Certain Class of Almost -Cosymplectic Manifolds
This paper concerned with almost -cosymplectic manifolds satisfying conformally flat condition. Firstly, we investigate Kaehler integral submanifolds of almost -cosymplectic manifolds. Next, we study conformally flat almost -cosymplectic manifolds of dim 5 whose integral submanifolds are Kaehler. Finally, an illustrative example is constructed to verify our result.
The notion of conformal flatness is one of the most primitive concepts in differential geometry. Notwithstanding this fact, most of the studies have been local character. However, Kulkarni classified conformally flat manifolds up to conformal equivalence .
On a Riemannian manifold , Weyl established a tensor of type (1, 3) which vanishes whenever the metric is (locally) conformally equivalent to a flat metric. Therefore, this tensor is called the conformal curvature tensor of the metric and defined byfor any vector fields on . Here, we denote by and the Riemann curvature tensor and scalar curvature of , respectively [2, 3]. A necessary condition being conformally flat for a Riemannian manifold is the vanishing of the Weyl curvature tensor. It is obvious that the Weyl tensor vanishes identically in 2 dimensions. In general, it is nonzero in dimensions 4. The metric is locally conformally flat provided that the Weyl tensor vanishes for 4 dimensions. In this case, the metric has a local coordinate system where it is proportional to a constant tensor. In dimension 3, we have
Equation (2) is a necessary and sufficient condition for three-dimensional Riemannian manifold being conformally flat. Here, is the divergence operator of .
A (2n + 1)-dimensional Riemannian manifold is conformally flat if and only if the Weyl conformal curvature tensor vanishes for any vector fields when or the tensor of type (1, 1) defined asfor any vector field , is of Codazzi type when . Here, is the Ricci operator associated with the Ricci tensor and is the scalar curvature .
In contact metric manifolds, Okumura obtained that a conformally flat Sasakian manifold of dimension 3 is locally isometric to the unit sphere . Later, this result was extended to the -contact manifolds by Tanno for dimension 3 Gosh and Sharma showed that a conformally flat contact strongly pseudo-convex integrable CR manifold is locally isometric to a unit sphere if the characteristic vector field is an eigenvector of the Ricci tensor at each point . Afterwards, Gosh et al. obtained that a conformally flat contact strongly pseudo-convex integrable CR manifold of dimension 3 is of constant curvature 1 .
Moreover, if an almost cosymplectic manifold with dimension (2n + 1) is conformally flat , then it is locally flat and cosymplectic . Conversely to this, there exist conformally flat almost cosymplectic manifolds with Kaehler leaves which are not locally flat and not cosymplectic in dimension 3 .
Recently, Blair et al. have focused on almost contact metric manifolds with conformally flat condition . The authors construct an illustrative example of 3-dimensional conformally flat almost -Kenmotsu manifold whose sectional curvature is nonconstant. Furthermore, they consider conformally flat almost contact metric manifolds which are -Einstein manifold with dim 5. Moreover, Wang investigated conformally flat almost Kenmotsu manifold of k-dimension 3 . After these studies, we also point out that Weyl conformal curvature tensor has been studied extensively by Venkatesha et al. .
In this paper, we study the geometry of conformally flat almost -cosymplectic manifolds. We aim at characterizing and classifying conformally flat almost -cosymplectic manifolds. Then, we obtain the curvature properties of almost -cosymplectic manifolds with Kaehler integral submanifolds and investigate almost -cosymplectic manifolds of dim 5 which are Kaehler integral submanifolds. Finally, we give a concrete example of 3-dimensional almost -Kenmotsu manifolds.
Let be a -dimensional smooth manifold endowed with a triple where is a type of (1, 1) tensor field, is a vector field, and is a 1-form on such that
If admits a Riemannian metric , defined bythen is called almost contact structure . Also, the fundamental 2-form of is defined by . If the Nijenhuis tensor vanishes, defined bythen is said to be normal . It is obvious that a normal almost Kenmotsu manifold is said to be Kenmotsu manifold. In other words, an almost contact metric manifold is known as Kenmotsu if and only if . An almost contact metric structure is cosymplectic if and only if and are closed .
In the light of the above definitions, the generalization of almost Kenmotsu manifold is called almost -Kenmotsu manifold if and , where is a nonzero real constant [13, 14]. If we combine almost -Kenmotsu and almost cosymplectic manifolds, then we introduce a new notion of an almost -cosymplectic manifold defined by and for any real number . A normal almost -cosymplectic manifold is said to be -cosymplectic manifold, and it is either cosymplectic or -Kenmotsu under the condition or , respectively [16–19].
Let be an almost -cosymplectic manifold endowed with . is the contact distribution of given by . Since , is integrable and the -dimensional distribution is defined by . Moreover, it is clear that is orthogonal to Assume that is a maximal integral submanifold of Therefore, the restricted to integral submanifold is the normal vector of Thus, there exists a Hermitian structure and the tensor field induces an almost complex structure defined by for any vector field tangent to [12, 13, 20].
Suppose that is the Riemannian metric induced on defined by . Then, has an almost Hermitian structure on given by for any vector field and tangent to The fundamental 2-form of induced on This means that , i.e., is the pull-back of the tensor field from to Since is closed, we obtain . Thus, the pair is an almost Kaehler structure on of When the structure is complex, becomes a Kaehler structure on If the structure is Kaehler on every integral submanifolds of the distribution this manifold is said to be an almost -cosymplectic manifold with Kaehler integral submanifolds.
Denote by and h the (1, 1) tensor fields on defined byrespectively. Here, is the Lie derivative of Obviously, and . Thus, we the following relations for any vector fields on :
3. Curvature Properties
This section deals with the fundamental curvature equations of almost -cosymplectic manifolds with Kaehler integral submanifolds. Let us give the basic propositions that we will use in later usage. The proof of some propositions are left to the reader for shortness.
Proposition 1. Let be an almost -cosymplectic manifold. Then, we havewhere is the Jacobi operator with respect to By direct computations, we have the following proposition.
Proposition 2. An almost -cosymplectic manifold with Kaehler integral submanifolds holds the following equation:
Remark 1. The Ricci operator does not have to commute with the basic collineation for a contact metric manifold. Now, we give this condition for almost -cosymplectic manifold with Kaehler integral submanifolds.
Proposition 3. Let be an almost -cosymplectic manifold with Kaehler integral submanifolds. The following identity is held:
Proof. Using (5) and (19), one obtainsPutting and in (19), we haveSubstituting of (21) into (22) yields immediatelyBy the help of (5), (23) can be written asContracting with respect to and , we obtainThe rest of the proof follows acting on the last equation.
Definition 1. and are the Ricci and -Ricci operators of defined byand denote by and the scalar and -scalar curvatures of , whererespectively .
Proposition 4. For the Ricci and -Ricci operators of almost -cosymplectic manifold with Kaehler integral submanifolds , we havewhere is an orthonormal frame at any point of .
Proof. Taking trace of (19) with respect to yieldsBy using the properties of in (29), this ends the proof.
Proposition 5. For the scalar and -scalar curvatures of almost -cosymplectic manifold with Kaehler integral submanifolds , we obtain
Proof. (30) is a direct consequence of (29) by means of (10) and the following equation:
Proposition 6. For the Ricci operator of almost -cosymplectic manifold with Kaehler integral submanifolds , we have
Proof. By using the projection of (28) onto ξ and , the proof is obvious.
Proposition 7. An almost -cosymplectic manifold with Kaehler integral submanifolds satisfies the following equation:
Proof. From (13), we getThe proof is clear from the right side of equation (34).
Proposition 8. Let be an almost -Kenmotsu manifold . Then, it has Kaehler integral submanifolds of the distribution if and only if it holdswhere
Proof. By using similar technique in , the proof is clear.
Proposition 9. Let be an almost -Kenmotsu manifold . If is conformally flat, then is a space of constant negative curvature
Proof. By the help of [11, 15], we can complete the proof.
4. Conformal Flatness Condition
This section is devoted to study conformally flat almost -cosymplectic manifolds whose integral submanifolds are Kaehler.
Theorem 1. If is a conformally flat almost -cosymplectic manifold with Kaehler integral submanifolds , then is the eigenvector of Ricci operator on
Proof. Substituting for in (19), we haveLet be an orthonormal basis of vector fields on . Taking in (36) for , then we getwhere . Since is conformally flat and is identically zero. So, we can writePutting , and in (38) and summing over we obtainFrom (37) and (39), we havewhere and Following from (40), we can complete the proof.
Theorem 2. If is a conformally flat almost -cosymplectic manifold with Kaehler integral submanifolds , then we have
Proof. By the hypothesis, we haveIt follows thatThen, making use of (32) and (43), we getSince from (30) and (44), we haveTaking into account (32) and (42), we deduceOn the other hand, using (28) and (30) in (46), we haveFrom (46) and (47), it follows thatTaking into account (45) and (48), one obtainsFinally, making use of (32) and (49) and Theorem 1, the proof ends.
Theorem 3. A conformally flat almost -cosymplectic manifold with Kaehler integral submanifolds satisfies the following equation:Proof. From (42), we havePutting in (13), we get
On the other hand, using (38) and (41), it follows that
The last two equalities lead to
In view of (28), (49), (51), and (54), one finds
Substituting for in (55) and using (41), we have
Finally, this proof ends using (55) and (56).
Theorem 4. Let be a conformally flat almost -cosymplectic manifold with Kaehler integral submanifolds . Then, the following relation is held:
Proof. The proof follows from (20) and Theorem 1 in .
Theorem 5. Let be an almost -cosymplectic manifold with Kaehler integral submanifolds . If is conformally flat, then is either locally flat and cosymplectic or is an -Kenmotsu manifold with constant negative curvature
Proof. Let be an almost -cosymplectic manifold with Kaehler integral submanifolds. Assume additionally that is conformally flat and At certain places of the main idea of the proof, we are inspired by the paper of Dacko and Olszak . Also, the almost cosymplectic case is clear by means of Theorem 1 in .
Now, we shall prove the assertion of our theorem in case of Since (50) holds. Thus, the auxiliary tensor has the following shape:From (50) and (58), can be written asIn view of (7), (10), and (59) with , we obtainwhere So, the last equation reduces toReplacing and by in (38) and using (50), we find thatThen, using (52) and (62), we haveFollowing from (61) and (63), we getAlso, using (18) and (64), one obtainswhich reduces to Finally, we can use Proposition 9 to complete the proof.
Remark 2. It is noted that Theorem 5 generalizes the result of Dacko and Olszak .
Example. Considering such that are the standard coordinates in the vector fields are where are given by with for constants , and Also, the set of is linearly independent at each point of . Let be the Riemannian tensor product given byLet be the 1-form defined by and be the (1, 1) tensor field defined byFurthermore, we can calculateThus, we can check the only nonzero components of . Namely, we getSince , it implies that on . Moreover, Nijenhuis torsion tensor of vanishes.
For the curvature operator , the nonzero components are as follows:Clearly, is not locally flat.
For the Ricci tensor , assume ; then, we obtain and Consequently, the Ricci operator satisfies the equationswith . For the auxiliary operator of , we havewith where is defined by .
To obtain the conformal flatness of , it remains to verify the Codazzi condition for Namely,for It is seen that the Codazzi condition does not hold. Thus, the manifold is not conformally flat and has constant sectional curvature
5. Conclusion and Discussion
A Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation [2, 3]. There exist conformally flat contact metric manifolds which are not constant curvature . However, this is an open problem in dimensions 5. In recent years, some authors have studied this area for almost contact metric manifolds [4, 8, 9].
This paper deals with the conformally flat almost -cosymplectic manifolds given by Kaehler integral submanifolds. Our main target is to make some generalizations and classifications on such manifolds, and certain results are proved in the last two sections. .
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this study.
This work was supported by Afyon Kocatepe University Scientific Research Coordination Unit with project no. 18.KARİYER.37.
R. S. Kulkarni, “Conformally flat manifolds,” Proceedings of the National Academy of Sciences, vol. 69, no. 9, pp. 2675-2676, 1972.View at: Publisher Site | Google Scholar
D. E. Blair, “Riemann geometry of contact and symplectic manifolds,” Progress in Mathematics, vol. 203, 2002.View at: Google Scholar
K. Yano and M. Kon, Structures on Manifolds, Series in Pure Mathematics, World Scientific Publishing Co., Singapore, 1984.
Y. Wang, “Conformally flat almost Kenmotsu 3-manifolds,” Mediterranean Journal of Maths, vol. 14, pp. 2759–2770, 2017.View at: Publisher Site | Google Scholar
M. Okumura, “Some remarks on space with a certain contact structure,” Tohoku Mathematical Journal, vol. 14, no. 2, pp. 135–145, 1962.View at: Publisher Site | Google Scholar
A. Ghosh and R. Sharma, “On contact strongly pseudo-convex integrableCR manifolds,” Journal of Geometry, vol. 66, no. 1-2, pp. 116–122, 1999.View at: Publisher Site | Google Scholar
A. Ghosh, T. Koufogiorgos, and R. Sharma, “Conformally flat contact metric manifolds,” Journal of Geometry, vol. 70, no. 1, pp. 66–76, 2001.View at: Publisher Site | Google Scholar
P. Dacko and Z. Olszak, “On conformally flat almost cosymplectic manifolds with Keahlerian leaves,” Rendiconti del Seminario Matematico Università e Politecnico di Torino, vol. 56, no. 1, pp. 89–103, 1998.View at: Google Scholar
D. E. Blair and H. Yıldırım, “On conformally flat almost contact metric manifolds,” Mediterranean Journal of Mathematics, vol. 13, no. 5, pp. 2759–2770, 2016.View at: Publisher Site | Google Scholar
V. Venkatesha, D. M Naik, and H. A. Kumara, “Conformal curvature tensor on paracontact metric manifolds,” Matematicki Vesnik, vol. 73, no. 3, pp. 215–225, 2020.View at: Google Scholar
K. Kenmotsu, “A class of contact Riemannian manifold,” Tohoku Mathematical Journal, vol. 24, pp. 93–103, 1972.View at: Publisher Site | Google Scholar
Z. Olszak, “Locally conformal almost cosymplectic manifolds,” Colloquium Mathematicum, vol. 57, no. 1, pp. 73–87, 1989.View at: Publisher Site | Google Scholar
D. Janssens and L. Vanhecke, “Almost contact structures and curvature tensors,” Kodai Mathematical Journal, vol. 4, pp. 1–27, 1981.View at: Publisher Site | Google Scholar
I. Vaisman, “Conformal changes of almost contact metric structures,” Lecture Notes in Mathematics, vol. 792, pp. 435–443, 1980.View at: Publisher Site | Google Scholar
T. W. Kim and H. K. Pak, “Canonical foliations of certain classes of almost contact metric structures,” Acta Mathematica Sinica, English Series, vol. 21, no. 4, pp. 841–846, 2005.View at: Publisher Site | Google Scholar
G. Dileo and A. M. Pastore, “Almost Kenmotsu manifolds and local symmetry,” Bulletin of the Belgian Mathematical Society Simon Stevin, vol. 14, pp. 343–354, 2007.View at: Publisher Site | Google Scholar
M. Falcitelli and M. Pastore, “Almost Kenmotsu f-manifolds,” Balkan Journal of Geometry and Its Applications, vol. 1, no. 12, pp. 32–43, 2007.View at: Google Scholar
D. M Naik, V. Venkatesha, and H. A. Kumara, “Some results on almost Kenmotsu manifolds,” Note Math, vol. 40, pp. 87–100, 2020.View at: Google Scholar
V. Venkatesha, D. M. Naik, and A. T. Vanli, “Second order parallel tensor on almost Kenmotsu manifolds,” Kyungpook Mathematical Journal, vol. 61, pp. 191–203, 2021.View at: Google Scholar
D. Chinea and C. Gonzalez, “A classification of almost contact metric manifolds,” Annali di Matematica pura ed applicata, vol. 156, no. 4, pp. 15–36, 1990.View at: Publisher Site | Google Scholar