Certain results on almost contact pseudo-metric manifolds

Abstract

We study the geometry of almost contact pseudo-metric manifolds in terms of tensor fields $h:=\frac{1}{2}\pounds _\xi \varphi$ and $\ell := R(\cdot ,\xi )\xi$, emphasizing analogies and differences with respect to the contact metric case. Certain identities involving $\xi$-sectional curvatures are obtained. We establish necessary and sufficient condition for a nondegenerate almost CR structure $(\mathcal {H}(M), J, \theta )$ corresponding to almost contact pseudo-metric manifold M to be CR manifold. Finally, we prove that a contact pseudo-metric manifold $(M, \varphi ,\xi ,\eta ,g)$ is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost CR structure $(\mathcal {H}(M), J)$ is integrable and J is parallel along $\xi$ with respect to the Bott partial connection.

1 Introduction

In 1969, Takahashi [22] initiated the study of contact structures associated with pseudo-Riemannian metrics. Afterwards, a number of authors studied such structures mainly focusing on a special case, namely Sasakian pseudo-metric manifolds. The case of contact Lorentzian structures $(\eta , g)$, where $\eta$ is a contact 1-form and g a Lorentzian metric associated to it, has a particular relevance for physics and was considered in [12] and [4]. A systematic study of almost contact pseudo-metric manifolds was undertaken by Calvaruso and Perrone [7] in 2010, introducing all the technical apparatus which is needed for further investigations, and such manifolds have been extensively studied under several points of view in [1,2,3, 6, 9, 10, 15,16,17, 24], and references cited therein.

The operators $h:=\frac{1}{2}\pounds _\xi \varphi$ and $\ell : = R(\cdot ,\xi )\xi$ play fundamental roles in the study of geometry of contact pseudo-metric manifolds. For contact metric manifolds, Sharma [21] obtained the following beautiful results (Theorem 1.1 in [21]):

• (a) a contact metric manifold is K-contact if and only if h is a Codazzi tensor;

• (b) a contact metric manifold is K-contact if and only if $\tau$, the tensor metrically equivalent to the strain tensor $\pounds _\xi g$ of M along $\xi$, is a Codazzi tensor;

• (c) the sectional curvatures of all plane sections containing $\xi$ vanish if and only if the tensor $\ell$ is parallel.

The proof of these results exploit, in an essential way, the fact that in the contact Riemannian case, the self-adjoint operator h vanishes if $h^2=0$. But in the contact pseudo-metric case the condition $h^2=0$ does not necessarily imply that $h=0$ (see [16]). So the corresponding results fail for general contact pseudo-metric structures.

Under these circumstances, becomes interesting to explore more the geometry of contact pseudo-metric manifolds. The paper is organized as follows. In Sect. 2, we give the basics of almost contact pseudo-metric manifolds. In Sect. 3, we study contact pseudo-metric manifold M with h satisfying Codazzi condition and we prove that M is Sasakian pseudo-metric manifold if and only if the Eq. (2.10) is satisfied and h is a Codazzi tensor. In Sect. 4, we investigate the Codazzi condition for the operator $\tau$, and we obtain a necessary and sufficient condition for $\tau$ to be a Codazzi tensor on contact pseudo-metric manifold. Moreover, if $\tau$ is a Codazzi tensor, then $h^2=0$ and the Ricci operator Q satisfies $Q\xi =2\varepsilon n\xi$, and we prove that M is a Sasakian pseudo-metric manifold if and only if the Eq. (2.10) is satisfied and $\tau$ is a Codazzi tensor. In Sect. 5, we obtain certain identities involving $\xi$-sectional curvatures of contact pseudo-metric manifolds. It is proved that the parallelism of the tensor $\ell$ together with the condition $\nabla _\xi h=0$ on a contact pseudo-metric manifold implies that all $\xi$-sectional curvatures vanish. At the end, we investigate the nondegenerate almost $\textit{CR}$ structure $(\mathcal {H}(M), J, \theta )$ corresponding to almost contact pseudo-metric manifold M, and establish a necessary and sufficient condition for an almost contact pseudo-metric manifold to be a $\textit{CR}$ manifold. Finally, we show that a contact pseudo-metric manifold $(M, \varphi ,\xi ,\eta ,g)$ is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost $\textit{CR}$ structure $(\mathcal {H}(M),J)$ is integrable and J is parallel along $\xi$ with respect to the Bott partial connection. We note that many of results of this paper are the corresponding results to those obtained in [14, 25] on almost paracontact metric manifolds.

2 Preliminaries

In this section, we briefly recall some general definitions and basic properties of almost contact pseudo-metric manifolds. For more information and details, we recommend the reference [7].

A $(2n+1)$-dimensional smooth connected manifold M is said to be an almost contact manifold if there exists on M a (1, 1) tensor field $\varphi$, a vector field $\xi$, and a 1-form $\eta$ such that

$$\begin{aligned} \varphi ^2 = -I + \eta \otimes \xi , \quad \eta (\xi ) = 1, \quad \varphi \xi =0, \quad \eta \circ \varphi =0. \end{aligned}$$

(2.1)

It is known that the first relation along with any one of the remaining three relations in (2.1) imply the remaining two relations. Also, for an almost contact structure, the rank of $\varphi$ is 2n. For more details, we refer to [5].

If an almost contact manifold is endowed with a pseudo-Riemannian metric g such that

$$\begin{aligned} g(\varphi X, \varphi Y)=g( X, Y)-\varepsilon \eta (X)\eta (Y), \end{aligned}$$

(2.2)

where $\varepsilon =\pm 1$, for all $X,Y\in TM$, then $(M, \varphi , \xi , \eta , g)$ is called an almost contact pseudo-metric manifold. The relation (2.2) is equivalent to

$$\begin{aligned} \eta (X)=\varepsilon g(X, \xi )\; \mathrm{along} \; \mathrm{with}\; g(\varphi X,Y)=-g(X,\varphi Y). \end{aligned}$$

(2.3)

In particular, in an almost contact pseudo-metric manifold, it follows that $g(\xi , \xi )=\varepsilon$ and so, the characteristic vector field $\xi$ is a unit vector field, which is either space-like or time-like, but cannot be light-like.

The fundamental 2-form of an almost contact pseudo-metric manifold $(M, \varphi , \xi , \eta , g)$ is defined by

$$\begin{aligned} \Phi (X,Y)=g(X, \varphi Y), \end{aligned}$$

which satisfies $\eta \wedge \Phi ^{n}\ne 0$. An almost contact pseudo-metric manifold is said to be a contact pseudo-metric manifold if $d\eta =\Phi$, where

$$\begin{aligned} d\eta (X, Y)=\frac{1}{2}(X\eta (Y)-Y\eta (X)-\eta ([X, Y])). \end{aligned}$$

The curvature operator R is given by

$$\begin{aligned} R(X, Y)=[\nabla _X, \nabla _Y]-\nabla _{[X, Y]}. \end{aligned}$$

This sign convention of R is opposite to the one used in [7, 9, 15,16,17]. The Ricci operator Q is determined by

$$\begin{aligned} S(X, Y ) = g(QX, Y ). \end{aligned}$$

In an almost contact pseudo-metric manifold $(M, \varphi , \xi , \eta , g)$ there always exists a special kind of local pseudo-orthonormal basis $\{e_i, \varphi e_i, \xi \}_{i=1}^n$, called a local $\varphi$-basis.

In a contact pseudo-metric manifold, the (1, 1) tensor $h=\frac{1}{2}\pounds _\xi \varphi$ is self-adjoint and satisfies

$$\begin{aligned} h\xi =0,\quad \varphi h + h\varphi =0, \quad \text {tr}(h) = \text {tr}(\varphi h)=0. \end{aligned}$$

Further, one has the following formulas:

$$\begin{aligned} \nabla _X \xi= & {} -\varepsilon \varphi X-\varphi h X, \end{aligned}$$

(2.4)

$$\begin{aligned} (\pounds _\xi g)(X, Y)= & {} 2g(h\varphi X, Y), \end{aligned}$$

(2.5)

$$\begin{aligned} (\nabla _\xi h)X= & {} \varphi X-h^2\varphi X+\varphi R(\xi , X)\xi , \end{aligned}$$

(2.6)

$$\begin{aligned} R(\xi , X)\xi -\varphi R(\xi ,\varphi X)\xi= & {} 2(h^2+\varphi ^2)X, \end{aligned}$$

(2.7)

$$\begin{aligned} \text {tr }\nabla \varphi= & {} 2n\xi . \end{aligned}$$

(2.8)

A contact pseudo-metric manifold M is said to be a K-contact pseudo-metric manifold if $\xi$ is a Killing vector field (or equivalently, $h=0$), and is said to be a Sasakian pseudo-metric manifold if the almost complex structure J on the product manifold $M\times \mathbb {R}$ defined by

$$\begin{aligned} J\left( X, f\frac{d}{dt}\right) =\left( \varphi X - f\xi , \eta (X)\frac{d}{dt}\right) , \end{aligned}$$

is integrable, where $X\in TM$, t is the coordinate on ${\mathbb {R}}$ and f is a $C^\infty$ function on $M\times {\mathbb {R}}$. It is well known that a contact pseudo-metric manifold M is a Sasakian pseudo-metric manifold if and only if

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

(2.9)

for all $X, Y\in TM$. A Sasakian pseudo-metric manifold is always K-contact pseudo-metric. A 3-dimensional K-contact pseudo-metric manifold becomes a Sasakian pseudo-metric manifold, which may not be true in higher dimensions. Further on a Sasakian pseudo-metric manifold we have

$$\begin{aligned} R(X,Y)\xi =\eta (Y)X-\eta (X)Y. \end{aligned}$$

(2.10)

In contact metric case, the condition (2.10) implies that the manifold is Sasakian, which is not true in contact pseudo-metric case [15]. However, we have the following:

Lemma 2.1

[15]. Let M be a K-contact pseudo-metric manifold. Then M is a Sasakian pseudo-metric manifold if and only if the curvature tensor R satisfies (2.10).

3 The Codazzi condition for h

A self-adjoint tensor A of type (1, 1) on a pseudo-Riemannian manifold is known to be a Codazzi tensor if

$$\begin{aligned} (\nabla _X A)Y=(\nabla _Y A)X \end{aligned}$$

(3.1)

for all $X, Y\in TM$. Now, we prove the following:

Theorem 3.1

Let M be a contact pseudo-metric manifold. Then the following statements are true:

• (i) If h is a Codazzi tensor, then $h^2=0$.

• (ii) M is a Sasakian pseudo-metric manifold if and only if M satisfies (2.10) and h is a Codazzi tensor.

Proof

(i) Suppose that h is a Codazzi tensor, that is,

$$\begin{aligned} (\nabla _X h)Y=(\nabla _Yh)X, \qquad X,Y \in TM. \end{aligned}$$

For $Y=\xi$, using (2.4) in the above equation, we obtain

$$\begin{aligned} (\nabla _\xi h)X=-\varepsilon \varphi h X- h^2 \varphi X. \end{aligned}$$

In view of (2.6), the above equation turns into

$$\begin{aligned} \varphi R(\xi , X)\xi =-\varepsilon \varphi h X- \varphi X. \end{aligned}$$

(3.2)

Operating $\varphi$ on both sides of (3.2), it follows that

$$\begin{aligned} R(\xi , X)\xi =\varphi ^2X-\varepsilon h X. \end{aligned}$$

(3.3)

Making use of (3.3) in (2.7), shows that $h^2=0$.

(ii) If M is a Sasakian pseudo-metric manifold, then $h=0$ and M satisfies (2.10); and the result is trivial. Conversely, suppose that (2.10) is true and h is a Codazzi tensor. From (2.10), we obtain that

$$\begin{aligned} R(\xi , X)\xi =\varphi ^2 X, \qquad X \in TM. \end{aligned}$$

(3.4)

Equations (3.3) and (3.4) imply that $h=0$, that is, M is a K-contact pseudo-metric manifold. Thus, the result follows from Lemma 2.1. $\square$

Remark 3.2

In a contact metric manifold, if h is a Codazzi tensor, then $h=0$, that is, the manifold becomes K-contact manifold [21]. In the Riemannian case, as $h^2=0$ implies $h=0$, Theorem 3.1 (a) holds in a stronger form, that is, M is K-contact if and only if h is a Codazzi tensor. But, in the case of M being contact pseudo-metric, the condition $h^2=0$ does not imply that $h=0$, because h may not be diagonalizable (see [16]). Note that the result (ii) of Theorem 3.1 is stronger than the Lemma 2.1 which was proved in [15].

In a contact Lorentzian manifold, just like the case of contact metric manifold, the condition $h^2=0$ implies $h=0$ (see [6]). Hence, we immediately have the following

Corollary 3.3

Let M be a contact Lorentzian manifold. If h is a Codazzi tensor, then $h=0$, that is, M is K-contact Lorentzian manifold.

4 The Codazzi condition for $\tau$

We denote by $\tau$, the tensor metrically equivalent to the strain tensor $\pounds _\xi g$ along $\xi$, that is,

$$\begin{aligned} g(\tau X, Y)=(\pounds _\xi g)(X,Y) \end{aligned}$$

for all $X, Y\in TM$. As pointed out in the introduction, in a contact metric manifold, if $\tau$ satisfies the Codazzi condition, then $h=0$, that is, the manifold is a K-contact manifold. This fact need not be true in the case of contact pseudo-metric manifolds. So, it is quite interesting to study contact pseudo-metric manifolds, which satisfy the Codazzi condition for $\tau$. Now we prove the following:

Lemma 4.1

In a contact pseudo-metric manifold, $\tau$ is a Codazzi tensor if and only if the curvature tensor R satisfies

$$\begin{aligned} R(\xi , X)Y=\varepsilon (\nabla _X \varphi )Y. \end{aligned}$$

(4.1)

Proof

Treating $\nabla \xi$ as a tensor of type (1, 1), that is $\nabla \xi : X \mapsto \nabla _X \xi$, one can see that

$$\begin{aligned} R(X, Y)\xi =(\nabla _X\nabla \xi )Y-(\nabla _Y\nabla \xi )X, \end{aligned}$$

which together with (2.4) gives

$$\begin{aligned} R(X,Y)\xi =-\varepsilon (\nabla _X\varphi )Y-(\nabla _X\varphi h)Y+\varepsilon (\nabla _Y\varphi )X+(\nabla _Y\varphi h)Y. \end{aligned}$$

(4.2)

On the other hand, if $\tau$ is a Codazzi tensor, then from (2.5) we have

$$\begin{aligned} (\nabla _X h\varphi )Y=(\nabla _Y h\varphi )X. \end{aligned}$$

Thus, (4.2) shows that $\tau$ is a Codazzi tensor if and only if

$$\begin{aligned} R(X,Y)\xi =\varepsilon \{(\nabla _Y\varphi )X-(\nabla _X\varphi )Y\}. \end{aligned}$$

(4.3)

Now if $\tau$ is a Codazzi tensor, then by using Bianchi identity and (4.3), we get

$$\begin{aligned} R(\xi , X, Y,Z)&=\varepsilon \{g(X, (\nabla _X\varphi )Y)-g(Y, (\nabla _X\varphi )Z)+g(Z, (\nabla _X\varphi )Y)\\&\quad -g(X, (\nabla _Z\varphi )Y)\}\\&=-2\varepsilon g((\nabla _X\varphi )Z, Y)+R(Z,Y,\xi , X), \end{aligned}$$

and so

$$\begin{aligned} R(\xi ,X,Y,Z)=-\varepsilon g((\nabla _X\varphi )Z, Y), \end{aligned}$$

which gives (4.1).

Conversely, if (4.1) is true, then from Bianchi identity we have

$$\begin{aligned} R(X, Y, \xi , Z)&=R(\xi , Z, X, Y)=-R(Z,X,\xi ,Y)-R(X, \xi , Z, Y)\\&=-R(\xi , Y, Z, X)+R(\xi , X, Z, Y)\\&=-\varepsilon \{g((\nabla _Y\varphi )Z, X)-g((\nabla _X\varphi )Z,Y)\}, \end{aligned}$$

which leads to (4.3), and hence $\tau$ is a Codazzi tensor. $\square$

Now we prove the contact semi-Riemannian version of Theorem 3.3 of [14].

Theorem 4.2

Let M be a contact pseudo-metric manifold. Then the following statements are true.

• (i) If $\tau$ is a Codazzi tensor, then $h^2=0$ and the Ricci operator Q satisfies

  $$\begin{aligned} Q\xi =2\varepsilon n\xi . \end{aligned}$$

  (4.4)

• (ii) M is Sasakian if and only if M satisfies (2.10) and $\tau$ is a Codazzi tensor.

Proof

(i). If $\tau$ is a Codazzi tensor, then (4.1) gives

$$\begin{aligned} R(\xi , X)\xi =\varepsilon (\nabla _X\varphi )\xi =\varphi ^2X-\varepsilon h X, \end{aligned}$$

where we used (2.4). This implies

$$\begin{aligned} \varphi R(\xi , \varphi X)\xi =-\varphi ^2X-\varepsilon h X, \end{aligned}$$

and so

$$\begin{aligned} R(\xi , X)\xi -\varphi R(\xi , \varphi X)\xi =2\varphi ^2X. \end{aligned}$$

(4.5)

Comparing (2.7) and (4.5), we obtain $h^2=0$.

Now, if $\{e_i\}_i^{2n+1}$ is any local pseudo-orthonormal basis, then considering (4.1) we get

$$\begin{aligned} S(X, \xi )&=\sum _{i=1}^{2n+1}\varepsilon _i R(e_i, X, \xi , e_i) = \varepsilon \sum _{i=1}^{2n+1}\varepsilon _i g((\nabla _{e_i}\varphi )e_i, X) \\&=\varepsilon g(\text {tr}(\nabla \varphi ), X), \end{aligned}$$

which by using (2.8) we have (4.4).

(ii). Suppose that M is a Sasakian pseudo-metric manifold, then M satisfies (2.10) and $h=0$.

Conversely, suppose that M satisfies (2.10) and $\tau$ is a Codazzi tensor. Then (4.1) shows that

$$\begin{aligned} g((\nabla _X\varphi )Y, Z)&=\varepsilon R(\xi , X, Y, Z)=-\varepsilon R(Z, Y, \xi , X)\\&=-\varepsilon \{ \eta (Y)g(Z,X)-\varepsilon g(Z, \xi )g(X, Y)\} \end{aligned}$$

which gives (2.9). Hence M becomes a Sasakian pseudo-metric manifold. $\square$

Corollary 4.3

Let M be a contact Lorentzian manifold. If $\tau$ is a Codazzi tensor, then $h=0$, that is, M is K-contact Lorentzian manifold.

5 $\xi$-Sectional curvatures

The $\xi$-sectional curvature $K(\xi , X )$ of a contact pseudo-metric manifold is defined by

$$\begin{aligned} K(\xi , X)=\varepsilon {\varepsilon _{\! X}} g(R(\xi ,X)X, \xi ), \end{aligned}$$

where X is a unit vector field such that $X\in \text {Ker}\, \eta$ and $g(X, X)={\varepsilon _{\! X}} =\pm 1$.

It is well known that a contact metric manifold is K-contact if and only if all $\xi$-sectional curvatures are equal to +1 (see [5]). We recall that for a K-contact pseudo-metric manifold, all $\xi$-sectional curvatures are equal to $\varepsilon$ (see [15, Theorem 3.3]).

Now we prove the following result which is related to Theorem 3.5 of [14].

Theorem 5.1

On a contact pseudo-metric manifold M, the $\xi$-sectional curvatures satisfy

$$\begin{aligned} K(\xi , X)= & {} \varepsilon \{1-{\varepsilon _{\! X}} g(h^2X, X)-{\varepsilon _{\! X}} g((\nabla _\xi h)X, \varphi X)\}, \end{aligned}$$

(5.1)

$$\begin{aligned} K(\xi , X)= & {} K(\xi , \varphi X)-2\varepsilon {\varepsilon _{\! X}} g((\nabla _\xi h)X , \varphi X) \end{aligned}$$

(5.2)

for any unit vector $X\in \text {Ker}\, \eta$.

Proof

Using (2.6), we have

$$\begin{aligned} K(\xi , X)&=-\varepsilon {\varepsilon _{\! X}} R(\xi , X, \xi , X)\\&=-\varepsilon {\varepsilon _{\! X}} g(-\varphi (\nabla _\xi h)X-X+h^2X, X)\\&=\varepsilon \{{\varepsilon _{\! X}} g(\varphi (\nabla _\xi h)X, X)+{\varepsilon _{\! X}}^2-{\varepsilon _{\! X}} g(h^2X, X)\}, \end{aligned}$$

which gives (5.1).

Now, plugging X by $\varphi X$ in (5.1) keeping $h \varphi =-\varphi h$ and $\nabla _\xi \varphi =0$ in mind, we obtain

$$\begin{aligned} K(\xi , \varphi X)=\varepsilon \{1-{\varepsilon _{\! X}} g(h^2X, X)+{\varepsilon _{\! X}} g((\nabla _\xi h)X, \varphi X)\}. \end{aligned}$$

(5.3)

Now, from (5.1) and (5.3), we get (5.2). $\square$

As we discussed in introduction, due to the fact that $h^2=0$ does not imply $h=0$ in a contact pseudo-metric manifold, the parallel condition of $\ell$ does not imply that $\xi$-sectional curvatures vanish. However, in the following we show that this is true with an additional assumption $\nabla _\xi h=0$.

Theorem 5.2

Let M be a contact pseudo-metric manifold with $\nabla _\xi h=0$. Then the following are true.

• (i) The tensor $h^2=0$ if and only if all $\xi$-sectional curvatures are equal to $\varepsilon$.

• (ii) If $\nabla \ell =0$, then all $\xi$-sectional curvatures vanish.

Proof

(i). Taking the inner product of the unit vector field $X\in \text {Ker}\, \eta$ with (2.7) yields the following formula for sectional curvatures:

$$\begin{aligned} K(\xi , X)+K(\xi , \varphi X)=2\varepsilon \{ 1-{\varepsilon _{\! X}} g(h^2 X, X)\}. \end{aligned}$$

(5.4)

Now, since $\nabla _\xi h=0$, (5.2) yields

$$\begin{aligned} K(\xi , X)=K(\xi , \varphi X) \end{aligned}$$

(5.5)

for any unit vector $X\in \text {Ker } \eta$. From (5.4) and (5.5) we see

$$\begin{aligned} K(\xi , X)=\varepsilon \text { if and only if } g(h^2 X, X)=0. \end{aligned}$$

This concludes the proof of (i).

(ii). Applying by $\varphi$ on both sides of (2.6) and using $\nabla _\xi h=0$, it follows that

$$\begin{aligned} \ell X = -h^2 X + X-\eta (X)\xi , \end{aligned}$$

(5.6)

for any $X\in TM$. Now, in view of $(\nabla _X \ell )\xi =0$ and (5.6), we have

$$\begin{aligned} \varepsilon h^2 \varphi X-h^3 \varphi X-\varepsilon \varphi X+h \varphi X=0. \end{aligned}$$

(5.7)

If $X\in \text {Ker } \eta$ is a unit vector field, then taking the inner product of $\varphi X$ with (5.7) leads to

$$\begin{aligned} \varepsilon g(h^2 X, X)+g(h^3 X, X)-\varepsilon g(X,X)-g(h X, X)=0. \end{aligned}$$

(5.8)

Now replacing X by $\varphi X$ in (5.7) and then taking inner product of X with the resulting equation gives

$$\begin{aligned} -\varepsilon g(h^2 X, X)+g(h^3 X, X)+\varepsilon g(X,X)-g(h X, X)=0. \end{aligned}$$

(5.9)

Now subtracting (5.8) from (5.9) yields

$$\begin{aligned} g(h^2 X, X)=g(X,X)={\varepsilon _{\! X}} \end{aligned}$$

(5.10)

for any unit vector $X\in \text {Ker}\, \eta$. Using (5.10) and $\nabla _\xi h=0$ in (5.1) we conclude that $K(\xi , X)=0$. $\square$

Corollary 5.3

A contact Lorentzian manifold is a K-contact Lorentzian manifold if and only if all $\xi$-sectional curvatures are equal to $-1$.

6 Almost $\textit{CR}$ structures

First, we recall few notions of almost $\textit{CR}$ structures (see [11, 16, 18]). Let M be a $(2n+1)$-dimensional (connected) differentiable manifold. Let $\mathcal {H}(M)$ be a smooth real subbundle of rank 2n of the tangent bundle TM (also called Levi distribution), and $J:\mathcal {H}(M)\rightarrow \mathcal {H}(M)$ be a smooth bundle isomorphism such that $J^2=-I$. Then the pair $(\mathcal {H}(M), J)$ is called an almost $\textit{CR}$ structure on M. An almost $\textit{CR}$ structure is called a $\textit{CR}$ structure if it is integrable, that is, the following two conditions are satisfied

$$\begin{aligned}{}[JX,Y]+[X,JX]\in & {} \mathcal {H}(M), \end{aligned}$$

(6.1)

$$\begin{aligned} J([JX,Y]+[X,JY])= & {} [JX,JY]-[X,Y] \end{aligned}$$

(6.2)

for all $X,Y\in \mathcal {H}(M)$.

On an almost $\textit{CR}$ manifold $(M,\mathcal {H}(M),J)$, we define a 1-form $\theta$ such that $\text {Ker}\, \theta =\mathcal {H}(M)$, and such a differential 1-form $\theta$ is called a pseudo-Hermitian structure on M. Then on $\mathcal {H}(M)$, the Levi form $L_\theta$ is defined by

$$\begin{aligned} L_\theta (X,Y)=d\theta (X,JY) \end{aligned}$$

for all $X,Y\in \mathcal {H}(M)$. Furthermore, we define a (0, 2)-tensor field on $\mathcal {H}(M)$ by

$$\begin{aligned} \alpha (X,Y)=(\nabla _X\theta )(JY)+(\nabla _{JX}\theta )(Y) \end{aligned}$$

for all $X,Y\in \mathcal {H}(M)$.

Then we have the following:

Proposition 6.1

For an almost $\textit{CR}$ structure $(\mathcal {H}(M),J,\theta )$, the following statements are equivalent:

• (i) $L_\theta$ is Hermitian, that is, $L_\theta (JX,JY)=L_\theta (X,Y)$;

• (ii) $L_\theta$ is symmetric, that is, $L_\theta (X,Y)=L_\theta (Y,X)$;

• (iii) $[JX,Y]+[X,JY]\in \mathcal {H}(M)$;

• (iv) $\alpha$ is symmetric, that is, $\alpha (X,Y)=\alpha (Y,X)$.

Proof

It is immediate that (i)$\Leftrightarrow$(ii) and (ii)$\Leftrightarrow$(iii) follows from the fact that

$$\begin{aligned} d\theta (X,Y)=-\frac{1}{2}\theta ([X,Y]) \end{aligned}$$

for all $X,Y\in \mathcal {H}(M)$. On the other hand, as in general

$$\begin{aligned} d\theta (X,Y)=\frac{1}{2}((\nabla _X\theta )Y-(\nabla _Y\theta )X), \end{aligned}$$

the condition (ii) is equivalent to

$$\begin{aligned} (\nabla _X\theta )(JY)+(\nabla _{JX}\theta )Y=(\nabla _Y\theta )(JX)+ (\nabla _{JY}\theta )X, \end{aligned}$$

and so (ii)$\Leftrightarrow$(iv). $\square$

The authors came to know from the reviewer that, the equivalence of (ii) with (iv) is related to Lemma 1 of [25], and the equivalence of (i), (ii) and (iii) can be found, for example, in [19] (see Proposition 6, p.17).

An almost pseudo-Hermitian $\textit{CR}$ structure $(\mathcal {H}(M), J,\theta )$ is said to be nondegenerate if the Levi form $L_\theta$ is a nondegenerate Hermitian form, and so the 1-form $\theta$ is a contact form.

Let $(M, \mathcal {H}(M),J,\theta )$ be a nondegenerate pseudo-Hermitian almost $\textit{CR}$ manifold. We extend the complex structure J to an endomorphism $\varphi$ of the tangent bundle TM in such a way that $\theta =J$ on $\mathcal {H}(M)$ and $\varphi \xi =0$, where $\xi$ is the Reeb vector field of $\theta$. Then the Webster metric $g_\theta$, which is a pseudo-Riemannian metric, is defined by

$$\begin{aligned} g_\theta (X,Y)=L_\theta (X,Y),\quad g_\theta (X,\xi )=0, \quad g_\theta (\xi ,\xi )=\varepsilon \end{aligned}$$

for all $X, Y\in \mathcal {H}(M)$. In this case, $(\varphi ,\xi ,\eta =-\theta , g=g_\theta )$ defines a contact pseudo-metric structure on M. Conversely, if $(\varphi ,\xi ,\eta ,g)$ is a contact pseudo-metric structure, then $(\mathcal {H}(M), J, \theta )$, where $\mathcal {H}(M)=\text {Ker}\, \eta$, $\theta =-\eta$, and $J=\varphi _{\arrowvert \mathcal {H}(M)}$, defines a nondegenerate almost $\textit{CR}$ structure on M. Thus, we have:

Proposition 6.2

([16]). The notion of nondegenerate almost $\textit{CR}$ structure $(\mathcal {H}(M), J, \theta )$ is equivalent to the notion of contact pseudo-metric structure $(\varphi ,\xi ,\eta ,g)$.

Now, we prove the following result which is related to Theorem 1 of [25].

Theorem 6.3

An almost contact pseudo-metric manifold M is $\textit{CR}$ manifold if and only if

$$\begin{aligned} (\nabla _X J)Y-(\nabla _{JX}J)JY=\alpha (X,Y)\xi \end{aligned}$$

(6.3)

for all $X,Y\in \mathcal {H}(M)$.

Proof

Applying J to (6.2) gives

$$\begin{aligned} (\nabla _YJ)X-(\nabla _XJ)Y=J(\nabla _{JX}J)Y-J(\nabla _{JY}J)X \end{aligned}$$

for all $X,Y\in \mathcal {H}(M)$. Since $J(\nabla _{JX}J)Y=-(\nabla _{JX}J)JY$, the above equation becomes

$$\begin{aligned} (\nabla _YJ)X-(\nabla _XJ)Y=(\nabla _{JY}J)JX-(\nabla _{JX}J)JY \end{aligned}$$

(6.4)

for all $X,Y\in \mathcal {H}(M)$. If we define a (0, 3)-tensor field A on $\mathcal {H}(M)$ as

$$\begin{aligned} A(X,Y,Z)=g((\nabla _{JX}J)JY-(\nabla _XJ)Y,Z) \end{aligned}$$

(6.5)

for all $X,Y\in \mathcal {H}(M)$, then from (6.4) one obtain

$$\begin{aligned} A(X,Y,Z)=A(Y,X,Z). \end{aligned}$$

(6.6)

Next, a simple computation shows that

$$\begin{aligned}&A(X,Y,Z)+A(X,Z,Y)\\&\quad =g((\nabla _{JX}J)JY-(\nabla _XJ)Y,Z)+g((\nabla _{JX}J)JZ - (\nabla _XJ)Z,Y) \\&\quad =-g((\nabla _{JX}J)Z,JY)+g((\nabla _{JX}J)JZ,Y)\\&\quad =-g(\nabla _{JX}JZ,JY)-g((\nabla _{JX}Z),J^2Y) \\&\qquad +g(\nabla _{JX}J^2Z,Y) - g(J(\nabla _{JX}JZ),Y)\\&\quad =0, \end{aligned}$$

where the skew-symmetry of J and $\nabla J$ are used. This together with (6.6) gives the following:

$$\begin{aligned} A(X,Y,Z)&=-A(X,Z,Y)=-A(Z,X,Y)=A(Z,Y,X)\\&=A(Y,Z,X)=-A(Y,X,Z)=-A(X,Y,Z). \end{aligned}$$

Hence it follows that $A=0$, and so (6.5) implies

$$\begin{aligned} (\nabla _{JX}J)JY-(\nabla _XJ)Y=\gamma (X,Y)\xi \end{aligned}$$

(6.7)

for all $X,Y\in \mathcal {H}(M)$, for certain (0,2)-tensor field $\gamma$ on $\mathcal {H}(M)$. It remains to show that $\gamma =\alpha$. From (6.7), it follows that

$$\begin{aligned} \gamma (X,Y)&=\varepsilon g((\nabla _{JX}J)JY-(\nabla _XJ)Y,\xi )\\&=\varepsilon \{- g((\nabla _{JX}J)\xi ,JY)+g((\nabla _XJ)\xi ,Y)\}\\&=\varepsilon \{ g(\nabla _{JX}\xi ,Y)-g(J\nabla _X\xi ,Y)\}\\&=(\nabla _{JX}\theta )Y+(\nabla _X\theta )JY\\&=\alpha (X,Y). \end{aligned}$$

Conversely, suppose that (6.3) holds true. Then projecting (6.3) onto $\xi$, it follows that $\alpha$ is symmetric and is equivalent to (6.1). The symmetry of $\alpha$ together with (6.3) gives (6.4), which yields

$$\begin{aligned} -[JX,Y]-[X,JY]=J[JX,JY]-J[X,Y], \end{aligned}$$

for all $X,Y\in \mathcal {H}(M)$, and so satisfies the Eq. (6.2). $\square$

Let $(M, \varphi ,\xi ,\eta ,g)$ be an almost contact pseudo-metric manifold with $(\mathcal {H}(M), J)$ as the corresponding almost $\textit{CR}$ structure. For $Y\in TM$, we denote $Y_{\arrowvert \mathcal {H}(M)}$ to the orthogonal projection on $\mathcal {H}(M)$. Then, the Bott partial connection $\breve{\nabla }$ on $\mathcal {H}(M)$ (along $\xi$) is the map $\breve{\nabla }:S(\xi )\times \mathcal {H}(M)\rightarrow \mathcal {H}(M)$ defined by

$$\begin{aligned} \breve{\nabla }_\xi X := (\pounds _\xi X)_{\arrowvert \mathcal {H}(M)}=[\xi , X]_{\arrowvert \mathcal {H}(M)} \end{aligned}$$

for any $X\in \mathcal {H}(M)$ (see, [20, p. 18]), where $S(\xi )$ is the 1-dimensional linear subspace of TM generated by $\xi$.

Theorem 6.4

Let $(M, \varphi ,\xi ,\eta ,g)$ be an almost contact pseudo-metric manifold, and $\xi$ a geodesic vector field. Then $h=0$ if and only if $\breve{\nabla }_\xi J=0$.

Proof

As $h\xi =0$, we may observe that, $h=0$ if and only if $hX=0$ for any $X\in \mathcal {H}(M)$.

Now using $\nabla _\xi \xi =0$, for any $X\in \mathcal {H}(M)$, we have

$$\begin{aligned} \eta ([\xi ,X])=\varepsilon g(\xi , \nabla _\xi X-\nabla _X \xi )=0, \end{aligned}$$

which means $\pounds _\xi X \in \mathcal {H}(M)$. Thus, we get

$$\begin{aligned} 2hX&=\pounds _\xi (\varphi X)-\varphi (\pounds _\xi X)=\breve{\nabla }_\xi (\varphi X)- \varphi (\breve{\nabla }_\xi X) \\&=\breve{\nabla }_\xi (JX)-J(\breve{\nabla }_\xi X)=(\breve{\nabla }_\xi J)X \end{aligned}$$

for any $X\in \mathcal {H}(M)$, completing the proof. $\square$

Note that the proof of Theorem 6.4 is related to the formula (4.3) of [14]. For contact pseudo-metric manifold, the structure vector field is geodesic. So we have the following:

Corollary 6.5

A contact pseudo-metric manifold is K-contact pseudo-metric if and only if $\breve{\nabla }_\xi J=0$.

For paracontact metric manifolds the result of the following Theorem was proved in Corollary 4.9 of [14].

Theorem 6.6

A contact pseudo-metric manifold $(M, \varphi ,\xi ,\eta ,g)$ is Sasakian pseudo-metric if and only if the corresponding nondegenerate almost $\textit{CR}$ structure $(\mathcal {H}(M), J)$ is integrable and $\breve{\nabla }_\xi J=0$.

Proof

First we observe that, following the same proof given in [23] for the Riemannian case, the integrable condition (that is, (6.1) and (6.2)) of the corresponding $\textit{CR}$ structure $(\mathcal {H}(M), J)$ is equivalent to

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

(6.8)

where

$$\begin{aligned} (\nabla _X\eta )\varphi Y=-g(X,Y)+\varepsilon \eta (X)\eta (Y)-\varepsilon g(hX,Y), \end{aligned}$$

and

$$\begin{aligned} \varphi (\nabla _X \xi )=\varepsilon X-\varepsilon \eta (X)\xi +hX. \end{aligned}$$

Thus, (6.8) becomes

$$\begin{aligned} (\nabla _X \varphi )Y=g(X+\varepsilon hX,Y)\xi -\varepsilon \eta (Y)(X+\varepsilon hX). \end{aligned}$$

(6.9)

If the contact pseudo-metric manifold $(M, \varphi ,\xi ,\eta ,g)$ is Sasakian, then (6.9) satisfies with $h=0$, and so corresponding nondegenerate almost $\textit{CR}$ structure $(\mathcal {H}(M), J)$ is integrable and $\breve{\nabla }_\xi J=0$.

Conversely, as $\breve{\nabla }_\xi J=0$ implies $h=0$, Eq. (6.9) reduces to (2.9), and so the structure is Sasakian pseudo-metric. $\square$