A New Class of Contact Pseudo Framed Manifolds with Applications

Duggal, K. L.

doi:https://doi.org/10.1155/2021/6141587

International Journal of Mathematics and Mathematical Sciences

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Research Article | Open Access

Volume 2021 | Article ID 6141587 | https://doi.org/10.1155/2021/6141587

A New Class of Contact Pseudo Framed Manifolds with Applications

K. L. Duggal¹

Academic Editor: Luca Vitagliano

Received26 Jun 2021

Revised05 Aug 2021

Accepted10 Aug 2021

Published27 Aug 2021

Abstract

In this paper, we introduce a new class of contact pseudo framed (CPF)-manifolds by a real tensor field of type , a real function such that where is its characteristic vector field. We prove in our main Theorem 2 that admits a closed 2-form if is constant. In 1976, Blair proved that the vector field of a normal contact manifold is Killing. Contrary to this, we have shown in Theorem 2 that, in general, of a normal CPF-manifold is non-Killing. We also have established a link of CPF-hypersurfaces with curvature, affine, conformal collineations symmetries, and almost Ricci soliton manifolds, supported by three applications. Contrary to the odd-dimensional contact manifolds, we construct several examples of even- and odd-dimensional semi-Riemannian and lightlike CPF-manifolds and propose two problems for further consideration.

1. Introduction

Let be an -dimensional real differential manifold. Recently, we introduced in [1] a new class of differentiable structure called pseudo Cauchy Riemann structure on defined by a real tensor field of type at every point of satisfyingwhere is a nonzero real function on . Suppose admits a real -dimensional distribution where is its metric tensor. Following are three mutually exclusive cases of the causal character (see O’Neill [2]) of : (a) is Riemannian, (b) is semi-Riemannian, and (c) is lightlike. If (a) or (b) holds, then following the terminology of Newlander and Nirenberg [3], a manifold with a PCR-structure admits a realizable PCR-structure (see Penrose [4]) if is invariant () with respect to and it is involutive, that is, . Then, is called a PCR-manifold satisfying

In this paper, we assume that PCR-structure is realizable. We quote following two results for any PCR-manifold.

Theorem 1 (see [1]). A PCR-manifold is not necessarily even-dimensional if is everywhere nonzero real function on .

Suppose admits a semi-Riemannian metric . Then is also an almost metric PCR-manifold if for a nonzero real function ,that is, compatible with is conformal or homothetic (in particular, isometric) if is constant (in particular, ), respectively.

Proposition 1 (see [1]). Suppose an almost metric PCR-manifold admits a 2-form defined by (3). Then, .

In support of the above results, we presented two odd-dimensional examples of almost metric PCR-manifolds and one spacetime model. Also, in same paper [1], we introduced a revised version of a contact manifold, called contact pseudo framed (CPF)-manifold by a real tensor field of type and a real function such that , and splits into a direct sum of two subbundles, namely, with a pseudo Cauchy Riemann (PCR)-structure and 1-dimensional . Contrary to the odd-dimensional contact manifolds, for the first time in the literature, we presented some examples of a class of even-dimensional CPF-manifolds and also shown that the metric of PCR and CPF-manifolds is not severely restricted.

The purpose of this paper is to study CPF-manifolds, with focus on their new even-dimensional class, and discuss the similarities and differences compared with the results of odd-dimensional contact manifolds [5] and physical applications compared with one of our paper on contact spacetime manifolds [6]. We also solve some problems proposed in our previous paper [1], propose new open problems, and suggest further research on this topic.

2. Contact Pseudo Framed (CPF) Manifolds

Recall from [1] that an -dimensional manifold is called an almost metric contact pseudo framed (CPF) manifold, where is a 1-form (called contact form); is a vector field, called characteristic vector field; is a semi-Riemannian metric; and f is a tensor field satisfyingwhere or as is spacelike or timelike and , , and are real nonzero functions. Also, as per previous section, there exists an -dimensional almost PCR-distribution given by which is not necessarily even-dimensional and is spacelike or timelike or lightlike, where is never a lightlike vector.

Example 1. Let be a basis for at a point of a 6- dimensional semi-Riemannian manifold , with a real tensor field of type and two real nonzero functions and . SupposeThen, it is easy to see that . Therefore, we have a 5-dimensional PCR-distribution generated by and a complementary 1-dimensional null operator such that splits into a direct sum of two subbundles, namely, (with a PCR-structure) and . Using the metric compatible equation (3) for the PCR-subspace, we getCancelling , we get . Similarly, for the vectors , we get . Here, we used the same symbol f for the PCD-subspace. Using the above data and taking , we haveTherefore, is an almost metric (Riemannian or Lorentzian as is 1 or , respectively) CPF-manifold.

Remark 1. The above example confirms that contrary to the odd-dimensional almost contact manifolds, there do exist a new class of even-dimensional almost Riemannian or Lorentzian CPF-manifolds. Secondly, if we take the triplet , then it is easy to construct an example of semi-Riemannian almost CPF-manifold. So, the metric of this new class is not very restrictive. Finally, we leave it an open problem of taking another set of data for for constructing more general examples of semi-Riemannian even- and odd-dimensional almost CPF-manifolds. In particular, if is global, then for the Lorentzian case (following the terminology used in [6]), we say that is a spacetime manifold. As an application, see reference [1] where it has been shown that 4-dimensional (in general -dimensional) de-Sitter and Robertson–Walker spacetimes are physical examples of almost CPF-spacetime manifolds. Moreover, all the physical applications presented in an earlier paper [6] on odd-dimensional contact spacetimes will also hold for even-dimensional almost CPF-spacetimes.
Suppose admits a 2-form defined byIt is easy to show that for admitting above 2-form , then . We say that is almost CPK-manifold (briefly, ACPK-manifold) if is closed, i.e., , and it is called metric CPK-manifold if its CPF-structure is normal, i.e., its torsion tensor is zero, i.e.,where is the pseudo Nijenhuis tensor of .

2.1. ACPF-Hypersurfaces

Let be an almost metric CPF-hypersurface of a metric PCR-manifold . Then, as explained in the previous section, there exists an -dimensional almost PCR-distribution given by , and is either nondegenerate or degenerate. In this subsection, we assume that is nondegenerate.

Mathematical Model. Let be a product manifold where is a 1-dimensional affine space and is an almost metric CPF-manifold with respect to a metric connection satisfying . Denote a vector field on by where is tangent to , is coordinate of , and is a real function on . Define a tensor field and a semi-Riemannian metric on bywhere is also a real function on . Using above two equations, it is easy to show that and . We further assume that . Therefore, is a metric PCR-manifold with respect to a metric connection satisfying .

Theorem 2. Let be an almost metric CPF-hypersurface of a metric PCR-manifold which satisfies (10) and (11). If admits a 2-form and is constant, then(a) is closed, that is, (b)(c) is Killing if is also constant

Proof 1. Using , a straightforward computation ofprovidesFrom the above two results with some computations, it is easy to show that(1) (2)For constant, Equation (1) implies if is constant, (a) holds. Now, again using the above two results and providesThen, (2) follows by using (15) and constant in (14). Therefore,Consequently, (b) holds and (c) is immediate.

Remark 2. Recall an open problem in [1] to find the condition on pair for almost Khler , and since are same for almost CPF-manifolds, in Theorem 2, we have shown that ACPF-manifold if is constant, which also holds for ACPK-manifold and CPK-manifold if its CPF-structure is normal. In 1990, Chinea and Gonzales [7] studied classification of the almost contact metric manifolds into several classes. We leave it an open problem to investigate similar classification of ACPF, ACPK, and CPK metric manifolds. Blair [5] has proved that a contact metric manifold is K-contact manifold if its characteristic vector field is Killing. On the contrary, we have shown in the above theorem that of a CPK-manifold is Killing only if is also constant. Later on we will discuss an application of the non-Killing condition (b) of this theorem.
Since Example 1 holds for , it does not support the condition of Theorem 2 with , for which we have following example.

Example 2. Suppose is a basis for at a point of a semi-Riemannian manifold with a real tensor field of type and two real nonzero functions and on . Consider, as per Theorem 2, is hypersurface of a metric PCR-manifold which satisfies (13) and (14) with constant. LetFrom the above, we get . Therefore, as explained in the previous example, we can use the metric compatible equation (3). Assume that admits a 2-form defined by (8) so .
Take , and , where , , and are real functions on . Then, using (3), we get . Thus, . Similarly, we get . With this data and, we haveThus, in support of Theorem 2, we have an example of an odd-dimensional semi-Riemannian ACPF-hypersurface , of a metric PCR-manifold , which admits a 2-form and spacelike or timelike as is 1 or . The case of even-dimensional semi-Riemannian ACPF-hypersurfaces which admit a 2-form is left open.

Application 1. Here, we present an application of the conclusion (b) of Theorem 2 for a -dimensional ACPF-hypersurface of a -dimensional metric PCR-manifold , which admits a closed 2-form if is constant. This means thatConsider a local coordinates system of at a point of . We know that for the tangent vector , there is its dual . If we set where is the ordinary derivative, then relating this with , we get . Putting this value of in (19), we getFor an application of the above data, we now prefer using the local coordinates system . Suppose, in general, the Ricci tensor of a CPF-hypersurface of a semi-Riemannian manifold is of the form as follows:where are constants and are functions on . A particular case of above expression (21) of Ricci tensor was used by Yano (see page 163 in [8]) for a real hypersurface of a Kaehlerian manifold for which , and he called pseudo-Einstein and Einstein only if . Now, we need following brief information on a symmetry called curvature collineation (CC) defined and studied by Katzin et al. [9] on the existence of a vector field of an -dimensional semi-Riemannian manifold () which leaves its curvature tensor invariant. This meansObviously, (22) implies, by contraction, that is also a Ricci collineation (RC) vector field, that is,However, it is easy to see that RC does not imply CC. In general, we setThe following form of the Lie-derivative of curvature tensor is well-known:For a CC vector field , using (22)–(25), the following holds:where and are the scalar curvature and the conformal curvature tensor, respectively, andIn particular, it follows from the above third identity that if is conformally flat, that is, if vanishes, then vanishes. However, the converse does not hold, so, in general, following problem remains open.
Condition(s) on are found with a CC symmetry such that vanishes.
Equation (25) raises the question of finding possible values for the tensor which represents the change in , with respect to vector field . For this purpose, recall that any curvature tensor satisfies following identity: . Taking Lie-derivative of this identity with respect to , using (22) and (25), we state the following.

Proposition 2. A necessary condition for a vector field to be a CC vector is that the following curvature identity holds:

Since the above identity places no restriction on CC vector field , it is reasonable to use it for the Ricci curvature of CPF-hypersurface , satisfying some prescribed value of (21). In support of this, we cite following two references of conformally flat vanishes and nonconformally flat () , respectively:(1)Set , , and in general equation (21). Using this and (20) in (21), we get Katzin et al. [9] have proved that the necessary condition for a non-Einstein conformally flat manifold to admit a curvature collineation vector field is where and are functions on . Thus, for , the above equation represents our prescribed relation (29) for a non-Einstein conformally flat CPF-hypersurface of an almost metric CPF-manifold such that is also a CC vector field.(2)Here, we set , , and in general equation (21). Using this and (20) in (21), we get which is a subcase of CC known as affine collineation (AC) [10], where the Ricci tensor of is covariant constant . Using this symmetry, Grycak [11] has proved that a non-Einstein conformally recurrent manifold (i.e., , where is the conformal curvature tensor and is an exact recurrent form) which is neither conformally flat nor recurrent admits an AC vector field satisfying (31) for a non-Einstein conformally recurrent CPF-hypersurface of an almost metric CPF-manifold . On conformally recurrent manifolds, we refer [12].

Application 2. Here, we use the following conformal collineation symmetry introduced by Tashiro [13]:

Definition 1. An dimensional semi-Riemannian manifold admits a conformal collineation symmetry defined by a vector field ifwhere denotes Christoffel’s symbols, is a function and , and is called an Affine Conformal Vector (ACV) field of .

Proposition 3 (see [10]). A vector field on a semi-Riemannian manifold is an ACV if and only ifwhere is a covariant constant () symmetric tensor.

An ACV reduces to a conformal Killing vector, briefly denoted by CKV, if K is proportional to . Thus, an ACV deviates from a CKV field if there exists a second-order covariant constant symmetric tensor . On the existence of an ACV and specific restrictions on the ambient manifold , we recall that, in 1932, Eisenhart [14] proved “If a Riemannian manifold admits such a tensor , independent of , then is reducible.” This means that is a product manifold of the form (). In 1926, H. Levy [15] proved that “A second-order covariant constant nonsingular symmetric tensor in a space of constant curvature is proportional to the metric tensor.” Thus, a semi-Riemannian manifold of constant curvature admits no ACV other than a CKV. So, with proper ACV is restricted to manifolds with nonconstant curvature. Physically, for example, Minkowski, de-Sitter, or anti-de-Sitter spacetimes do not admit an ACV. In particular, an ACV is called an affine vector, briefly denoted by AV, if . For basic details on ACV, see Tashiro [13], Paterson [16], Duggal [10], Mason-Martens [17], and others referred therein. For detailed information on curvature, affine, and conformal collineations, we refer [18]. Following is a reference having a link of CPF-hypersurfaces with conformal collineation manifolds.

(3) We know that is Ricci symmetric if . Levine–Katzin [19] have proved that if a conformally flat manifold admits a conformal collineation symmetry, with CKV vector field , then for some constants and and is Ricci symmetric. Thus, for and , the ACV vector field satisfies the prescribed relation (33) for a Ricci symmetric CPF-hypersurface of an almost metric CPF-manifold . Based on the above three references, we state following Corollary of Theorem 2:

Corollary 1. Let be an almost metric CPF-hypersurface of a metric PCR-manifold , with constant and admits 2-form . Then, admits curvature or affine or conformal collineations symmetries for prescribed values of the general equation (21).

The picture views of this link with three curvature symmetries are as follows: where and denote CPF-hypersurface and conformally flat manifold, respectively, and where denotes conformally recurrent manifold. where denotes reducible conformally recurrent manifold.

Application 3. Here, we need the following brief on the concept of Ricci flow for Riemannian manifolds introduced by Hamilton [20] in 1982. Let be an -dimensional Riemannian manifold. The Ricci flow on is defined by the following evolving equation: where is a vector field on and is a constant. The vector field , satisfying equation (37), is called Ricci soliton vector and is called Ricci soliton (briefly RS) manifold which is said to be shrinking, steady, or expanding as is positive, zero, or negative, respectively. The Ricci soliton manifolds are natural extension of Einstein manifolds and are self-similar (homothetic) solutions, called Ricci solitons. Basic details and a collection of research papers on this area for the Riemannian case are available in [21, 22], respectively. There are few papers on Ricci solitons for semi-Riemannian (in particular, Lorentzian) manifolds (for example, see Crasmareanu [23], Brozos-Vzquez et al. [24], and Onda [25]). In year 2011, Pigola et al. [26] introduced a modified concept of the Ricci solitons equation called almost Ricci solitons (briefly, ARS) by allowing the soliton constant to be a variable function. So far, we know following references on ARS-manifolds: Barros et al. [27], Barros–Riberiro [28], Sharma [29], Wang [30], and Duggal [31]. Following is a link between the CC symmetry of CPF-hypersurfaces and ARS and RS evolving equation (37) as is a variable function or a constant.(4)Grycak [11] has proved that a non-Einstein conformally recurrent manifold (i.e., , where is the conformal curvature tensor and is an exact recurrent form) which is neither conformally flat nor recurrent admits a vector field satisfying (30) such thatAnd is a constant. This relates with the Ricci soliton equation (37) if we take so is a RS-manifolds. Although Grycak did not discuss a link of his result with a symmetry vector, but the above relation is a particular case of the general equation (21), it is reasonable to assume that this reference has a link with CC or AC symmetries. Based on the above, we state following Corollary of Theorem 2:

Corollary 2. Let be an almost metric CPF-hypersurface of a metric PCR-manifold , with constant and admits 2-form . Then, is also almost Ricci soliton or Ricci soliton for some prescribed values of the general equation (21) as is a variable function or a constant.

The picture views of this link with the almost Ricci soliton and Ricci soliton manifolds are as follows:where , , and denote CPF-hypersurface, almost Ricci soliton, and Ricci soliton manifolds, respectively.

Remark 3. Although we have some references with specific prescriptions for the unknown tensor linking the CPF-hypersurfaces with three types of curvature symmetries and almost Ricci soliton manifold, it is reasonable to assume that such a link may also hold for some other types of CPF-hypersurfaces of semi-Riemannian manifolds. Therefore, we leave it an open problem to research on deeper study of possible solutions of equation (21) linking the CPF-manifolds with a variety of other types of semi-Riemannian manifolds. Presently, three curvature symmetry manifolds and almost Ricci soliton manifolds are among most important topics of research in differential geometry so their link established with semi-Riemannian CPF-hypersurfaces is expected to produce substantial useful original results on geometry and physics of this new class of ACPF-manifolds with applications.

3. Lightlike ACPF-Manifolds

For this section, we need the following brief information on lightlike manifolds taken from the study by Duggal–Sahin [32] (pages 30–40). Let be an almost metric CPF-manifold which admits a 2-form . Suppose its PCR-distribution is lightlike having a degenerate induced metric . Let be the lightlike PCR-subspace of with respect to its distribution . This means that there exists a vector field say in such that . Let be the radical distribution of , with respect to . Then,where is nondegenerate complementary (but not orthogonal) screen distribution of in . Let . Then, and . In this way, is called an -dimensional -lightlike ACPF-manifold. In this section, we show that there do exist even- and odd-dimensional lightlike ACPF-manifolds with degenerate induced metric which admit a 2-form . Following is an even-dimensional example.

Example 3. Suppose is a basis for at a point of a -dimensional () metric manifold with a real tensor field of type and two real nonzero functions and on . LetThen, it is easy to see that . Therefore, we have a -dimensional PCR-distribution generated by and a complementary 1-dimensional null operator such that splits into a direct sum of two subbundles, namely, (with a PCR-structure) and . Let be the PCR-subspace of , with respect to its distribution D. Therefore, we can use the metric compatible equation (3) for the PCR-subspace of . Assume admits a 2-form defined by (8) so that . From , we get . This implies that so is a null vector. Let and for each , where each and are real nonzero functions on . Then, implies that , . Using this data and , we haveThus, is a 2n-dimensional 1-lightlike ACPF-manifold with 1-dimensional radical distribution and -dimensional screen distribution such thatwhere is nondegenerate complementary (but not orthogonal) screen distribution of in . also admits a 2-form .
Following is an example of odd-dimensional 2-lightlike ACPF-manifold.

Example 4. Suppose is a basis for of a -dimensional () metric manifold with a real tensor field of type and two real nonzero functions and on . LetThen, it is easy to see that . Therefore, we have a -dimensional PCR-distribution generated by and a complementary 1-dimensional null operator such that splits into a direct sum of two subbundles, namely, (with a PCR-structure) and . Let be the PCR-subspace of , with respect to its distribution D. Therefore, we use the metric compatible equation (3) for the PCR-subspace of and admits a 2-form defined by (8) so that . From , we get . This implies that so is a null vector. Similarly, so is also a null vector. Let and for each , where each and are real nonzero functions on . Then, implies , . Using this and , we haveThus, is a -dimensional 2-lightlike ACPF-manifold which admits a 2-dimensional radical distribution and -dimensional screen distribution such thatwhere is nondegenerate complementary (but not orthogonal) screen distribution of in . also admits a 2-form .
Finally, we present a physical model of 1-lightlike ACPF-hypersurfaces of a metric PCR-globally hyperbolic spacetime. A product manifold is called a globally hyperbolic spacetime manifold if is a compact Riemannian manifold and is time oriented by a timelike global vector field . Simple examples are Minkowski, de-Sitter, and Einstein static spacetimes. We refer [33] for details on physical importance of globally hyperbolic spacetimes.
Physical Model. Let be an -dimensional globally hyperbolic PCR-spacetime with its metric given bywith respect to a coordinate system on . Take two null coordinates and such that and . Thus, the above metric transforms into a nonsingular metric: . The absence of and in this metric implies that { constant.} and { constant.} are lightlike hypersurfaces of . Let ( constant.) be one of this lightlike pair with 1-dimensional distribution generated by the null vector {} in and -dimensional Riemannian screen distribution with metric . In particular, there will be many global timelike vector fields in globally hyperbolic spacetimes . If we take a fixed global time function, then its gradient is a global timelike vector field in a given . With this choice of a global timelike vector field, we conclude that corresponding lightlike hypersurface admits a global null vector field. Using the Hopf–Rinow theorem, one may choose a screen distribution whose leaf is a complete Riemannian hypersurface of . Thus, we have a physical model of a global 1-lightlike hypersurface of a globally hyperbolic PCR-spacetime manifold. Then, proceeding exactly as in Example 3, it is straight forward to show that ( constant.) is a global 1-lightlike APCF-hypersurface of a globally hyperbolic PCR-spacetime manifold .

Remark 4. Examples 3 and 4 show that there exist even- and odd-dimensional -lightlike ACPF-manifolds which admit a 2-form . Also, Examples 2–4 confirm that there do exist a new class of odd-dimensional semi-Riemannian and even- or odd-dimensional r-lightlike ACPF-manifolds, in particular, ACPF-hypersurfaces of a metric PCR-manifold which admit 2-form . However, we still do not have any example of even-dimensional semi-Riemannian ACPF-manifold with 2-form . We leave this as open problem and propose following for further consideration:(1)The results presented on ACPF-hypersurfaces in this paper can be extended for general theory of ACPF-submanifolds of a variety of metric PCR-manifolds by revising the definition of Riemannian contact CR-submanifolds [8] and contact CR-lightlike submanifolds (see chapter 7 in [32]).(2)Considerable work has been done on submanifolds of normal contact (also called Sasakian) manifolds (see [8, 34]). We propose an extension of their results for submanifolds of normal CPF-manifolds.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The author declares that there are no conflicts of interest regarding the publication of this article.

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Copyright © 2021 K. L. Duggal. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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