)$ are 5-dimensional nearly Sasakian (non-Sasakian) manifolds.

• (c) If $p>0$, the distribution $\overline{\mathcal {D}}=\left[ \xi \right] \oplus \mathcal {D}(-\lambda _{1}^2)\oplus \cdots \oplus \mathcal {D}(-\lambda _{r}^2)$ is integrable with totally geodesic leaves.

Theorem 2.3

For a nearly Sasakian manifold $(M,\phi ,\xi ,\eta ,g)$ of dimension $2n+1\ge 5$ the 1-form $\eta$ is a contact form.

Before listing some known results on nearly cosymplectic manifolds, we recall that an almost contact metric manifold $(M,\phi ,\xi ,\eta ,g)$ is said to be a coKähler manifold if $d\eta =0$, $d\Phi =0$ and $N_{\phi }\equiv 0$. Equivalently, one can require $\nabla \phi =0$. It is known that a coKähler manifold is locally the Riemannian product of the real line and a Kähler manifold, which is an integral submanifold of the distribution $\mathcal {D}=\mathrm {Ker}(\eta )$. Note that some authors call cosymplectic the class of manifold that we denominate coKähler (see [6] for details).

A nearly cosymplectic manifold is an almost contact metric manifold $(M,\phi , \xi ,\eta ,g)$ such that

$$\begin{aligned} (\nabla _X\phi )Y+(\nabla _Y\phi )X=0 \end{aligned}$$

(9)

for all vector fields X, Y. Clearly, this condition is equivalent to (2). It is known that in a nearly cosymplectic manifold the Reeb vector field $\xi$ is Killing and satisfies $\nabla _\xi \xi =0$ and $\nabla _\xi \eta =0$. The tensor field h of type (1, 1) defined by

$$\begin{aligned} \nabla _X\xi =hX \end{aligned}$$

(10)

is skew-symmetric and anticommutes with $\phi$. It satisfies $h\xi =0$, $\eta \circ h=0$ and

$$\begin{aligned} \nabla _\xi \phi =\phi h=\frac{1}{3}{\mathcal L}_\xi \phi . \end{aligned}$$

The following formulas hold ([9, 10]):

$$\begin{aligned} g((\nabla _X\phi )Y, hZ)&=\eta (Y)g(h^2X,\phi Z)-\eta (X)g(h^2Y,\phi Z), \end{aligned}$$

(11)

$$\begin{aligned} (\nabla _Xh)Y&=g(h^2X,Y)\xi -\eta (Y)h^2X, \end{aligned}$$

(12)

$$\begin{aligned} \mathrm {tr}(h^2)&=\mathrm {constant}. \end{aligned}$$

(13)

3 Nearly Sasakian manifolds

We start by computing the covariant derivatives of the structure endomorphisms $\phi$ and h on a nearly Sasakian manifold.

Proposition 3.1

Let $(M,\phi ,\xi ,\eta ,g)$ be a nearly Sasakian manifold of dimension $2n+1\ge 5$. Then for all vector fields X, Y on M one has

$$\begin{aligned} (\nabla _{X}\phi )Y= & {} \eta (X) \phi h Y - \eta (Y) (X + \phi h X) + g(X+\phi h X, Y)\xi , \end{aligned}$$

(14)

$$\begin{aligned} (\nabla _{X}h)Y= & {} \eta (X)\phi h Y - \eta (Y) (h^2 X + \phi h X) + g(h^{2}X + \phi h X,Y)\xi , \end{aligned}$$

(15)

$$\begin{aligned} (\nabla _{X}\phi h)Y= & {} g(\phi h^2 X - hX,Y)\xi + \eta (X)(\phi h^2 Y - hY) - \eta (Y) (\phi h^2 X - hX). \end{aligned}$$

(16)

Proof

From (6), for all vector fields X, Y, Z we have

$$\begin{aligned} g((\nabla _X\phi )Y, hZ)=-\eta (Y)g(\phi h X, hZ)+\eta (X)g(\phi hY,hZ)-\eta (Y)g(X,hZ), \end{aligned}$$

which is coherent with (14). On the other hand,

$$\begin{aligned} g((\nabla _X\phi )Y, \xi )&=-g(Y, (\nabla _X\phi )\xi )=g(Y,\phi \nabla _X\xi )=g(Y,-\phi ^2X+\phi hX)\\&=g(X+\phi hX,Y)-\eta (X)\eta (Y). \end{aligned}$$

Now, assume that $\text {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}$ and consider the distribution $\overline{\mathcal {D}}=\left[ \xi \right] \oplus \mathcal {D}(-\lambda _{1}^2)\oplus \cdots \oplus \mathcal {D}(-\lambda _{r}^2)$. In order to complete the proof of (14), it remains to show that

$$\begin{aligned} g((\nabla _X\phi )Y, V)=-\eta (Y)g(X,V) \end{aligned}$$

(17)

for every $X,Y\in {\mathfrak {X}}(M)$ and $V\in {\mathcal D}_0$. Since the distribution $\overline{\mathcal {D}}$ is integrable with totally geodesic leaves, if $X,Y\in \overline{\mathcal {D}}$, then $(\nabla _X\phi )Y\in \overline{\mathcal {D}}$ and both sides in (17) vanish. Now consider $X\in \mathcal {D}_0$ and $Y\in \overline{\mathcal {D}}$. Then

$$\begin{aligned} g((\nabla _X\phi )Y, V)=-g(Y, (\nabla _X\phi )V)=-\eta (Y)g(X,V), \end{aligned}$$

where we applied the fact that the distribution ${\mathcal D}(0)=[\xi ]\oplus \mathcal {D}_0$ is integrable with totally geodesic leaves, and the induced almost contact metric structure on each leaf is Sasakian, so that $(\nabla _X\phi )V=g(X,V)\xi -\eta (V)X$. On the other hand, if we take $X\in \overline{\mathcal {D}}$ and $Y\in \mathcal {D}_0$, then $g((\nabla _Y\phi )X, V)=-\eta (X)g(Y,V)$, and applying (4), we have

$$\begin{aligned} g((\nabla _X\phi )Y, V)=-g((\nabla _Y\phi )X+\eta (X)Y, V)=0. \end{aligned}$$

Finally, taking $X,Y\in \mathcal {D}_0$, (17) is verified because of (3) and the fact that the vector fields X, Y, V are orthogonal to $\xi$.

As regards (15), it follows from (8) and (14). Finally, a straightforward computation using (14) and (15) gives (16). $\square$

We will write $\epsilon _{d\eta }$ for the operator on $\Omega ^*(M)$ defined by $\omega \mapsto {d\eta }\wedge \omega$.

Proposition 3.2

Let $(M,\eta )$ be a contact manifold of dimension $2n+1$. Then, the operator

$$\begin{aligned} \epsilon _{d\eta }:\Omega ^2(M)&\rightarrow \Omega ^{4}(M)\\ \beta&\mapsto d\eta \wedge \beta \end{aligned}$$

is injective for $n\ge 3$.

Proof

Since $\text {d}\eta$ is a nondegenerate 2-form on the distribution ${\mathcal D}=\mathrm {Ker}(\eta )$, the assumption $n\ge 3$ implies that the operators

$$\begin{aligned} \epsilon _{d\eta }:\Omega ^1({\mathcal D}) \rightarrow \Omega ^{3}({\mathcal D})\qquad \alpha \mapsto d\eta \wedge \alpha \end{aligned}$$

(18)

and

$$\begin{aligned} \epsilon _{d\eta }:\Omega ^2({\mathcal D}) \rightarrow \Omega ^{4}({\mathcal D})\qquad \beta \mapsto d\eta \wedge \beta . \end{aligned}$$

(19)

are injective. For every $k\ge 1$ we have

$$\begin{aligned} \Omega ^k(M)=\Omega ^k({\mathcal D})\oplus \eta \wedge \Omega ^{k-1}({\mathcal D}). \end{aligned}$$

(20)

Indeed, every k-form $\omega$ on M can be decomposed as

$$\begin{aligned} \omega =i_\xi (\eta \wedge \omega )+\eta \wedge i_\xi \omega . \end{aligned}$$

On the other hand, if a k-form $\omega$ belongs to the intersection of the two subspaces, that is $\omega \in \Omega ^k({\mathcal D})$ and $\omega =\eta \wedge \sigma$, with $\sigma \in \Omega ^{k-1}({\mathcal D})$, then

$$\begin{aligned} \sigma =i_\xi (\eta \wedge \sigma )+\eta \wedge i_\xi \sigma =i_\xi \omega =0, \end{aligned}$$

and thus $\omega =0$. This shows that the sum in (20) is direct.

Now, let $\omega =\beta +\eta \wedge \alpha$, with $\beta \in \Omega ^2({\mathcal D})$ and $\alpha \in \Omega ^1({\mathcal D})$, be a 2-form on M such that $\text {d}\eta \wedge \omega =0$. Then, owing to (20) for $k=4$, we have $d\eta \wedge \beta =0$ and $\text {d}\eta \wedge \eta \wedge \alpha =0$, which also gives $\text {d}\eta \wedge \alpha =0$. Finally, we deduce from the injectivity of the operators in (18) and (19) that both the forms $\beta$ and $\alpha$ vanish, and thus $\omega =0$. $\square$

Now we are able to prove our main result.

Theorem 3.3

Every nearly Sasakian manifold of dimension $2n+1>5$ is Sasakian.

Proof

Let M be a nearly Sasakian manifold with structure $(\phi ,\xi ,\eta ,g)$, of dimension $2n+1$. We consider the 2-forms H and $\Phi _k$, $k=1,2$, defined by

$$\begin{aligned} H(X,Y)=g(hX,Y),\qquad \Phi _k(X,Y)=g(\phi h^k X,Y). \end{aligned}$$

We shall prove that

$$\begin{aligned} \text {d}H&=3\eta \wedge \Phi _1, \end{aligned}$$

(21)

$$\begin{aligned} \text {d}\Phi _1&=3\eta \wedge (\Phi _2-H). \end{aligned}$$

(22)

From (15), we have that for all vector fields X, Y, Z,

$$\begin{aligned} g((\nabla _{X}h)Y,Z)&= \eta (X)g(\phi h Y,Z) - \eta (Y) g(h^2 X + \phi h X,Z) + \eta (Z)g(h^{2}X + \phi h X,Y)\\&= \eta (X)g(\phi h Y,Z) +\eta (Y) g(\phi h Z,X)+\eta (Z)g(\phi h X,Y)\\&\quad - \eta (Y) g(h^2 Z,X)+ \eta (Z)g(h^{2}X ,Y). \end{aligned}$$

Therefore,

$$\begin{aligned} \text {d}H(X,Y,Z)&=g((\nabla _{X}h)Y,Z)+g((\nabla _{Y}h)Z,X)+g((\nabla _{Z}h)X,Y)\\&=3\left( \eta (X)g(\phi h Y,Z) +\eta (Y) g(\phi h Z,X)+\eta (Z)g(\phi h X,Y)\right) \\&= 3\eta \wedge \Phi _1(X,Y,Z). \end{aligned}$$

Analogously, from (16), we have

$$\begin{aligned} g((\nabla _{X}\phi h)Y,Z)&= \eta (X)g(\phi h^2 Y-hY,Z) - \eta (Y) g(\phi h^2 X - h X,Z)\\ {}&\quad + \eta (Z)g(\phi h^2X -h X,Y)\\&= \eta (X)g(\phi h^2 Y,Z) +\eta (Y) g(\phi h^2 Z,X)+\eta (Z)g(\phi h^2 X,Y)\\&\quad -\eta (X)g( hY,Z) -\eta (Y) g(hZ,X)-\eta (Z)g(h X,Y). \end{aligned}$$

Hence,

$$\begin{aligned} \text {d}\Phi _1(X,Y,Z)&=g((\nabla _{X}\phi h)Y,Z)+g((\nabla _{Y}\phi h)Z,X)+g((\nabla _{Z}\phi h)X,Y)\\&=3\left( \eta (X)g(\phi h^2 Y,Z) +\eta (Y) g(\phi h^2 Z,X)+\eta (Z)g(\phi h^2 X,Y)\right) \\&\quad -3\left( \eta (X)g( hY,Z)+ \eta (Y) g(hZ,X)+\eta (Z)g(h X,Y)\right) \\&= 3\,\eta \wedge \Phi _2(X,Y,Z)-3\,\eta \wedge H(X,Y,Z). \end{aligned}$$

Now, from (21) and (22), we have

$$\begin{aligned} 0=\text {d}^2H=3\,\text {d}\eta \wedge \Phi _1-3\eta \wedge \text {d}\Phi _1=3\,\text {d}\eta \wedge \Phi _1. \end{aligned}$$

If we assume that the dimension of M is $2n+1>5$, $\eta$ being a contact form, the fact that $\text {d}\eta \wedge \Phi _1=0$ implies $\Phi _1=0$, by Proposition 3.2. Therefore $h=0$, and the structure is Sasakian. $\square$

4 Nearly cosymplectic manifolds

In this section we will classify nearly cosymplectic manifolds of dimension higher than five. In the following, given a nearly cosymplectic manifold $(M,\phi ,\xi ,\eta ,g)$, we shall denote by h the operator defined in (10).

Proposition 4.1

Let $(M,\phi ,\xi ,\eta ,g)$ be a nearly cosymplectic manifold. Then $h=0$ if and only if M is locally isometric to the Riemannian product $\mathbb {R}\times N$, where N is a nearly Kähler manifold.

Proof

For every vector fields X, Y we have

$$\begin{aligned} \text {d}\eta (X,Y)=g(\nabla _X\xi ,Y)-g(\nabla _Y\xi ,X)=2g(hX,Y). \end{aligned}$$

(23)

Therefore, if $h=0$, the distribution $\mathcal D=\mathrm {Ker}(\eta )$ is integrable. Denoting by N an integral submanifold of ${\mathcal D}$, it is a totally geodesic hypersurface of M. Indeed, for every $X,Y\in \mathcal {D}$, we have $g(\nabla _XY,\xi )=-g(Y,hX)=0$. Being also $\nabla _\xi \xi =0$, M turns out to be locally isometric to the Riemannian product $\mathbb {R}\times N$. Further, the almost contact metric structure induces on N an almost Hermitian structure which is nearly Kähler.

Conversely, if M is locally isometric to the Riemannian product $\mathbb {R}\times N$, where N is a nearly Kähler manifold, then $\text {d}\eta (X,Y)=0$ for all vector fields X, Y orthogonal to $\xi$. By (23) and $h\xi =0$, we deduce that $h=0$. $\square$

As a consequence of the above proposition, a nearly cosymplectic manifold $(M,\phi ,\xi ,\eta ,g)$ is coKähler if and only if $h=0$ and the leaves of the distribution $\mathcal D$ are Kähler manifolds. Recall that 4-dimensional nearly Kähler manifolds are Kähler (see [12, Theorem 5.1]), and this implies that if M is a 5-dimensional nearly cosymplectic manifold with $h=0$, then it is a coKähler manifold.

We shall now study the spectrum of the symmetric operator $h^2$.

Proposition 4.2

The eigenvalues of the symmetric operator $h^2$ are constant.

Proof

From (12) it follows that

$$\begin{aligned} (\nabla _Xh^2)Y=g(X,h^3Y)\xi -\eta (Y)h^3X. \end{aligned}$$

(24)

Let us consider an eigenvalue $\mu$ of $h^2$ and a local unit vector field Y, orthogonal to $\xi$, such that $h^2Y=\mu Y$. Applying (24) for any vector field X, we have

$$\begin{aligned} 0&=g((\nabla _Xh^2)Y,Y)\\&=g(\nabla _X(h^2Y),Y)-g(h^2(\nabla _XY),Y)\\&=X(\mu )g(Y,Y)+\mu g(\nabla _XY,Y)-g(\nabla _XY,h^2Y)\\&=X(\mu )g(Y,Y) \end{aligned}$$

which implies that $X(\mu )=0$. $\square$

Since h is skew-symmetric, the nonvanishing eigenvalues of $h^2$ are negative. Therefore, the spectrum of $h^2$ is of type

$$\begin{aligned} \text {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}, \end{aligned}$$

where we can assume that each $\lambda _i$ is a positive real number and $\lambda _i\ne \lambda _j$ for $i\ne j$. Notice that if X is an eigenvector of $h^2$ with eigenvalue $-\lambda _i^2$, then X, $\phi X$, hX, $h\phi X$ are orthogonal eigenvectors of $h^2$ with eigenvalue $-\lambda _i^2$. Since $h(\xi )=0$, we get the eigenvalue 0 has multiplicity $2p+1$ for some integer $p\ge 0$.

We denote by ${\mathcal D}(0)$ the distribution of the eigenvectors with eigenvalue 0, and by ${\mathcal D}_0$ the distribution of the eigenvectors in ${\mathcal D}(0)$ orthogonal to $\xi$, so that ${\mathcal D}(0)=[\xi ]\oplus {\mathcal D}_0$. Let ${\mathcal D}(-\lambda _i^2)$ be the distribution of the eigenvectors with eigenvalue $-\lambda _i^2$. We remark that the distributions ${\mathcal D}_0$ and ${\mathcal D}(-\lambda _i^2)$ are $\phi$-invariant and h-invariant.

Proposition 4.3

Let $(M,\phi ,\xi ,\eta ,g)$ be a nearly cosymplectic manifold and let $\mathrm {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}$ be the spectrum of $h^2$. Then,

• (a) for each $i=1,\ldots , r$, the distribution $[\xi ]\oplus \mathcal D(-\lambda _i^2)$ is integrable with totally geodesic leaves.

Assuming that the eigenvalue 0 is not simple,

• (b) the distribution $\mathcal D_0$ is integrable with totally geodesic leaves, and each leaf of $\mathcal D_0$ is endowed with a nearly Kähler structure;

• (c) the distribution $[\xi ]\oplus \mathcal D(-\lambda _1^2)\oplus \ldots \oplus \mathcal D(-\lambda _r^2)$ is integrable with totally geodesic leaves.

Proof

Consider an eigenvector X of $h^2$ with eigenvalue $-\lambda _i^2$. Then $\nabla _X\xi =hX\in {\mathcal D}(-\lambda _i^2)$. On the other hand, (24) implies that $\nabla _\xi h^2=0$, and thus $\nabla _\xi X$ is also an eigenvector with eigenvalue $-\lambda _i^2$. Now, taking $X,Y\in {\mathcal D}(-\lambda _i^2)$ and applying (24), we get

$$\begin{aligned} h^2(\nabla _XY)=-\lambda _i^2\nabla _XY-(\nabla _Xh^2)Y=-\lambda _i^2\nabla _XY+\lambda _i^2 g(X,hY)\xi . \end{aligned}$$

Therefore,

$$\begin{aligned} h^2(\phi ^2\nabla _XY)=\phi ^2(h^2\nabla _XY)=-\lambda _i^2\phi ^2(\nabla _XY). \end{aligned}$$

Thus $\phi ^2\nabla _XY\in {\mathcal D}(-\lambda _i^2)$. It follows that $\nabla _XY=-\phi ^2 \nabla _XY +\eta (\nabla _XY)\xi$ belongs to the distribution $[\xi ]\oplus {\mathcal D}(-\lambda _i^2)$. This proves (a).

As regards (b), applying again (24), we have $(\nabla _Xh^2)Y=0$ for every $X,Y\in {\mathcal D}_0$, so that $h^2(\nabla _XY)=0$. Moreover,

$$\begin{aligned} g(\nabla _XY,\xi )=-g(Y,\nabla _X\xi )=-g(Y,hX)=0. \end{aligned}$$

Hence, ${\mathcal D}_0$ is integrable with totally geodesic leaves. Since the leaves of ${\mathcal D}_0$ are $\phi$-invariant, the nearly cosymplectic structure induces a nearly Kähler structure on each integral submanifold of ${\mathcal D}_0$.

Finally, in order to prove (c), owing to (a), we only have to show that

$$\begin{aligned} g(\nabla _XY,Z)=0 \end{aligned}$$

for every $X\in {\mathcal D}(-\lambda _i^2)$, $Y\in {\mathcal D}(-\lambda _j^2)$, $i\ne j$, and $Z\in {\mathcal D}_0$. In fact, from (24), we have

$$\begin{aligned} g(\nabla _XY,Z)&=-\frac{1}{\lambda _j^2}g(\nabla _X(h^2Y),Z)\\&=-\frac{1}{\lambda _j^2}g((\nabla _Xh^2)Y+h^2(\nabla _XY),Z)\\&=-\frac{1}{\lambda _j^2}\eta (Z)g(X,h^3Y)-\frac{1}{\lambda _j^2}g(\nabla _XY,h^2Z) \end{aligned}$$

which vanishes since $\eta (Z)=0$ and $h^2Z=0$. $\square$

Theorem 4.4

Let $(M,\phi ,\xi ,\eta ,g)$ be a nearly cosymplectic manifold such that 0 is a simple eigenvalue of $h^2$. Then M is a 5-dimensional manifold.

Proof

First we show that

$$\begin{aligned} (\nabla _X\phi )Y&=g(\phi hX,Y)\xi +\eta (X)\phi hY-\eta (Y)\phi hX, \end{aligned}$$

(25)

$$\begin{aligned} (\nabla _X\phi h)Y&=g(\phi h^2X,Y)\xi +\eta (X)\phi h^2Y-\eta (Y)\phi h^2X \end{aligned}$$

(26)

for all vector fields X and Y. Applying (10) we have

$$\begin{aligned} g((\nabla _X\phi )Y,\xi )=-g(Y,(\nabla _X\phi )\xi )=g(Y,\phi \nabla _X\xi )= g(Y,\phi hX). \end{aligned}$$

Taking a vector field U orthogonal to $\xi$, then $U=hZ$ for some vector field Z. Then, by applying (11) and recalling that $\phi$ anticommutes with h, we get

$$\begin{aligned} g((\nabla _X\phi )Y,U)&= \eta (Y)g(h^2X,\phi Z)-\eta (X)g(h^2Y,\phi Z)\\&= \eta (Y)g(hX,\phi hZ)-\eta (X)g(hY,\phi hZ)\\&= {}-\eta (Y)g(\phi hX,U)+\eta (X)g(\phi hY,U) \end{aligned}$$

which completes the proof of (25). From (12) and (25) we easily get (26).

We consider now the 2-forms $\Phi _k$, $k=0,1,2$, defined by

$$\begin{aligned} \Phi _k(X,Y)=g(\phi h^k X,Y). \end{aligned}$$

In particular, $\Phi _0=-\Phi$. We prove that

$$\begin{aligned} \text {d}\Phi _0=3\eta \wedge \Phi _1,\qquad \text {d}\Phi _1=3\eta \wedge \Phi _2. \end{aligned}$$

(27)

From (25), for all vector fields X, Y, Z we have

$$\begin{aligned} g((\nabla _X\phi )Y,Z)=\eta (X)g(\phi hY,Z)+\eta (Y)g(\phi hZ,X)+\eta (Z)g(\phi hX,Y), \end{aligned}$$

which implies that $\text {d}\Phi _0=3\eta \wedge \Phi _1.$ Analogously, from (26), we have

$$\begin{aligned} g((\nabla _X\phi h)Y,Z)=\eta (X)g(\phi h^2Y,Z)+\eta (Y)g(\phi h^2Z,X)+\eta (Z)g(\phi h^2X,Y), \end{aligned}$$

so that $\text {d}\Phi _1=3\eta \wedge \Phi _2.$ From (27),

$$\begin{aligned} 0=\text {d}^2\Phi _0=3\,\text {d}\eta \wedge \Phi _1-3\eta \wedge \text {d}\Phi _1=3\,\text {d}\eta \wedge \Phi _1. \end{aligned}$$

Next we show that if 0 is a simple eigenvalue, then $\eta$ is a contact form. This, by an argument similar to the one in the proof of Theorem 3.3, will imply that $\dim M=5$.

First we assume that $\text {Spec}(h^2)=\{0,-\lambda ^2\}$, with $\lambda >0$, 0 being a simple eigenvalue. This is equivalent to require that

$$\begin{aligned} h^2=-\lambda ^2(I-\eta \otimes \xi ). \end{aligned}$$

Let us take the tensor fields

$$\begin{aligned} \tilde{\phi }=-\frac{1}{\lambda }h,\quad \tilde{\xi }=\frac{1}{\lambda }\xi ,\quad \tilde{\eta }=\lambda \eta ,\quad \tilde{g}=\lambda ^2 g. \end{aligned}$$

One can verify that $(\tilde{\phi }, \tilde{\xi }, \tilde{\eta }, \tilde{g})$ is an almost contact metric structure. Moreover, from (23) we have

$$\begin{aligned} \text {d}\tilde{\eta }(X,Y)=2\lambda g(hX,Y)=\frac{2}{\lambda }\,\tilde{g}(hX,Y)=2\,\tilde{g}(X,-\frac{1}{\lambda }hY)=2\,\tilde{g}(X,\tilde{\phi }Y). \end{aligned}$$

Therefore $(\tilde{\phi }, \tilde{\xi }, \tilde{\eta }, \tilde{g})$ is a contact metric structure. In particular, both the forms $\tilde{\eta }$ and $\eta$ are contact forms. Hence, in this case M is a 5-dimensional manifold and the multiplicity of the eigenvalue $-\lambda ^2$ is 4.

We assume now that

$$\begin{aligned} \text {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}, \end{aligned}$$

where $\lambda _i$ is a positive real number and $\lambda _i\ne \lambda _j$ for $i\ne j$. From Proposition 4.3, we know that for each $i=1,\ldots , r$, the distribution $[\xi ]\oplus {\mathcal D}(-\lambda _i^2)$ is integrable with totally geodesic leaves. Each integral submanifold of this distribution is endowed with an induced almost contact metric structure, here again denoted by $(\phi ,\xi ,\eta ,g)$, whose structure tensor field h satisfies

$$\begin{aligned} h^2=-\lambda _i^2(I-\eta \otimes \xi ). \end{aligned}$$

We deduce that $\eta$ is a contact form on the leaves of the distribution. In particular, each eigenvalue $-\lambda _i^2$ of $h^2$ has multiplicity 4.

Notice that, taking two distinct eigenvalues $-\lambda _{i}^{2}$ and $-\lambda _{j}^{2}$, for every $X\in {\mathcal D}(-\lambda _{i}^{2})$ and $Y\in {\mathcal D}(-\lambda _{j}^{2})$, we have

$$\begin{aligned} \text {d}\eta (X,Y)=2g(hX,Y)=0, \end{aligned}$$

(28)

since the operator h preserves the distributions ${\mathcal D}(-\lambda _{i}^2)$ and ${\mathcal D}(-\lambda _{j}^2)$, which are mutually orthogonal.

Now, fix a point $x \in M$. Since $\eta$ is a contact form on the leaves of each distribution $[\xi ]\oplus {\mathcal D}(-\lambda _{i}^2)$, for any $i\in \left\{ 1,\ldots ,r\right\}$ one can find a basis $(v_{1}^{i}, v_{2}^{i}, v_{3}^{i}, v_{4}^{i})$ of ${\mathcal D}_{x}(-\lambda _{i}^{2})$ such that

$$\begin{aligned} \eta \wedge (\text {d}\eta )^{2}(\xi _{x}, v_{1}^{i}, v_{2}^{i}, v_{3}^{i}, v_{4}^{i})\ne 0. \end{aligned}$$

(29)

Therefore, putting $n=2r$, the dimension of M is $2n+1$ and

$$\begin{aligned}&\eta \wedge (\text {d}\eta )^{n} \left( \xi _{x}, v_{1}^{1}, v_{2}^{1}, v_{3}^{1}, v_{4}^{1}, \ldots , v_{1}^{r}, v_{2}^{r}, v_{3}^{r}, v_{4}^{r}\right) \\&\quad = \eta (\xi _{x}) (\text {d}\eta )^{2}(v_{1}^{1}, v_{2}^{1}, v_{3}^{1}, v_{4}^{1}) \ldots (\text {d}\eta )^{2}(v_{1}^{r}, v_{2}^{r}, v_{3}^{r}, v_{4}^{r}) \ne 0. \end{aligned}$$

This proves that $\eta$ is a contact form. $\square$

Theorem 4.5

Let $(M,\phi ,\xi ,\eta ,g)$ be a nearly cosymplectic non-coKähler manifold of dimension $2n+1>5$. Then M is locally isometric to one of the following Riemannian products:

$$\begin{aligned} \mathbb {R}\times N^{2n}, \qquad M^5\times N^{2n-4}, \end{aligned}$$

where $N^{2n}$ is a nearly Kähler non-Kähler manifold, $N^{2n-4}$ is a nearly Kähler manifold, and $M^5$ is a nearly cosymplectic non-coKähler manifold.

Proof

If $h=0$, then M is locally isometric to the Riemannian product $\mathbb {R}\times N^{2n}$, where $N^{2n}$ is a nearly Kähler non-Kähler manifold.

If $h\ne 0$, then $h^2$ admits nonvanishing eigenvalues and we can assume $\text {Spec}(h^2)=\{0,-\lambda _1^2,\ldots ,-\lambda _r^2\}$, where each $\lambda _i$ is a positive real number. Since $\dim M>5$, owing to Theorem 4.4, the eigenvalue 0 is not a simple eigenvalue. From (b) and (c) of Proposition 4.3, M is locally isometric to the Riemannian product $M^{\prime }\times N$, where $M^{\prime }$ is an integral submanifold of the distribution $[\xi ]\oplus \mathcal D(-\lambda _1^2)\oplus \ldots \oplus \mathcal D(-\lambda _r^2)$, and N is an integral submanifold of ${\mathcal D}_0$, which is endowed with a nearly Kähler structure. Now, $M^{\prime }$ is endowed with an induced nearly cosymplectic structure for which 0 is a simple eigenvalue of the operator $h^2$. Therefore, by Theorem 4.4, we have that $\lambda _1=\ldots =\lambda _r$ and $M'$ is a 5-dimensional nearly cosymplectic non-coKähler manifold. Consequently, the dimension of N is $2n-4$. $\square$

Remark 4.6

Note that if the manifold M in Theorem 4.5 is assumed to be complete and simply connected, then, by the de Rham decomposition theorem, the isometry becomes global as the involved distributions are parallel with respect to the Levi-Civita connection. Note also that the nearly Kähler factor can be further decomposed. See Theorem 1.1 and Proposition 2.1 in [14] for details.