Weak Quasi-Contact Metric Manifolds and New Characteristics of K-Contact and Sasakian Manifolds

Abstract

Quasi-contact metric manifolds (introduced by Y. Tashiro and then studied by several authors) are a natural extension of contact metric manifolds. Weak almost-contact metric manifolds, i.e., where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor, have been defined by the author and R. Wolak. In this paper, we study a weak analogue of quasi-contact metric manifolds. Our main results generalize some well-known theorems and provide new criterions for K-contact and Sasakian manifolds in terms of conditions on the curvature tensor and other geometric objects associated with the weak quasi-contact metric structure.

1. Introduction

Contact Riemannian geometry plays an important role in both mathematics and physics. It considers a $(2n+1)$-dimensional smooth manifold M equipped with an almost-contact metric (a.c.m.) structure $(f,\xi ,\eta ,g)$, where g is a Riemannian metric, f is a $(1,1)$-tensor, $\xi$ is a vector field and $\eta$ is a 1-form satisfying

$$\begin{array}{cc}\hfill & f^2=-\mathrm{id}+\eta \otimes \xi ,\eta \left(\xi \right)=1,\hfill \\ \hfill & g(fX,fY)=g(X,Y)-\eta \left(X\right)\eta \left(Y\right),\eta \left(X\right)=g(\xi ,X),\hfill \end{array}$$

where $X,Y\in {\mathfrak{X}}_M$ and ${\mathfrak{X}}_M$ are the Lie algebra of smooth vector fields on M. In [1], D. Chinea and C. Gonzalez obtained a classification of a.c.m. manifolds which was analogous to the classification of almost-Hermitian manifolds established by A. Gray and H.M. Hervella; see [2]. Let $M^{2n+1}(f,\xi ,\eta ,g)$ be an almost-contact metric manifold. The tensor ${\mathcal{N}}^{\left(1\right)}$ is defined by

$${\mathcal{N}}^{\left(1\right)}=[f,f]+2d\eta \otimes \xi ,$$

where $2d\eta (X,Y)=X\left(\eta \left(Y\right)\right)-Y\left(\eta \left(X\right)\right)-\eta \left([X,Y]\right)=\left({\nabla }_X\eta \right)Y-\left({\nabla }_Y\eta \right)X$, and

$$[f,f](X,Y)=f^2[X,Y]+[fX,fY]-f[fX,Y]-f[X,fY],$$

is the Nijenhuis torsion of f. The (1,1)-tensor h is defined by

$$h=(1/2){\mathcal{L}}_{\xi }f,$$

where $\left({\mathcal{L}}_{Z}f\right)X=[Z,fX]-f[Z,X]$ is the Lie derivative, as in [3]. On a contact metric manifold, h vanishes if and only if $\xi$ is a Killing vector field. A contact metric manifold $M(f,\xi ,\eta ,g)$ for which $\xi$ is a Killing vector field is called a K-contact manifold, as in [3]. The following classes of a.c.m. manifolds are well known:

(1)

$\mathcal{M}$—normal a.c.m. manifolds, characterized by the equality ${\mathcal{N}}^{\left(1\right)}=0$.

(2)

$\mathcal{S}$—Sasakian manifolds, characterized by the equality; see ([3] Theorem 6.3),

$$\left({\nabla }_{X}f\right)Y=g(X,Y)\xi -\eta \left(Y\right)X,X,Y\in {\mathfrak{X}}_M.$$

(1)

(3)

$\mathcal{C}$—contact metric manifolds, characterized by the equality $d\eta =\Phi$, where $\Phi (X,Y)=g(X,fY)$.

(4)

$\mathcal{N}$—nearly-Sasakian manifolds, characterized by the equality; see [4]:

$$\left({\nabla }_{Y}f\right)Y=g(Y,Y)\xi -\eta \left(Y\right)Y,Y\in {\mathfrak{X}}_M,$$

(5)

$\mathcal{Q}$—quasi-contact metric (q.c.m.) manifolds, characterized by the equality

$$\left({\nabla }_{X}f\right)Y+\left({\nabla }_{fX}f\right)fY=2g(X,Y)\xi -\eta \left(Y\right)\left(X+hX+\eta \left(X\right)\xi \right),X,Y\in {\mathfrak{X}}_M.$$

(2)

Thus, $\mathcal{N}\subset \mathcal{S}=\mathcal{M}\cap \mathcal{C}$ and $\mathcal{C}\subset \mathcal{Q}$. For the last inclusion, see ([3] Lemma 7.3); the inverse is true for dimension 3, but for dimensions $>3$ it is an open question; see [5,6,7].

By means of the almost-Hermitian cone (see Section 2), a.c.m. manifolds and almost-Hermitian manifolds correspond to each other.

(1)

$M(f,\xi ,\eta ,\overline{g})$ belongs to $\mathcal{M}$ if and only if $\overline{M}(J,\overline{g})$ belongs to $\mathcal{H}$—Hermitian manifolds, defined by $[J,J]=0$.

(2)

$M(f,\xi ,\eta ,\overline{g})$ belongs to $\mathcal{S}$ if and only if $\overline{M}(J,\overline{g})$ belongs to $\mathcal{K}$—Kähler manifolds, defined by $\overline{\nabla }J=0$, where $\overline{\nabla }$ is the Levi–Civita connection for $\overline{g}$.

(3)

$M(f,\xi ,\eta ,\overline{g})$ belongs to $\mathcal{C}$ if and only if $\overline{M}(J,\overline{g})$ belongs to $\mathcal{AK}$—almost-Kähler manifolds, defined by $d\Omega =0$, where $\Omega (X,Y)=\overline{g}(X,JY)$.

(4)

$M(f,\xi ,\eta ,\overline{g})$ belongs to $\mathcal{N}$ if and only if $\overline{M}(J,\overline{g})$ belongs to $\mathcal{NK}$—nearly-Kähler manifolds, defined by $\left({\overline{\nabla }}_{X}J\right)X=0$.

(5)

$M(f,\xi ,\eta ,\overline{g})$ belongs to $\mathcal{Q}$ if and only if $\overline{M}(J,\overline{g})$ belongs to $\mathcal{QK}$—quasi-Kähler manifolds, defined by $\left({\overline{\nabla }}_{X}J\right)Y+\left({\overline{\nabla }}_{JX}J\right)JY=0$.

Thus, $\mathcal{AK}\cap \mathcal{NK}=\mathcal{K}=\mathcal{QK}\cap \mathcal{H}$ and $\mathcal{AK},\mathcal{NK}\subset \mathcal{QK}$.

In [8,9,10,11,12,13], we introduced and studied metric structures on a smooth manifold that generalized the a.c.m. structures. These so-called “weak” structures (where the linear complex structure on the contact distribution is replaced by a nonsingular skew-symmetric tensor) made it possible to take a new look at the classical theory and find new applications.

One may consider classes $w\mathcal{M}$, $w\mathcal{S}, w\mathcal{C}, w\mathcal{N} \mathrm{and} w\mathcal{Q}$ of weak structures, defined similarly to the above classes, $\mathcal{M}$, $\mathcal{S}, \mathcal{C}, \mathcal{N} \mathrm{and} \mathcal{Q}$. Our previous works [8,9,10,11,12,13] are devoted to the classes $w\mathcal{M}$, $w\mathcal{S}, w\mathcal{C} \mathrm{and} w\mathcal{N}$; see survey [14]. This paper continues our study of the geometry of weak a.c.m. manifolds and discusses how the above weak a.c.m. structures relate to each other. The above open question motivates us to study class $w\mathcal{Q}$ of weak q.c.m. manifolds. Our overall goal is to demonstrate that weak a.c.m. structures allow us to look at the theory of contact metric manifolds in a new way. To do this, we successfully extended classical theorems from contact geometry to the more general setting of weak q.c.m. manifolds $w\mathcal{Q}$.

This paper is organized as follows. In Section 2, following the introductory Section 1, we review the basics of weak a.c.m. manifolds and prove Lemma 1. Section 3 contains our main contributions—Propositions 1, 2, 3, 4 and five theorems—where we generalize some well-known results and provide new criterions for K-contact manifolds (Theorems 1, 3 and 4) and Sasakian manifolds (Theorems 2 and 5).

2. Preliminaries

A weak a.c.m. structure on a smooth manifold $M^{2n+1}$ is defined by a $(1,1)$-tensor f, a nonsingular $(1,1)$-tensor Q, a vector field $\xi$, a 1-form $\eta$, and a Riemannian metric g such that

$$\begin{array}{c}{f}^2=-Q+\eta \otimes \xi ,\eta \left(\xi \right)=1,\hfill \\ g(fX,fY)=g(X,QY)-\eta \left(X\right)\eta \left(Y\right),\eta \left(X\right)=g(\xi ,X)(X,Y\in {\mathfrak{X}}_M).\hfill \end{array}$$

(3)

The following equalities are true for a.c.m. manifolds; see ([9] Proposition 1(a)):

$$\begin{array}{c}f\xi =0,\eta \circ f=0,\eta \circ Q=\eta ,[Q,f]=[\tilde{Q},f]=0,\eta \circ \tilde{Q}=0,\tilde{Q}\xi =0,\hfill \end{array}$$

(4)

where $\tilde{Q}=Q-{\mathrm{id}}_{TM}$ is a “small” tensor. According to the above, $f(ker\eta )\subset ker\eta$ and $\mathrm{rank}f=2n$. In this case, f is skew-symmetric, and Q is self-adjoint and positive definite.

A weak a.c.m. structure satisfying (2) is called a weak q.c.m. structure. A weak a.c.m. structure satisfying $d\eta =\Phi$ is called a weak contact metric structure. For a weak contact metric structure, the 1-form $\eta$ is contact; see [12]. A 1-form $\eta$ on $M^{2n+1}$ is said to be contact if $\eta \wedge {\left(d\eta \right)}^n\ne 0$, e.g., [3]. A weak almost-contact structure is said to be normal if ${\mathcal{N}}^{\left(1\right)}=0$. A normal weak contact metric structure is called a weak Sasakian structure. Recall that a weak a.c.m. structure is weak Sasakian if and only if it is a Sasakian structure; see ([8] Theorem 3).

The following tensors on a.c.m. manifolds are well known; see [3]:

$$\begin{array}{ccc}& & {\mathcal{N}}^{\left(2\right)}(X,Y)=\left({\pounds }_{fX}\eta \right)\left(Y\right)-\left({\pounds }_{fY}\eta \right)\left(X\right)=2d\eta (fX,Y)-2d\eta (fY,X),\hfill \\ & & {\mathcal{N}}^{\left(3\right)}\left(X\right)=\left({\pounds }_{\xi }f\right)X=[\xi ,fX]-f[\xi ,X],\hfill \\ & & {\mathcal{N}}^{\left(4\right)}\left(X\right)=\left({\pounds }_{\xi }\eta \right)\left(X\right)=\xi \left(\eta \left(X\right)\right)-\eta \left([\xi ,X]\right)=2d\eta (\xi ,X).\hfill \end{array}$$

For a weak a.c.m. structure, the following equality is true; see [7] for $Q=\mathrm{id}$:

$$\begin{array}{c}{\mathcal{N}}^{\left(2\right)}(X,Y)=\left({\nabla }_{fX}\eta \right)\left(Y\right)-\left({\nabla }_Y\eta \right)\left(fX\right)-\left({\nabla }_{fY}\eta \right)\left(X\right)+\left({\nabla }_X\eta \right)\left(fY\right).\hfill \end{array}$$

(5)

Let $M^{2n+1}(f,Q,\xi ,\eta ,g)$ be a weak a.c.m. manifold. Define (1,1)-tensors J, $\mathcal{P}$ and a Riemannian metric $\overline{g}$ on the product $\overline{M}=M^{2n+1}\times \mathbb{R}$ for $X,Y\in TM$ and $t\in \mathbb{R}$:

$$\begin{array}{ccc}& & J(X,0)=(fX,-\eta \left(X\right){\partial }_t),J(0,{\partial }_t)=(\xi ,0),\mathcal{P}(X,0)=(QX,0),\mathcal{P}(0,{\partial }_t)=(0,{\partial }_t).\hfill \\ & & \overline{g}((X,0),(Y,0))=e^{-2t}g(X,Y),\overline{g}((X,0),(0,{\partial }_t))=0,\overline{g}((0,{\partial }_t),(0,{\partial }_t))=e^{-2t}.\hfill \end{array}$$

Then, $J^2=-\mathcal{P}$ will hold and $\overline{M}(J,\mathcal{P},\overline{g})$ will be a weak almost-Hermitian manifold. The tensors ${\mathcal{N}}^{\left(i\right)}(i=1,2,3,4)$ appear when we use the integrability condition $[J,J]=0$ of J to express the normality condition ${\mathcal{N}}^{\left(1\right)}=0$ of the weak a.c.m. structure; see [10]. For weak contact metric manifolds, we have ${\mathcal{N}}^{\left(2\right)}={\mathcal{N}}^{\left(4\right)}=0$ and the trajectories of $\xi$ are geodesics, i.e., ${\nabla }_{\xi }\xi =0$; moreover, ${\mathcal{N}}^{\left(3\right)}=0$ if and only if $\xi$ is a Killing vector field; see [10].

The following result generalizes Proposition 2.6 in [7].

Lemma 1. 

For a weak q.c.m. structure $(f,Q,\xi ,\eta ,g)$, the following equalities are true:

$$\begin{array}{cc}\hfill & \left({\nabla }_X\eta \right)\left(QY\right)+\left({\nabla }_{fX}\eta \right)\left(fY\right)+2g(fX,Y)=0,\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & {\nabla }_{\xi }f=0,\hfill \end{array}$$

(7)

$$\begin{array}{cc}\hfill & {\nabla }_{\xi }\xi =0,{\nabla }_{\xi }\eta =0,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & Q\nabla \xi =(\nabla \xi )Q=-f-fh,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\pounds }_{\xi }Q={\nabla }_{\xi }Q=0,\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & hf+fh=0,\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & hQ-Qh=0.\hfill \end{array}$$

(12)

Proof. 

Setting $X=\xi$ in (2) and using $f\xi =0$ and $h\xi =0$, we obtain (7). The equality (8) follows directly from (7). From (2) with $Y=\xi$, we obtain $f{\nabla }_X\xi =X-\eta \left(X\right)\xi +hX$. Multiplying this by f and using $f^2=-Q+\eta \otimes \xi$, see (3), we obtain $Q\nabla \xi =-f-fh$; see (9).

Replacing X and Y with $fX$ and $fY$, respectively, in (2) gives

$$\left({\nabla }_{fX}f\right)fY+\left({\nabla }_{f^{2}X}f\right)f^{2}Y=2g(fX,fY)\xi ,$$

and then using $f^2=-Q+\eta \otimes \xi$, see (3), we obtain

$$\begin{array}{cc}\hfill & \left({\nabla }_{QX}f\right)\left(QY\right)+\left({\nabla }_{fX}f\right)fY=2g(QX,Y)\xi -2\eta \left(X\right)\eta \left(Y\right)\xi \hfill \\ \hfill & +\eta \left(Y\right)\left({\nabla }_{QX}f\right)\xi +\eta \left(X\right)\left({\nabla }_{\xi }f\right)QY-\eta \left(X\right)\eta \left(Y\right)\left({\nabla }_{\xi }f\right)\xi .\hfill \end{array}$$

Subtracting from this (2), we obtain

$$\begin{array}{c}\left({\nabla }_{QX}f\right)\left(QY\right)-\left({\nabla }_{X}f\right)Y+2g(\tilde{Q}X,Y)\xi \hfill \\ =\eta \left(Y\right)\left({\nabla }_{QX}f\right)\xi +\eta \left(X\right)\left({\nabla }_{\xi }f\right)QY-\eta \left(X\right)\eta \left(Y\right)\left({\nabla }_{\xi }f\right)\xi \hfill \\ -\eta \left(X\right)\eta \left(Y\right)\xi +\eta \left(Y\right)X+\eta \left(Y\right)hX.\hfill \end{array}$$

(13)

From (7) and the definition of h, we obtain

$$\begin{array}{c}2hX=f{\nabla }_X\xi -{\nabla }_{fX}\xi .\hfill \end{array}$$

(14)

Thus, from (13) and (7) we obtain

$$\begin{array}{cc}\hfill & \left({\nabla }_{QX}f\right)\left(QY\right)-\left({\nabla }_{X}f\right)Y+2g(\tilde{Q}X,Y)\xi \hfill \\ \hfill & =\eta \left(Y\right)\left(X-f{\nabla }_{QX}\xi -\eta \left(X\right)\xi +(1/2)f{\nabla }_X\xi -(1/2){\nabla }_{fX}\xi \right).\hfill \end{array}$$

Setting $Y=\xi$ in the above equation, we obtain

$$\begin{array}{c}2X-2\eta \left(X\right)\xi -{\nabla }_{fX}\xi =f{\nabla }_X\xi .\hfill \end{array}$$

Applying f to the above equation and using (3) and $\eta \left({\nabla }_X\xi \right)=0$, we obtain

$$\begin{array}{c}2fX=f{\nabla }_{fX}\xi +f^2{\nabla }_X\xi =f{\nabla }_{fX}\xi -Q{\nabla }_X\xi .\hfill \end{array}$$

Multiplying this by Y, we obtain (6).

Using (14) and $f^2=-Q+\eta \otimes \xi$, see (3), we find

$$\begin{array}{c}2(hf+fh)X=f^2{\nabla }_X\xi -{\nabla }_{f^{2}X}\xi ={\nabla }_{\tilde{Q}}\xi -\tilde{Q}{\nabla }_X\xi =\left({\pounds }_{\xi }\tilde{Q}\right)X.\hfill \end{array}$$

Using (7) and (8), we find ${\nabla }_{\xi }\tilde{Q}=0$. We can also calculate the following, using (7) and (9):

$$\begin{array}{cc}\hfill & 2hX=\left({\pounds }_{\xi }\tilde{Q}\right)X=f{\nabla }_X\xi -{\nabla }_{fX}\xi \hfill \\ \hfill & =-Q^{-1}{f}^2-Q^{-1}{f}^{2}h+Q^{-1}{f}^2-Q^{-1}f(fh+(1/2){\pounds }_{\xi }\tilde{Q})X\hfill \\ \hfill & =2hX-(1/2)Q^{-1}f\left({\pounds }_{\xi }\tilde{Q}\right)X.\hfill \end{array}$$

Thus, ${\pounds }_{\xi }\tilde{Q}=0$, which proves (10) and (11). Multiplying (11) by f and using (11) gives (12). Using the above, we can complete the proof of (9): $(\nabla \xi )QX=Q^{-1}(-fQX-fhQX)=-fX-fhX$. □

Remark 1. 

For weak contact metric manifolds, we generally have ${\nabla }_{\xi }f\ne 0$; see ([8] Corollary 1), which differs from (7). Thus, the class $w\mathcal{C}$ is not contained in $w\mathcal{Q}$, although $\mathcal{C}\subset \mathcal{Q}$.

3. Main Results

The next theorem completes ([10] Theorem 2) and characterizes K-contact manifolds among weak q.c.m. manifolds satisfying (10) by using the following property of K-contact manifolds; see [3]:

$$\nabla \xi =-f.$$

(15)

Theorem 1. 

Let $M(f,Q,\xi ,\eta ,g)$ be a weak q.c.m. manifold and (15) be valid. Then, $Q=\mathrm{id}$, and $M(f,\xi ,\eta ,g)$ is a K-contact manifold.

Proof. 

Let a weak q.c.m. manifold $M(f,Q,\xi ,\eta ,g)$ satisfy the condition (15). From (9) and (15), we obtain $f(h-\tilde{Q})=0$. Since f is non-degenerate on $ker\eta$ and $h\xi =\tilde{Q}\xi =0$ is true, we find $h=\tilde{Q}$. On the other hand, using (10), we obtain ${\pounds }_{\xi }\tilde{Q}=0$. Therefore, from (11) and $[Q,f]=0$, see (4), we find $f\tilde{Q}=0$. From this, since f is non-degenerate on $ker\eta$, we obtain $\tilde{Q}=0$. Therefore, $M(f,\xi ,\eta ,g)$ is a q.c.m. manifold. Since $h=0$ also holds, according to ([7] Theorem 3.2), $M(f,\xi ,\eta ,g)$ is a contact metric manifold. Next, using (15) and the skew-symmetry of f, we can conclude that $\xi$ is a Killing vector field:

$$\begin{array}{c}g({\nabla }_X\xi ,Y)+g({\nabla }_Y\xi ,X)=-g(fX,Y)-g(fY,X)=0.\hfill \end{array}$$

Therefore, $M(f,\xi ,\eta ,g)$ is a K-contact manifold. □

The curvature tensor is given by $R_{X,Y}Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$.

The following result generalizes ([3] Proposition 7.1) on contact metric manifolds.

Proposition 1. 

For a weak q.c.m. structure $(f,Q,\xi ,\eta ,g)$, the following equalities are true:

$$\begin{array}{cc}\hfill & \left({\nabla }_{\xi }h\right)X=Q^{-1}\left(fX-h^{2}fX\right)-f{R}_{X,\xi }\xi ,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & Q{R}_{\xi ,X}\xi -f{R}_{\xi ,fX}\xi =2h^{2}X+(Q+Q^{-1})f^{2}X.\hfill \end{array}$$

(17)

Proof. 

Using (9) and (7), we can compute

$$\begin{array}{c}Q{R}_{\xi ,X}\xi =f{\nabla }_{\xi }(-X-hX)+f[\xi ,X]+fh[\xi ,X].\hfill \end{array}$$

Applying f and using (3), (12) and (9), we obtain

$$\begin{array}{cc}Qf{R}_{\xi ,X}\xi \hfill & =f^2{\nabla }_{\xi }(-X-hX)+f^2[\xi ,X]+f^{2}h[\xi ,X]\hfill \\ \hfill & =Q{\nabla }_{\xi }(X+hX)-\eta \left({\nabla }_{\xi }(X+hX)\right)\xi -Q[\xi ,X]+\eta \left([\xi ,X]\right)\xi -Qh[\xi ,X]\hfill \\ \hfill & =Q\left(\left({\nabla }_{\xi }h\right)X+{\nabla }_X\xi +h{\nabla }_X\xi \right).\hfill \end{array}$$

According to the above, since Q is nonsingular, the following is true:

$$\begin{array}{c}f{R}_{\xi ,X}\xi =\left({\nabla }_{\xi }h\right)X+{\nabla }_X\xi +h{\nabla }_X\xi .\hfill \end{array}$$

Using (9), (11) and (12), we obtain the following formula (16):

$$\begin{array}{c}f{R}_{\xi ,X}\xi =\left({\nabla }_{\xi }h\right)X+Q^{-1}\left(-fX+h^{2}fX\right).\hfill \end{array}$$

(18)

Next, applying f to (16) and using $f^{2}X=-QX+\eta \left(X\right)\xi$, see (3), we find

$$\begin{array}{c}Q{R}_{\xi ,X}\xi =h^{2}X+Q{f}^{2}X-f\left({\nabla }_{\xi }h\right)X.\hfill \end{array}$$

(19)

Using (7) and (11), we obtain $\left({\nabla }_{\xi }h\right)fX=-f\left({\nabla }_{\xi }h\right)X$. From (18), using $h^{2}f=f{h}^2$, see (12), we obtain

$$\begin{array}{c}f{R}_{\xi ,fX}\xi =\left({\nabla }_{\xi }h\right)fX+Q^{-1}(-f^{2}X+h^2{f}^{2}X)=-h^{2}X-Q^{-1}{f}^{2}X-f\left({\nabla }_{\xi }h\right)X.\hfill \end{array}$$

(20)

Subtracting (20) from (19) yields (17). □

The following result generalizes Proposition 2.3 in [7] for $Q=\mathrm{id}$.

Proposition 2. 

Let a weak a.c.m. structure $(f,Q,\xi ,\eta ,g)$ satisfy ${\nabla }_{\xi }f=0$; see (7). Then,

$$\begin{array}{c}g(hX,Y)-g(hY,X)=-(1/2){\mathcal{N}}^{\left(2\right)}(X,Y);\hfill \end{array}$$

(21)

hence, h is self-adjoint if and only if ${\mathcal{N}}^{\left(2\right)}\equiv 0$.

Proof. 

Note that $h\xi =2{\mathcal{N}}^{\left(3\right)}\left(\xi \right)=0$ and $hX\perp \xi$ for $X\perp \xi$. Using ${\nabla }_{\xi }f=0$, we obtain

$$\begin{array}{c}2hX=-{\nabla }_{fX}\xi +f{\nabla }_X\xi .\hfill \end{array}$$

(22)

From (5) and (22), we obtain (21), which completes the proof. □

The following result generalizes Theorem 4.2 in [6].

Proposition 3. 

Let $M(f,Q,\xi ,\eta ,g)$ be a weak q.c.m. manifold. If ξ is a Killing vector field, then the tensor h is skew-symmetric (and $h^2$ is nonpositive definite); if the contact distribution $ker\eta$ is integrable, then h is self-adjoint (and $h^2$ is nonnegative definite).

Proof. 

Using (9) and (11)–(12), we can calculate

$$\begin{array}{cc}\hfill & g({\nabla }_X\xi ,Y)\pm g({\nabla }_Y\xi ,X)=g(Q^{-1}(-f-fh)X,Y)\pm g(Q^{-1}(-f-fh)Y,X)\hfill \\ \hfill & =-g(Q^{-1}f(h\pm h^{*})X,Y)=g((h\pm h^{*})X,Q^{-1}fY),\hfill \end{array}$$

where $h^{*}$ is the conjugate tensor to h. According to the above, if $\xi$ is a Killing vector field, that is, $g({\nabla }_X\xi ,Y)+g({\nabla }_Y\xi ,X)=0$ for all $X,Y$, then $h+h^{*}=0$, and if the distribution $ker\eta$ is integrable then $g([Y,X],\xi )=g({\nabla }_X\xi ,Y)-g({\nabla }_Y\xi ,X)=0$ □

For a weak a.c.m. manifold $M(f,Q,\xi ,\eta ,g)$, we will build an f-basis consisting of mutually orthogonal nonzero vectors at a point $x\in M$. Let $e_1\in {(ker\eta )}_x$ be a unit eigenvector of the self-adjoint operator $Q>0$ with the eigenvalue ${\lambda }_1>0$. Then, $f{e}_1\in {(ker\eta )}_x$ is orthogonal to $e_1$ and $Q\left(f{e}_1\right)=f\left(Q{e}_1\right)={\lambda }_{1}f{e}_1$. Thus, the subspace orthogonal to the plane $span\{e_1,f{e}_1\}$ is Q-invariant. There exists a unit vector $e_2\in {(ker\eta )}_x$ such that $e_2\perp span\{e_1,f{e}_1\}$ and $Q{e}_2={\lambda }_2{e}_2$ for some ${\lambda }_2>0$. Obviously, $Q\left(f{e}_2\right)=f\left(Q{e}_2\right)={\lambda }_{2}f{e}_2$. All five vectors $\{\xi ,e_1,f{e}_1,e_2,f{e}_2\}$ are nonzero and mutually orthogonal. Continuing in the same manner, we find a basis $\{\xi ,e_1,f{e}_1,\dots ,e_n,f{e}_n\}$ of $T_{x}M$ consisting of mutually orthogonal vectors; see [12]. Note that $g(f{e}_i,f{e}_i)=g(Q{e}_i,e_i)={\lambda }_i$ and $\mathrm{trace}Q=2{\sum }_{i=1}^n{\lambda }_i$.

The following condition is stronger than ${\nabla }_{\xi }Q=0$, see (10), and is valid when $Q=\mathrm{id}$:

$$\begin{array}{c}\left({\nabla }_{X}Q\right)Y=0(X,Y\in ker\eta ).\hfill \end{array}$$

(23)

The exterior derivative of $\Phi$ is given by

$$\begin{array}{ccc}d\Phi (X,Y,Z)\hfill & & =(1/3)\{X\Phi (Y,Z)+Y\Phi (Z,X)+Z\Phi (X,Y)\hfill \\ & & -\Phi \left(\right[X,Y],Z)-\Phi \left(\right[Z,X],Y)-\Phi \left(\right[Y,Z],X)\}.\hfill \end{array}$$

The next result completes Theorem 3.2 in [7].

Proposition 4. 

Let a weak q.c.m. manifold satisfy the condition (23). If h is self-adjoint, then η is a contact form and $d\Phi =0$.

Proof. 

According to Lemma 1, conditions (6)–(8) are true. From (6), we obtain

$$\begin{array}{c}\left({\nabla }_X\eta \right)\left(QY\right)-\left({\nabla }_Y\eta \right)\left(QX\right)+\left({\nabla }_{fX}\eta \right)\left(fY\right)-\left({\nabla }_{fY}\eta \right)\left(fX\right)=-4g(fX,Y).\hfill \end{array}$$

(24)

According to (7) and Proposition 2, (21) is true; thus, ${\mathcal{N}}^{\left(2\right)}\equiv 0$ holds by symmetry of h. Using (5), we obtain

$$\begin{array}{c}\left({\nabla }_{fX}\eta \right)\left(Y\right)-\left({\nabla }_Y\eta \right)\left(fX\right)-\left({\nabla }_{fY}\eta \right)\left(X\right)+\left({\nabla }_X\eta \right)\left(fY\right)=0.\hfill \end{array}$$

(25)

Replacing X with $fX$ in (25) and using ${\nabla }_{\xi }\eta =0$ (since ${\nabla }_{\xi }\xi =0$; see (8)) and $\left({\nabla }_Y\eta \right)\left(\xi \right)=0$, we obtain

$$\begin{array}{c}\left({\nabla }_Y\eta \right)\left(QX\right)-\left({\nabla }_{QX}\eta \right)\left(Y\right)-\left({\nabla }_{fY}\eta \right)\left(fX\right)+\left({\nabla }_{fX}\eta \right)\left(fY\right)=0.\hfill \end{array}$$

(26)

Subtracting (26) from (24) gives

$$\begin{array}{c}2\left({\nabla }_X\eta \right)\left(Y\right)-2\left({\nabla }_Y\eta \right)\left(X\right)+\left({\nabla }_X\eta \right)\left(\tilde{Q}Y\right)-2\left({\nabla }_Y\eta \right)\left(\tilde{Q}X\right)+\left({\nabla }_{\tilde{Q}X}\eta \right)\left(Y\right)=4\Phi (X,Y).\hfill \end{array}$$

Using (23), we find $\left({\nabla }_X\eta \right)\left(\tilde{Q}Y\right)=\left({\nabla }_Y\eta \right)\left(\tilde{Q}X\right)=0$; therefore,

$$\begin{array}{c}\left({\nabla }_X\eta \right)\left(Y\right)-\left({\nabla }_Y\eta \right)\left(X\right)+(1/2)\left({\nabla }_{\tilde{Q}X}\eta \right)\left(Y\right)=2\Phi (X,Y).\hfill \end{array}$$

Hence, using $\left({\nabla }_X\eta \right)\left(Y\right)-\left({\nabla }_Y\eta \right)\left(X\right)=2d\eta (X,Y)$ and $\left({\nabla }_{\tilde{Q}X}\eta \right)\left(Y\right)=2d\eta (\tilde{Q}X,Y)$, we obtain

$$\begin{array}{c}d\eta (X+{\displaystyle \frac{1}{2}}\tilde{Q}X,Y)=\Phi (X,Y).\hfill \end{array}$$

In particular, $d\Phi =0$. In terms of the f-basis, we obtain ${\displaystyle \frac{1}{2}}({\lambda }_i+1)d\eta (e_i,Y)=\Phi (e_i,Y)$ and $\Phi (e_i,e_j)=0$, $\Phi (e_i,f{e}_j)=-{\lambda }_i{\delta }_{ij}$, where ${\lambda }_i>0$. Therefore,

$$\eta \wedge {\left(d\eta \right)}^n(\xi ,e_1,f{e}_1,\dots ,e_n,f{e}_n)={\left(d\eta \right)}^n(e_1,f{e}_1,\dots ,e_n,f{e}_n)\ne 0,$$

and $\eta$ is a contact form. □

Theorem 2. 

If a weak q.c.m. manifold $M(f,Q,\xi ,\eta ,g)$ satisfies the condition (1), then $Q=\mathrm{id}$ and $M(f,\xi ,\eta ,g)$ is a Sasakian manifold.

Proof. 

Using (1) in (2), we obtain the following for any $X,Y\in TM$:

$$\begin{array}{c}g(\tilde{Q}X,Y)\xi +\eta \left(Y\right)hX=0.\hfill \end{array}$$

Since $hX\perp \xi$, we obtain $\tilde{Q}=0$ and $h=0$; therefore, $M(f,\xi ,\eta ,g)$ is a quasi-contact metric manifold. According to ([5] Theorem 2.2) (h is self-adjoint), $M(f,\xi ,\eta ,g)$ is a contact metric manifold. According to the conditions and ([3] Theorem 6.3), $M(f,\xi ,\eta ,g)$ is a Sasakian manifold. □

On a contact metric manifold $M^{2n+1}$, we have $Ric(\xi ,\xi )=2n-\mathrm{trace}{h}^2$; see ([3] Corollary 7.1). The following theorem generalizes this property and generalizes ([5] Theorem A).

Theorem 3. 

Let $M(f,Q,\xi ,\eta ,g)$ be a weak q.c.m. manifold such that $K(\xi ,X)+K(\xi ,fX)\ge 0$ for all nonzero $X\perp \xi$. Then,

$$\begin{array}{c}\underset{1\le i\le n}{max}\left\{{\lambda }_i\right\}·Ric(\xi ,\xi )\ge n-\mathrm{trace}{h}^2+{\displaystyle \frac{1}{4n}}{(\mathrm{trace}Q-1)}^2.\hfill \end{array}$$

(27)

If $\mathrm{trace}\tilde{Q}=0$ and the equality in (27) holds, then $\tilde{Q}=0$, ${\lambda }_i=1(1\le i\le n)$ and $Ric(\xi ,\xi )=2n-\mathrm{trace}{h}^2$; moreover, if ξ is a Killing vector field, then $M(f,\xi ,\eta ,g)$ is a K-contact manifold.

Proof. 

From (17) with $X=e_i$ and then $X=f{e}_i$, we can obtain formulas with sectional curvature,

$$\begin{array}{cc}\hfill & {\lambda }_i\left(K(\xi ,e_i)+K(\xi ,f{e}_i)\right)={\lambda }_i^2+1-2g(h^2{e}_i,e_i),\hfill \\ \hfill & {\lambda }_i\left(K(\xi ,f{e}_i)+K(\xi ,e_i)\right)={\lambda }_i^2+1-2g(h^{2}f{e}_i/\parallel f{e}_i\parallel ,f{e}_i/\parallel f{e}_i\parallel )),\hfill \end{array}$$

using an f-basis $\{\xi ,e_1,\dots ,e_n,f{e}_1,\dots ,f{e}_n\}$. From the above, we obtain

$$\begin{array}{c}{\sum }_{i=1}^n{\lambda }_i\left(K(\xi ,e_i)+K(\xi ,f{e}_i)\right)=n-\mathrm{trace}{h}^2+{\sum }_{i=1}^n{\lambda }_i^2.\hfill \end{array}$$

(28)

Using the equality $\mathrm{trace}Q=1+2{\sum }_{i=1}^n{\lambda }_i$ and the well-known inequality ${\sum }_{i=1}^n{\lambda }_i^2\ge {\displaystyle \frac{1}{n}}{\left({\sum }_{i=1}^n{\lambda }_i\right)}^2$, from (28), we obtain (27). If the equality in (27) is true, then ${\lambda }_1=\dots ={\lambda }_n$. Note that $\mathrm{trace}Q=2n+1+\mathrm{trace}\tilde{Q}$. According to the condition $\mathrm{trace}\tilde{Q}=0$, we can obtain ${\lambda }_i=1$ and $\tilde{Q}=0$; thus, $M(f,\xi ,\eta ,g)$ is a q.c.m. manifold and $Ric(\xi ,\xi )=2n-\mathrm{trace}{h}^2$. If $\xi$ is a Killing vector field, then according to ([5] Theorem A), $M(f,\xi ,\eta ,g)$ is a K-contact metric manifold. □

It is well known that a K-contact structure satisfies the following condition:

$$R_{X,\xi }\xi =-X-\eta \left(X\right)\xi ,$$

(29)

thus, the sectional curvature of all planes containing $\xi$ is equal to 1.

The following result complements ([3] Theorem 7.2) and ([5] Theorem A) on K-contact manifolds.

Theorem 4. 

Let a weak q.c.m. manifold $M(f,Q,\xi ,\eta ,g)$ satisfy (29). If ξ is a Killing vector field, then, $Q=\mathrm{id}$ and $M(f,\xi ,\eta ,g)$ is a K-contact manifold.

Proof. 

According to (29), we obtain $K_{\xi ,X}=1$ for nonzero vectors $X\perp \xi$. Using this in (17), we obtain

$$\begin{array}{c}Q(-X)-f(-fX)=2h^{2}X+(Q+Q^{-1})f^{2}X.\hfill \end{array}$$

Using (3) and $Q=\mathrm{id}+\tilde{Q}$, we simplify the above equation with $X\perp \xi$ to the following:

$$\begin{array}{cc}2h^{2}X\hfill & =f^{2}X-QX-(Q+Q^{-1})f^{2}X\hfill \\ \hfill & =-QX+\eta \left(X\right)\xi -QX+(Q+Q^{-1})QX-\eta \left(X\right)(Q+Q^{-1})\xi \hfill \\ \hfill & =-2QX+Q^{2}X+X=-2\tilde{Q}X+Q^{2}X-X\hfill \\ \hfill & =-2\tilde{Q}X+{\tilde{Q}}^{2}X+2\tilde{Q}X={\tilde{Q}}^{2}X.\hfill \end{array}$$

Therefore, $2h^2={\tilde{Q}}^2\ge 0$, and consequently, $\mathrm{trace}{h}^2\ge 0$. On the other hand, by Proposition 3 and the conditions, h is skew-symmetric and $h^2\le 0$. Therefore, $\tilde{Q}=0$, and we conclude that $M(f,\xi ,\eta ,g)$ is a q.c.m. manifold with a Killing vector field $\xi$. Hence, according to ([5] Theorem A), $M(f,\xi ,\eta ,g)$ is a K-contact manifold. □

It is well known that a contact metric structure is Sasakian if and only if

$$R_{X,Y}\xi =\eta \left(Y\right)X-\eta \left(X\right)Y.$$

(30)

The following result complements ([3] Proposition 7.6) on contact metric manifolds and ([6] Theorem 4.3) on q.c.m. manifolds.

Theorem 5. 

Let a weak q.c.m. manifold $M(f,Q,\xi ,\eta ,g)$ satisfy conditions $\mathrm{trace}{h}^2\le 0$ and (30). Then, $Q=\mathrm{id}$ and $M(f,\xi ,\eta ,g)$ is a Sasakian manifold.

Proof. 

According to (30), we can obtain $R_{\xi ,X}\xi =-X$ for $X\perp \xi$. As in the proof of Theorem 4, we conclude that $2h^2={\tilde{Q}}^2\ge 0$. From this and the conditions, we obtain $\mathrm{trace}{\tilde{Q}}^2=0$; hence, $\tilde{Q}=0$ and $M(f,\xi ,\eta ,g)$ is a q.c.m. manifold with condition (30). Hence, according to ([6] Theorem 4.3), $M(f,\xi ,\eta ,g)$ is a Sasakian manifold. □

4. Conclusions

This paper contains substantial new mathematics that successfully extends important concepts and theorems about contact Riemannian manifolds for the case of weak q.c.m. manifolds and provides new tools for studying K-contact and Sasakian structures.