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.
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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
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
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
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
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
The curvature operator R is given by
This sign convention of R is opposite to the one used in [7, 9, 15,16,17]. The Ricci operator Q is determined by
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
Further, one has the following formulas:
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
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
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
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
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,
For \(Y=\xi \), using (2.4) in the above equation, we obtain
In view of (2.6), the above equation turns into
Operating \(\varphi \) on both sides of (3.2), it follows that
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
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,
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
Proof
Treating \(\nabla \xi \) as a tensor of type (1, 1), that is \(\nabla \xi : X \mapsto \nabla _X \xi \), one can see that
which together with (2.4) gives
On the other hand, if \(\tau \) is a Codazzi tensor, then from (2.5) we have
Thus, (4.2) shows that \(\tau \) is a Codazzi tensor if and only if
Now if \(\tau \) is a Codazzi tensor, then by using Bianchi identity and (4.3), we get
and so
which gives (4.1).
Conversely, if (4.1) is true, then from Bianchi identity we have
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
where we used (2.4). This implies
and so
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
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
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
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
for any unit vector \(X\in \text {Ker}\, \eta \).
Proof
Using (2.6), we have
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
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:
Now, since \(\nabla _\xi h=0\), (5.2) yields
for any unit vector \(X\in \text {Ker } \eta \). From (5.4) and (5.5) we see
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
for any \(X\in TM\). Now, in view of \((\nabla _X \ell )\xi =0\) and (5.6), we have
If \(X\in \text {Ker } \eta \) is a unit vector field, then taking the inner product of \(\varphi X\) with (5.7) leads to
Now replacing X by \(\varphi X\) in (5.7) and then taking inner product of X with the resulting equation gives
Now subtracting (5.8) from (5.9) yields
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
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
for all \(X,Y\in \mathcal {H}(M)\). Furthermore, we define a (0, 2)-tensor field on \(\mathcal {H}(M)\) by
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
for all \(X,Y\in \mathcal {H}(M)\). On the other hand, as in general
the condition (ii) is equivalent to
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
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
for all \(X,Y\in \mathcal {H}(M)\).
Proof
Applying J to (6.2) gives
for all \(X,Y\in \mathcal {H}(M)\). Since \(J(\nabla _{JX}J)Y=-(\nabla _{JX}J)JY\), the above equation becomes
for all \(X,Y\in \mathcal {H}(M)\). If we define a (0, 3)-tensor field A on \(\mathcal {H}(M)\) as
for all \(X,Y\in \mathcal {H}(M)\), then from (6.4) one obtain
Next, a simple computation shows that
where the skew-symmetry of J and \(\nabla J\) are used. This together with (6.6) gives the following:
Hence it follows that \(A=0\), and so (6.5) implies
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
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
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
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
which means \(\pounds _\xi X \in \mathcal {H}(M)\). Thus, we get
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
where
and
Thus, (6.8) becomes
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. \(\square \)
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Acknowledgements
The authors would like to thank the reviewer for careful and thorough reading of this manuscript and thankful for helpful suggestions towards the improvement of this paper. The second author (D.M.N.) is grateful to University Grants Commission, New Delhi (Ref. No.: 20/12/2015(ii)EU-V) for financial support in the form of Junior Research Fellowship.
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Venkatesha, V., Naik, D.M. & Tripathi, M.M. Certain results on almost contact pseudo-metric manifolds. J. Geom. 110, 41 (2019). https://doi.org/10.1007/s00022-019-0498-7
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DOI: https://doi.org/10.1007/s00022-019-0498-7