1. Introduction
The concept of warped products was proposed by Bishop and O’Neill [
1] to investigate Riemannian manifolds of negative sectional curvatures. Such products, which are generalizations of direct product Riemannian manifolds, play an important role in differential geometry and also have many applications in physics, particularly in the theory of relativity. In fact, many exact solutions to Einstein’s field equation can be expressed in terms of Lorentzian warped products [
2]. The last theory demands a larger class of manifolds, and then the idea of doubly warped products was introduced as a generalization of warped product manifolds. For two given pseudo-Riemannian manifolds,
and
, and for two positive smooth functions,
b and
f, on
B and
F, respectively, the doubly warped product
is the product manifold
equipped with the metric
. The smooth functions
b and
f are called warping functions,
B is called base manifold, and
F is called fiber manifold.
The concept of Poisson structure first appeared with Poisson [
3] in order to obtain new integrals of motions in Hamiltonian mechanics. Later, Lichnerowicz [
4] introduced the notion of the Poisson manifold as a smooth manifold equipped with a Poisson structure. In [
5], Vaisman introduced the idea of a contravariant derivative on the Poisson manifold. Then, Fernandes discussed many properties of the Poisson manifold with contravariant connections [
6]. It is worth noting that Poisson structures play an important role in Hamiltonian mechanics and also have interaction with the theory of relativity. Recently, [
7,
8] studied contravariant gravity on Poisson manifolds equipped with a Riemannian metric using Levi–Civita contravariant connections. The compatibility between a Poisson structure and a pseudo-Riemannian metric was investigated on different classes of smooth manifolds by many authors [
9,
10,
11,
12]. In [
13], Nasri and Mustapha studied some of the geometric properties of the product Riemannian Poisson manifold. After that, [
14] introduced the concept of the warped product Poisson manifold as a generalization of the product Riemannian Poisson manifold. The contravariant curvatures of the warped product Poisson manifold were studied in [
15].
Inspired by these studies, we introduce the notion of the doubly warped product Poisson manifold , associated with components and , where and are Poisson tensors on B and F, respectively, and we investigate some geometric structures on , such as the Levi–Civita contravariant connection, curvature tensor, sectional curvature, qualar, and null sectional curvatures. Some interesting consequences of the sectional curvature of a doubly warped product Poisson manifold are given.
This paper is organized as follows:
Section 2 represents some Poisson-tensor-, contravariant-connection-, and curvature-related formulas on a Poisson manifold equipped with a pseudo-Riemannian metric. In
Section 3, we derive the Levi–Civita contravariant connection, curvature tensor, and sectional contravariant curvature of a doubly warped product Poisson manifold
in terms of Levi–Civita connections, curvatures, and sectional curvatures of components
and
. Then, using warping functions
f,
b, and sectional curvatures of
and
, we investigate the sectional contravariant curvature of
. In
Section 4, we compute the qualar and null sectional contravariant curvatures of a degenerate plane at a point in
. In the final section, we give an example of a four-dimensional Lorentzian doubly warped product Poisson manifold, where Levi–Civita connection, qualar, and sectional curvatures are obtained.
2. Preliminaries
2.1. Poisson Structure
A Poisson bracket on a smooth manifold M is a Lie bracket on the space of a real-valued smooth function on
which satisfies the following Leibniz identity:
From the Leibniz identity, for any smooth function
on a Poisson manifold
, the map
is a derivation, and thus, there exists a unique vector field
on
M such that:
This vector
is called a Hamiltonian vector field of
and if
and the smooth function
is called a Casimir function on
M.
It also follows from the Leibniz identity that, for each Poisson structure
on
M, there exists a bivector field
called a Poisson tensor, on
M such that for any
A Poisson manifold
is a smooth manifold
M equipped with a Poisson tensor
.
2.2. Contravariant Connection and Curvature Tensor
The notion of contravariant connection defined on the Poisson manifold is similar to the notion of usual covariant connection. This connection was introduced by Vaismann in [
5] and analyzed in detail by Fernandes [
6].
Let
be a Poisson manifold. For each Poisson tensor
we associate the anchor map
defined by
and the Koszul bracket
defined on the space of differential 1-forms
by,
A contravariant connection
on
M with respect to the Poisson tensor
is a
bilinear map
satisfying the following properties:
- (i)
The mapping
is
-linear, i.e.,
- (ii)
The mapping
is a derivation in the following sense:
The torsion
and the curvature
of a contravariant connection
are
- and
-type tensor fields, defined, respectively, by:
where
When
(respectively,
,) the connection
is called torsion-free (respectively, flat).
Now, let
be a smooth manifold equipped with a covariant pseudo-Riemannian metric
Using the metric
, we can define the isomorphisms,
and its inverse
called musical isomorphisms. The contravariant metric
g associated with
is defined for any 1-forms
by
For a Poisson tensor
on
M, there exists a unique contravariant connection
associated with
, called the Levi–Civita contravariant connection, such that the metric
g is parallel with respect to
i.e.,
and that
is torsion-free, i.e.,
The contravariant connection
is the analog of the usual covariant Levi–Civita connection and can be expressed by the Koszul formula:
For any smooth function
we can write
, and for any 1-form
we have:
The relation between the Poisson tensor
and the metric
g is defined by the field endomorphism
as follows:
The contravariant Laplacian operator
of any tensor field
T on an
dimensional manifold
M, associated with the Levi–Civita connection
, is defined by [
16]:
where
is a local
orthonormal coframe field of
M.
For any smooth function
on
M, the contravariant Hessian
of
associated with
is given by [
14]:
From Equations (
4) and (
5), for any
, the contravariant Laplacian of
is given by:
If
is the curvature of the connection
and
is a non-degenerate plane at the point
spanned by two non-parallel cotangent vectors
and
, the sectional contravariant curvature of
is the number given by:
where
Note that, the connection
is flat if and only if the sectional contravariant curvature is identically zero.
2.3. Qualar and Null Sectional Contravariant Curvatures
The sectional covariant curvature function is defined only on non-degenerate planes. Then, a new type of curvature is needed for degenerate planes. Therefore, Harris [
17] proposed what is called the null sectional curvature of a degenerate plane in order to study Lorentzian manifold. The geometrical interpretation of this curvature was studied by Albujer and Haesen in [
18]. Some relations between qualar and null sectional curvatures are given by Gülbahar in [
19].
By analogy with the covariant case, we define the qualar and null sectional contravariant curvatures on a Poisson manifold equipped with a pseudo-Riemannian metric.
Firstly, we need to define the notions of timelike, null, and spacelike one forms. Using Equation (
2), a one form
is said to be:
- (i)
Timelike, if
- (ii)
Null, if and
- (iii)
Spacelike, if
Let
be an m-dimensional Poisson manifold equipped with a pseudo-Riemannian metric
g of index
s, and let
be a local
orthonormal basis, such that:
The qualar contravariant curvature at a point
is defined by
Using timelike and spacelike one forms, we can define null one forms as follows:
where
and
.
For any
the plane spanned by
and
is a degenerate plane, and its null sectional contravariant curvature
is defined by:
where
.
2.4. Vertical and Horizontal Lifts
In this subsection, we recall the notions of vertical and horizontal lifts of tensor fields on the product manifold (see [
14,
20]).
Let B and F be two smooth manifolds. We denote by and the spaces of vector fields on B and F, respectively, and by and the first and the second projection of on B and F, respectively.
Let be a smooth function on The vertical lift of f to is the smooth function on
Let
and
For any
the vertical lift of
to
is the unique tangent vector field
in
such that
We can define similarly the horizontal lift
of a function
and the horizontal lift
of a vector field
on
B to
by using the first projection
Now, let
be a smooth 1-form on
F; then, its pullback
by the second projection
, is a smooth 1-form
on
called the vertical lift of
to
such that for any
, we have,
Similarly, we can define the horizontal lift
of a smooth 1-form
by using the first projection
Lemma 1 ([
21]).
For any smooth functions , and for any vector fields and , we have: 2.5. Doubly Warped Product Poisson Manifold
Let
and
be Poisson tensors on
B and
F, respectively. The product Poisson structure on the product manifold
is the unique Poisson structure
such that for any
and
, we have [
14]:
Proposition 1 ([
14]).
Let , , and . Then, we have:- 1.
,
- 2.
- 3.
Now, let
and
be pseudo-Riemannian metrics on
B and
F, respectively. The doubly warped product
is the product manifold
equipped with the metric,
where
and
are smooth positive functions on
F and
B, respectively, called warping functions.
The doubly warped metric is defined explicitly, for any
and
, by:
The contravariant pseudo-Riemannian metric
g associated with
is given explicitly, for any
and
, by [
20]:
Definition 1. Let and be two pseudo-Riemannian manifolds equipped with Poisson tensor and , respectively, and let b and f be two smooth positive functions on B and F, respectively. A doubly warped product manifold equipped with the product Poisson structure will be called doubly warped product Poisson manifold of and .
3. Sectional Curvature of Doubly Warped Product Poisson Manifolds
In this section, we calculate the Levi–Civita connection, curvature tensor and sectional contravariant curvature of a doubly warped product Poisson manifold . Then, using sectional contravariant curvatures and warping functions of components and , we discuss the sectional curvature of .
Proposition 2. Let and be the Levi–Civita contravariant connections associated, respectively, with , and . Then, for any and , we have:
- 1.
- 2.
- 3.
Proof. 1.
Using Lemma 1, Proposition 1, Equation (
9) and taking
in the Koszul Formula (
3), we obtain:
Similarly, taking
in Equation (
3), we obtain:
Therefore,
- 2
Using Lemma 1, Proposition 1, Equation (
9) and taking
in (
3), we obtain:
Similarly, taking
in (
3), we obtain:
Then, the first part of 2 follows.
Moreover, since is torsion-free and then .
- 3
This is analogous to the proofs of 1. □
Lemma 2. Let , and be the curvatures tensors of , and , respectively. Then, for any and , we have:
Proof. 1.
Taking
and
in Equation (
1), we obtain,
Using Lemma 1 and Proposition 2 in the first term
of (
10), we obtain:
Interchanging
and
in the previous equation, the second term
of (
10) is given by:
Using Proposition 2, the third term
of (
10) is given by:
Using the above terms in Equation (
10), we obtain:
and the first part of the lemma follows.
- 2.
Taking
and
in (
1), we obtain,
Applying Proposition 2 in the first term
of (
11), we obtain:
Interchanging
and
in the previous equation, the second term
of (
11) is given by:
Applying Proposition 2 in the third term
of (
11), we obtain:
Replacing the above terms in (
11) and after some calculations the result follows.
- 3
Taking
and
in (
1), we obtain,
Applying Proposition 2 in the first term
of (
12), we obtain:
Applying Proposition 2 in the second term
of (
12), we obtain:
Using the above terms in (
12), we obtain:
and the third part of the lemma follows.
The proofs of Parts , and 6 are analogous to Proofs and 1, respectively.
□
Theorem 1. Let , and be the sectional contravariant curvatures of , and , respectively. Then, for any and , we have:
- 1.
- 2.
- 3.
Proof. 1.
Using Lemma 2 and taking
in Equation (
6), we obtain:
and
Hence,
- 2
Similarly, using Lemma 2 and taking
in (
6), we obtain:
and
Hence,
- 3.
This is analogous to the proofs of 1. □
Corollary 1. If and are flat then, is flat if and only if b and f are Casimir functions.
Proof. First, note that
b (respectively,
f) is a Casimir function if and only if
(respectively,
). Now, if
and
are flat, then from Theorem 1, the sectional curvature of
becomes:
⇒ Assume that
is flat. Using the first and third equations of (
13), we obtain:
Thus
b and
f are Casimir functions.
⇐ Assume that
b and
f are Casimir functions. Taking
in Equation (
13), we obtain
, and then
is flat. □
Corollary 2. If and are Riemannian manifolds, then is flat if and only if , are flat and b, f are Casimir functions.
Proof. If
is flat, using the fact that
B and
F are Riemannian manifolds in Part 2 of Theorem 1, we obtain:
Then,
Now, using Equation (
14) in Parts 1 and 3 of Theorem 1, we obtain:
Hence, from (
14) and (
15), we deduce that
b and
f are Casimir functions and
and
are flat.
Converse of this corollary directly follows from Corollary 1. □
Corollary 3. If has a positive sectional curvature, then and have positive sectional curvatures.
Proof. If
then from Theorem 1, for any
and
, we have:
and the corollary follows. □
Corollary 4. If b and f are Casimir functions on B and F, respectively, then the triplet has non-negative (respectively, non-positive) sectional curvature if and only if and have non-negative (respectively, non-positive) sectional curvatures.
Proof. If
f and
g are Casimir functions, then from Theorem 1, the sectional contravariant curvature
of
becomes:
and the corollary follows. □
Corollary 5. If and are Riemannian manifolds, then has non-negative sectional curvature if and only if , have non-negative sectional curvatures and b, f are Casimir functions.
Proof. If
,
B and
F are Riemannian manifolds, then from Part 2 of Theorem 1, we obtain:
Using Equation (
17) in Parts 1 and 3 of Theorem 1, for any
and
, we obtain:
Hence, from (
17) and (
18), we deduce that
b and
f are Casimir functions and
,
.
Converse of this corollary follows from Corollary 4. □
Corollary 6. If has a nonzero constant sectional curvature c and f is a Casimir function on F then, both and have constant sectional curvatures given, respectively, by:Furthermore, if b is a Casimir function, then Proof. By Part 1 of Theorem 1, if
f is a Caismir function and
, then for any 1-forms
, we have
. Hence,
By Part 3 of Theorem 1 and Equation (
19), for any
, we have
, hence
□
4. Qualar and Null Sectional Contravariant Curvatures of Doubly Warped Product Poisson Manifold
Let be a local orthonormal basis on a doubly warped product Poisson manifold , where is a orthonormal basis on B of index and is a orthonormal basis on F of index .
From Equations (
7) and (
8), the qualar contravariant curvature at point
and the null sectional contravariant curvature of
are defined, respectively, by:
and
where
,
and
such that:
Theorem 2. Let be a doubly warped product Poisson manifold of and , and let π be a degenerate plane spanned by a null 1-form and a unit 1-form . Then, the null sectional contravariant curvature of π is given by:
- 1.
for any , and where . - 2.
for any , and where . - 3.
for any , and - 4.
for any , and
Proof. Using Lemma 2, Theorem 1 and Equation (
21), we obtain this theorem. For example, consider Part 2 of the theorem for all
. We obtain:
and
In the same way, we can prove the remaining parts of the theorem. □
Theorem 3. Let be a doubly warped product Poisson manifold of and . Then, the qualar curvature at a point is given by:
Proof. Using Theorem 1 and Equation (
20), for any
, we obtain:
and the theorem follows. □
Corollary 7. Let be a doubly warped product manifold such that the fiber F is a Riemannian manifold. Then, the qualar curvature at point becomes: Proof. Taking in the Theorem 3, the corollary follows directly. □
5. Example
Let
be a pseudo-Riemannian manifold equipped with a Poisson tensor
, and let
be the Levi–Civita connection associated with
. If
is a local coordinates on a neighborhood
in
M, then we can define the Christoffel symbols
by [
6]:
Since
and
, then from (
3) and (
23), we have:
Multiplying both sides of the previous equation by
we obtain:
where
gkn is the inverse matrix of
gkn.
Now, we take
with the Lorentzian metric
, where
is an open interval and
with the metric
, and let
and
be positive smooth functions on
and
, respectively. Then, the product manifold
equipped with the following metric:
is a 4-dimensional Lorentzian doubly warped product manifold, where
,
and
are the coordinates on
,
, and
, respectively. Furthermore, if
g is the contravariant metric of
, then its local components are given by:
Let
and
be Poisson tensors on
B and
F, respectively, then
is a Poisson tensor on
and
is a 4-dimensional Lorentzian doubly warped product Poisson manifold.
Using (
24), the Christoffel symbols of
and
are given, respectively, by:
and
Hence, from (
23) and above equations of Christoffel symbols, we obtain:
and
Using Equation (
6), the sectional contravariant curvatures of the non-degenerated planes
and
are given, respectively, by:
Using the fact that
and
are, respectively,
- and
-orthonormal bases, we obtain the following identities:
Therefore, according to Theorem 1, the sectional curvature of the non-degenerate plane
is given by:
Using the above equations in (
22), the qualar curvature of
at point
is given by:
Remark 1. If and , then Equations (24) and (25) reduced to andFurthermore, if b and f are positive constants with , then 6. Conclusions
This research paper delved into the investigation of the sectional contravariant curvature of doubly warped product manifolds equipped with a product Poisson structure, which we called doubly warped product Poisson manifolds. Through our comprehensive investigation, we have uncovered several important results and made significant contributions to the understanding of some geometric structures, such as the Levi–Civita contravariant connection, curvature tensor, sectional curvature, qualar, and null sectional curvatures of this class of manifolds.
One of the main achievements of this paper is the establishment of various key findings regarding the properties of sectional contravariant curvature of doubly warped product Poisson manifolds. After expressing the Levi–Civita contravariant connection and the curvature tensor, we have established a relationship between the sectional contravariant curvature of a doubly warped product Poisson manifold and those of its factor manifolds. Using this relationship, we have explored some of its geometric properties, such as the determination of a necessary and sufficient condition for a doubly warped product Poisson manifold to be flat and the discussion of its sign in terms of signs of sectional curvatures of basic manifolds.
Additionally, we have calculated the qualar and null sectional contravariant curvatures of this class of manifolds. Moreover, we enhanced our results by providing an example of a four-dimensional Lorentzian doubly warped product Poisson manifold, in which Levi–Civita connection, qualar, and sectional curvatures are obtained.
The findings presented in this paper have implications for various areas of differential geometry and mathematical physics, contributing to the understanding of warped product manifolds, which are crucial for theoretical physics, especially in the theory of relativity. It fills a specific gap by extending the study of sectional contravariant curvatures to doubly warped product Poisson manifolds.
In conclusion, this paper expands our knowledge about some geometric structures on doubly warped product Poisson manifolds. The insights gained from this research, along with the expressions of the Levi–Civita contravariant connection and the curvature tensor, provide a solid foundation for further investigations and applications in related fields. In future research, we will explore how our results can be used to study the Ricci and scalar curvatures, Einstein’s manifolds, contravariant Einstein’s equation, and the cosmological constant on this class of manifolds.
Author Contributions
Conceptualization, methodology, software, validation, formal analysis, F.A., S.K.H. and I.A.-D.; investigation, resources, data curation, S.K.H.; writing—original draft preparation, writing—review and editing, F.A.; visualization, I.A.-D.; supervision, project administration, funding acquisition, F.A. All authors have read and agreed to the published version of the manuscript.
Funding
This work was supported and funded by the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University (IMSIU) (Grant Number IMSIU-RP23074).
Data Availability Statement
No new data were created or analyzed in this study. Data sharing is not applicable to this article.
Conflicts of Interest
The authors declare no conflict of interest.
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