Critical point equation on almost f -cosymplectic manifolds

Purpose – Besse first conjectured that the solution of the critical point equation (CPE) must be Einstein. The CPE conjecture on some other types of Riemannian manifolds, for instance, odd-dimensional Riemannian manifolds has considered by many geometers. Hence, it deserves special attention to consider the CPE on a certain class of almost contact metric manifolds. In this direction, the authors considered CPE on almost f -cosymplectic manifolds. Design/methodology/approach – The paper opted the tensor calculus on manifolds to find the solution of the CPE. Findings – In this paper, in particular, the authors obtained that a connected f -cosymplectic manifold satisfying CPE with \lambda 5 \tilde{f} is Einstein. Next, the authors find that a three dimensional almost f -cosymplectic manifold satisfying the CPE is either Einstein or its scalar curvature vanishes identically if its Ricci tensor is pseudo anti-commuting. Originality/value – The paper proved that the CPE conjecture is true for almost f -cosymplectic manifolds.


Introduction
One of the natural ways of finding canonical Riemannian metric, that is, Riemannian metrics with constant curvature in various form on a smooth manifold is to look for metrics which are critical points of a natural functional on the space of all metrics on a given manifold. In this context, it is very interesting to investigate the critical points of total scalar curvature functional S : M → R given by SðgÞ ¼ Z M r g dv g ; (1.1) Let C ⊂ M be the subset of metrics with constant scalar curvature. If we consider the functional in Eqn (1.1) restricted to C, then it is not difficult to see that the Euler-Lagrangian equation is given by, Hess g λ À ðΔ g λÞ À λRic g ¼ Ric g À r n g; (1.2) for some smooth function λ on M. Here Hess, Δ g , Ric and r stands for the Hessian form, the Laplacian, the Ricci tensor and the scalar curvature on M, respectively. Moreover, taking trace in Eqn (1.2), we obtain Δ g λ þ rλ n À 1 ¼ 0: We notice that if λ is constant in Eqn (1.2), then λ 5 0 and g becomes Einstein. Therefore, we have the following definition: Besse first conjectured that the solution of the CPE must be Einstein [1]. Since then, we find many articles regarding the solution of the CPE. In [2], Barros and Ribeiro proved that the CPE conjecture is true under the assumption of half conformally flat spaces. Recently, Hwang [3] proved that the CPE conjecture is also true under certain condition on the bounds of the potential function λ. A necessary and sufficient condition for the norm of the gradient of the potential function for a CPE metric to be the Einstein metric was obtained by Neto [4]. It is very interesting to consider the CPE on odd-dimensional Riemannian manifolds. In this direction, Ghosh and Patra considered the K-contact metrics that satisfy the CPE [5], and proved that the CPE conjecture is true for this class of metric. Patra et al. in [6], and De and Mandal in [7] independently considered an almost Kenmotsu manifold with CPE. Recently, present authors in [8], and Blaga and Dey in [9] studied CPE on cosymplectic manifold and three dimensional α-cosymplectic manifold, respectively.
As the generalization of almost Kenmotsu and almost cosymplectic manifolds, the results obtained in [6][7][8][9] motivates us to consider almost f-cosymplectic manifolds. In this paper, we classify an almost f-cosymplectic manifold which satisfies CPE.
On such a manifold, the fundamental two-form Φ of M is defined by for any vector field X and Y on M. One can define an almost complex structure J on M 3 R by where t is the coordinate of R and u is a smooth function. If the aforesaid structure J is integrable, then we call an almost contact structure as normal, and this is equivalent to require where [w, w] indicates the Nijenhuis tensor of w. An almost contact metric manifold M is said to be almost cosymplectic if dη 5 0 and dΦ 5 0, where d is the exterior differential operator, and it is said to be cosymplectic if in addition the almost contact structure is normal. An almost α-Kenmotsu manifold is an almost contact metric manifold, in which dη 5 0 and dΦ 5 2αη ∧ Φ, for a nonzero constant α. More generally, if the constant α is any real number, then almost contact structure is said to be almost α-cosymplectic [10]. Moreover, the authors in [11] generalizes the almost α-cosymplectic manifold by allowing the real number α to any smooth function f, and it is called as an almost f-cosymplectic manifold, which is an almost contact metric manifold M such that dΦ 5 2fη ∧ Φ and dη 5 0 for a smooth function f satisfying df ∧ η 5 0. In addition, a normal almost f-cosymplectic manifold is said to be f-cosymplectic manifold. In particular, M is an almost cosymplectic manifold under the condition f(constant) 5 0 and an almost α-Kenmotsu manifold if (α 5 f ≠ 1).
Besides, we recall that there is an operator h ¼ 1 2 £ ξ w, which is a self-dual operator. We denote by R and Ric the Riemannian curvature tensor and Ricci tensor, respectively. For an almost f-cosymplectic manifold M, the following equations were proved [11]:

CPE on normal almost f-cosymplectic manifolds
In this section, we aim to study CPE on normal almost f-cosymplectic manifold. We are aware that if almost contact metric manifold is normal then h 5 0. Hence, as a result of Proposition 9 and Proposition 10 of [11] we have the following identities, which are valid on f-cosymplectic manifolds; CPE on almost f-cosymplectic manifolds RðX ; Y Þξ ¼f fηðX ÞY À ηðY ÞX g; where Q is the Ricci operator of M. Now, we will give some properties, which will be used in the proof of our results.
Proof. Differentiation of Eqn (3.2), and utilization of first term of Eqn (3.1) provides Eqn (3.4). Now differentiating Eqn (3.3) along Z leads to Taking X 5 Z 5 E i in the above equation and then summing over i shows that One can easily deduce from second Bianchi identity that X Feeding Eqn (3.7) into Eqn (3.6) and with the help of Eqn (3.4), we obtain which proves Eqn (3.5). , be a nontrivial solution of the CPE (Eqn 1.2) on n-dimensional Riemannian manifold M. Then the curvature tensor R can be expressed as In the following, we will consider an f-cosymplectic manifold M satisfying a CPE and assume that the function f satisfies ξðf Þ ¼ 0.
is a solution of the CPE (Eqn 1.2), then one of the following statement holds: (1) M is Einstein AJMS (2) M is locally the product of a K€ ahler manifold and an interval or unit circle S 1 .
Proof. Taking scalar product of Eqn (3.8) with ξ and making use of Eqns (3.2) and (3.3), we obtain Replacing X by wX and Y by ξ in above relation, we get According to Proposition 2.1 of Chen [12], it is know that if ðξf Þ ¼ 0, thenf is constant. So that, Eqn (3.9) implies ð2n þ 1Þf wDλ ¼ wDν: The scalar curvature r of g is constant (as (g, λ) is a solution of the CPE). For a (2n þ 1)- Plugging X 5 ξ in Eqn (3.8) and calling back Lemma 3.1, we aimed at obtaining Since ∇ ξ ξ 5 0 and (ξλ) 5 g(ξ, Dλ), taking into account ∇ X Dλ 5 (λ þ 1)QX þ νX, we deduce Next we assume r ≠ − 2nð2n þ 1Þf, then from Eqn (3.11) we have wDλ 5 0. Action of w on this equation gives Dλ 5 (ξλ)ξ. Differentiating this along X, calling back Eqn (3.1) furnishes On the other hand, from Eqn (1.2) we can easily find that CPE on almost f-cosymplectic manifolds Comparing aforementioned equation with Eqn (3.13), we get Taking X 5 ξ in the above equation and making use of Eqns (3.2) and (2.1), we obtain Taking trace of CPE (1.2), we obatin 2nΔλ 5 Àλr, and this together with Eqn (3.19) gives that r ¼ −2nð2n þ 1Þf, which is contradictory to our assumption. Hence f u 0, and so M is cosymplectic. According to Blair's [13] result, we can easily conclude that M is locally the product of a K€ ahler manifold and an interval or unit circle S 1 . This finishes the proof. , In particular, when dimension of M is three, due to Theorem 3.3 we have the following outcome: It is known that an α-cosymplectic manifold is actually an f-cosymplectic manifold with f constant. By the reason of this, we obtain the following conclusion from Theorem 3.3.  Proof. One can easily obtain from Eqn (3.9) that fð4n þ 1Þλ þ 2ngwDλ ¼ wDν; where we applied our assumption λ ¼f. Uptaking ν ¼ −rf λ 2n þ 1 2nþ1 g in the above relation implies Suppose that ð4n þ 1Þλ þ 2n þ r 2n u 0. Due to constancy of r, we see that λ is constant. Next, we assume that ð4n þ 1Þλ þ 2n þ r 2n u 0$ in a neighborhood O of M. Consequently, one can gets wDλ 5 0. Applying w to this equation implies Dλ 5 (ξλ)ξ. In this context (3.13) holds, from which we can get Differentiating this along ξ gives (2λ þ 1)(ξλ) 5 ξ(ξλ)f þ (ξλ)(ξf ), due to our assumption Suppose that f 5 0, then from Eqn (3.20), we have λðλ þ 1Þ þ r 2nð2nþ1Þ ¼ 0, which means that λ is constant. In the following we suppose f ≠ 0, then as a result of Eqns (3.16), (3.20) and (3.21), we find Substitute this into 2nΔλ 5 Àλr to obtain −2n ðξλÞ f À 2nð2n þ 1Þλ ¼ r: Differentiating the aforesaid relation along ξ, remembering r is constant and applying Eqn (3.21), we reach at 2n þ 3 þ 1 f 2 ðξλÞ ¼ 0: If (ξλ) 5 0, then we have Dλ 5 0, which means λ is constant. Suppose (ξλ) ≠ 0, then we get

CPE on non-normal almost f-cosymplectic manifolds
Here, we consider a three dimensional almost f-cosymplectic manifold M with pseudo anticommuting Ricci tensor, that is, wQ þ Qw ¼ 2κw; κ is constant: This notion was introduced by Jeong and Suh [14], and they made use of this condition to classify a real hypersurface of complex two-plane Grassmannians. At first, we have the following lemma: CPE on almost f-cosymplectic manifolds Lemma 4.1.
[15] For a three dimensional almost f-cosymplectic manifold with pseudo anticommuting Ricci tensor the following formula holds: where a 5 g(Qξ, ξ).
where b is a function defined by b 5 g(∇ ξ we, e) and s is a (1,1) tensor field defined by se 5 e, swe 5 Àwe and sξ 5 0.
Next suppose a 2 4f 2 þ r 2 − a ≠ 0, then from Eqn (4.14) it can easily conclude that λ is constant. Case 2. If (eλ) ≠ 0 on a neighborhood O of M, then from (4.9) and (4.10) we extract 3a À r From the preceding equation, we can easily observe that b is constant because of μ and f are constant. It is easy to seen from Eqns (4.9) and (4.10) that (eλ) and (weλ) are constants in O.
From the above detailed discussion, we have concluded that M is Einstein or its scalar curvature vanishes, and so we state following result: Theorem 4.4. Let M be a three dimensional almost f-almost cosymplectic manifold satisfying the CPE Eqn (1.2), if its Ricci tensor is pseudo anti-commuting, then M is either Einstein or its scalar curvature vanishes identically.