Critical point equation on almost f-cosymplectic manifolds

H. Aruna Kumara (Department of Mathematics, Kuvempu University, Shimoga, India)
V. Venkatesha (Department of Mathematics, Kuvempu University, Shimoga, India)
Devaraja Mallesha Naik (Department of Mathematics, CHRIST (Deemed to be University), Bengaluru, India)

Arab Journal of Mathematical Sciences

ISSN: 1319-5166

Article publication date: 7 May 2021




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.


The paper opted the tensor calculus on manifolds to find the solution of the CPE.


In this paper, in particular, the authors obtained that a connected f-cosymplectic manifold satisfying CPE with \lambda=\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.


The paper proved that the CPE conjecture is true for almost f-cosymplectic manifolds.



Kumara, H.A., Venkatesha, V. and Naik, D.M. (2021), "Critical point equation on almost f-cosymplectic manifolds", Arab Journal of Mathematical Sciences, Vol. ahead-of-print No. ahead-of-print.



Emerald Publishing Limited

Copyright © 2021, H. Aruna Kumara, V. Venkatesha and Devaraja Mallesha Naik


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1. 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:MR given by

defined on a compact orientable Riemannian n-manifold (M, g), where M denotes set of all Riemannian metrics on (M, g) of unit volume, rg is the scalar curvature and dvg is the volume form. The functional S in Eqn (1.1) restricted over M is known as Einstein–Hilbert functional and its critical points are the Einstein metric (see chapter 2 in [1]).

Let CM 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,

for some smooth function λ on M. Here Hess, Δg, 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

We notice that if λ is constant in Eqn (1.2), then λ = 0 and g becomes Einstein. Therefore, we have the following definition:

Definition 1.1.

A compact Riemannian manifold (M, g) of dimension n > 3 with constant scalar curvature and unit volume together with a smooth potential function λ satisfying (Eqn 1.2), is called critical point equation (shortly, CPE).

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 [69] motivates us to consider almost f-cosymplectic manifolds. In this paper, we classify an almost f-cosymplectic manifold which satisfies CPE.

2. Preliminaries

Let M be a smooth differentiable manifold of dimension 2n + 1 equipped with a triple (ϕ, ξ, η), where ϕ is a (1, 1)-tensor field, ξ is a Reeb vector and η is a one-form such that

which implies ϕ(ξ) = 0, η(ϕ) = 0 and rank(ϕ) = 2n. If M admits a Riemannian metric g such that
for any vector fields X, Y, then M is said to have an almost contact metric structure (ϕ, ξ, η, g). 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×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 [ϕ, ϕ] indicates the Nijenhuis tensor of ϕ.

An almost contact metric manifold M is said to be almost cosymplectic if  = 0 and dΦ = 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  = 0 and dΦ = 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 = 2 ∧ Φ and  = 0 for a smooth function f satisfying dfη = 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) = 0 and an almost α-Kenmotsu manifold if (α = f ≠ 1).

Besides, we recall that there is an operator h=12 £ξ ϕ, 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]:

for any vector fields X, Y on M, where f̃=ξ(f)+f2.

3. 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 = 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;

where Q is the Ricci operator of M.

Now, we will give some properties, which will be used in the proof of our results.

Lemma 3.1.

An f-cosymplectic manifold M of dimension 2n + 1 satisfies

(3.4) (XQ)ξ=fQX2n(Xf̃)ξ2nf̃fX,
(3.5) (ξQ)X=2fQX(2n1)(Xf̃)ξ(ξf̃)X4nf̃fX.

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 = Z = Ei in the above equation and then summing over i shows that

One can easily deduce from second Bianchi identity that

Feeding Eqn (3.7) into Eqn (3.6) and with the help of Eqn (3.4), we obtain

which proves Eqn (3.5).□
Lemma 3.2.

[5] Let (g, λ) 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

for any vector fields X, Y on M, where ν=r(λn1+1n).

In the following, we will consider an f-cosymplectic manifold M satisfying a CPE and assume that the function f satisfies ξ(f̃)=0.

Theorem 3.3.

Let M be an f-cosymplectic manifold of dimension 2n + 1 with ξ(f̃)=0. If (g, λ) is a solution of the CPE (Eqn 1.2), then one of the following statement holds:

  1. M is Einstein

  2. M is locally the product of a Kähler manifold and an interval or unit circle S1.

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 ϕX and Y by ξ in above relation, we get


According to Proposition 2.1 of Chen [12], it is know that if (ξf̃)=0, then f̃ is constant. So that, Eqn (3.9) implies


The scalar curvature r of g is constant (as (g, λ) is a solution of the CPE). For a (2n + 1)-dimensional f-cosymplectic manifold, we have ν=r(λ2n+12n+1), therefore from Eqn (3.10) it appears that


From Eqn (3.11), we have either r=2n(2n+1)f̃ or ϕDλ = 0.

First suppose that r=2n(2n+1)f̃, then we have Dν=(2n+1)f̃Dλ. Plugging X = ξ in Eqn (3.8) and calling back Lemma 3.1, we aimed at obtaining


From Eqn (3.3), we deduce R(X,ξ)Y=f̃{g(X,Y)ξη(Y)X}, by virtue of this the foregoing equation reduces to


Making use of Dν=(2n+1)f̃Dλ in Eqn (3.12) we reach at


Since ∇ξξ = 0 and (ξλ) = g(ξ, ), taking into account ∇X = (λ + 1)QX + νX, we deduce ξ(ξλ)=f̃λ. If possible, let (ξλ) = f(λ + 1) in some open set O of M, then we have f̃λ=((ξf)+f2)(λ+1). By virtue of f̃=(ξf)+f2, one can see λ = λ + 1, that is, 1 = 0, which is absurd. Hence QX=2nf̃X and M is Einstein.

Next we assume r2n(2n+1)f̃, then from Eqn (3.11) we have ϕDλ = 0. Action of ϕ on this equation gives = (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


Comparing aforementioned equation with Eqn (3.13), we get


Taking X = ξ in the above equation and making use of Eqns (3.2) and (2.1), we obtain


Contraction of Eqn (3.13) with respect to X brings into view


Unifying this with Eqn (3.15) implies


Differentiating Eqn (3.17) along ξ, keeping in mind that f̃ and r are constants, we obtain ξ(ξλ)f+(ξλ)(ξf)=f̃(ξλ), and further, it implies

where we used f̃=(ξf)+f2.

If f ≢ 0, then we can assume f ≠ 0 on some neighborhood O of M. Thus, Eqn (3.18) implies ξ(ξλ) = (ξλ)f on O. Inserting this into Eqn (3.16), we find Δλ = (2n + 1)f(ξλ). Moreover, applying Eqn (3.17) in the previous relation shows that


Taking trace of CPE (1.2), we obatin 2nΔλ = −λr, and this together with Eqn (3.19) gives that r=2n(2n+1)f̃, which is contradictory to our assumption. Hence f ≢ 0, and so M is cosymplectic. According to Blair's [13] result, we can easily conclude that M is locally the product of a Kähler manifold and an interval or unit circle S1. This finishes the proof.□

In particular, when dimension of M is three, due to Theorem 3.3 we have the following outcome:

Corollary 3.4.

Let M be an f-cosymplectic manifold of dimension three satisfying CPE Eqn (1.2). If (ξf̃)=0, then M is either locally the product of a Kähler manifold and an interval or unit circle S1 or M has constant negative sectional curvature f̃.

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.

Corollary 3.5.

Let M be an α-cosymplectic manifold of dimension 2n + 1 with ξ(f̃)=0. If (g, λ) is a solution of the CPE Eqn (1.2), then M is either Einstein or locally the product of a Kähler manifold and an interval or unit circle S1.

Now we consider CPE with λ=f̃, and obtain the following result.

Theorem 3.6.

If a connected f-cosymplectic manifold M satisfying CPE Eqn (1.2) with λ=f̃, then M is Einstein.

Proof. One can easily obtain from Eqn (3.9) that

where we applied our assumption λ=f̃. Uptaking ν=r{λ2n+12n+1} in the above relation implies

Suppose that (4n+1)λ+2n+r2n ≢ 0. Due to constancy of r, we see that λ is constant. Next, we assume that (4n+1)λ+2n+r2n ≢ 0$ in a neighborhood O of M. Consequently, one can gets ϕDλ = 0. Applying ϕ to this equation implies = (ξλ)ξ. In this context (3.13) holds, from which we can get


Differentiating this along ξ gives (2λ + 1)(ξλ) = ξ(ξλ)f + (ξλ)(ξf), due to our assumption λ=f̃=(ξf)+f2 which further implies


Suppose that f = 0, then from Eqn (3.20), we have λ(λ+1)+r2n(2n+1)=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 = −λr to obtain


Differentiating the aforesaid relation along ξ, remembering r is constant and applying Eqn (3.21), we reach at


If (ξλ) = 0, then we have  = 0, which means λ is constant. Suppose (ξλ) ≠ 0, then we get f2=12n+3. Due to f ≠ 0, which shows (ξf) = 0. This together with λ=f̃=(ξf)+f2 yields λ = f2, showing λ constant. In a word, we have proved that λ is always constant in the neighborhood O of M, thus λ = constant in M. Hence, the proof completes from Eqn (1.2).

4. CPE on non-normal almost f-cosymplectic manifolds

Here, we consider a three dimensional almost f-cosymplectic manifold M with pseudo anti‐commuting Ricci tensor, that is,


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:

Lemma 4.1.

[15] For a three dimensional almost f-cosymplectic manifold with pseudo anticommuting Ricci tensor the following formula holds:

where a = g(, ξ).

Let U be the open subset where the tensor h ≠ 0 and U be the open subset such that h is identically zero. Thus, UU is open dense in M. There exists a local orthonormal frame field E = {ξ, e, ϕe} such that he = μe and hϕe = −μϕe, where μ is a positive nonvanishing smooth function of M. The following proposition is obtain from Proposition 12 and Proposition 14 of Öztürk et al. [10]:

Proposition 4.2.

For a three dimensional almost f-cosymplectic manifold, the following relations hold:

(4.1) h2f2ϕ2=a2ϕ2,
(4.2) ξh=2bhϕ+(ξμ)s,
where b is a function defined by b = g(∇ξϕe, e) and s is a (1,1) tensor field defined by se = e, sϕe = −ϕe and  = 0.

From this onwards, we assume that a three dimensional almost f-cosymplectic manifold M satisfies the CPE Eqn (1.2), then its scalar curvature r is constant. As a result of Lemma 4.1, it can be seen that a is constant. Chen in [15] obtained the relation (r2κ2f̃2a)ϕX=2ϕh2X. From this we have (2f̃a)ϕX=2ϕh2X, that is, 2h2X=(2f̃+a)ϕ2X as  = 0. Further, from Eqn (4.1) we find (ξf) = 0, due to dfη = 0 we obtain f is constant. Moreover, we can also find a = −2(f2 + μ2) from Eqn (4.1). Thus μ is constant. In view of Lemma 2 of [10], we have the following (also see [15]):

Lemma 4.3.

Let M be a three dimensional almost f-almost cosymplectic manifold satisfying the CPE Eqn (1.2). If the Ricci tensor is pseudo anticommuting, then with respect to E the Levi-Civita connection ∇ is given by

(4.3) ξe=bϕe,ξϕe=be,ξξ=0,eξ=feμϕe,ee=fξ,eϕe=μξ,ϕeξ=μe+fϕe,ϕeϕe=fξ,ϕee=μξ.

In view of Eqn (4.2), the relation Eqn (2.5) implies


By virtue of Eqn (3.8), we obtain


Substituting the above relation into Eqn (4.4), we obtain


Employing X by ϕX in Eqn (4.6) we reach at


In orthonormal frame field E, the gradient vector field can be written as


Thus from Eqn (4.7), one can obtain


First we assume f ≠ 0, because of f is constant and we shall divide this discussion into two cases:

Case 1. If () = 0, then from Eqn (4.10) we can observe (ϕeλ) = 0. This together with (4.8) yields = (ξλ)ξ. Differentiating this along X, using Eqn (2.2) gives


Employing X = ξ in the above equation and remembering ξDλ=(λ+1)Qξ+(Δλr3)ξ, we aimed at obtaining


One can find from Eqn (4.11) and second term of Eqn (2.2) that


By virtue of the foregoing relation, Eqn (4.12) transforms into (ξλ)=r6f(λ+1)a2f, which further gives

where we applied f and a are constants. Also, it is know that Δλ=λr2. Thus Eqn (4.12) transforms into

If a24f2+r2a=0, then we have ra12f2r3a24f2+a=0, which further shows that r6(a2f2+1)=0. The former case implies that the scalar curvature of M vanishes identically. In the latter case, we have a = −2f2, which together with a = −2(f2 + μ2) implies μ = 0, which is not possible. Next suppose a24f2+r2a0, then from Eqn (4.14) it can easily conclude that λ is constant.

Case 2. If () ≠ 0 on a neighborhood O of M, then from (4.9) and (4.10) we extract


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 () and (ϕeλ) are constants in O.

By the support of Eqn (4.3), we may easily compute that


Thus Δλ = 2f(ξλ) + ξ(ξλ). So, utilization of ξDλ=(λ+1)Qξ+(Δλr3)ξ followed from (3.14), shows that


As followed by Case 1, we can conclude that either r vanishes or λ is constant.

Next we assume f = 0, then from Eqns (4.9) and (4.10) we find b()(ϕeλ) = 0 as μ > 0, which further implies either ()(ϕeλ) = 0 or b = 0. We shall also discuss this matter into two cases.

Subcase i. If b = 0, then from Eqn (4.10) we find r = 3a. For three dimensional case, it is known that the Riemannian curvature is


From this, we have


This together with Eqn (4.4) yields QX = aX, which means M is Einstein.

Subcase ii. If b ≠ 0 on some neighborhood O of M, then we find ()(ϕeλ) = 0 on O. If possible, let () = 0 = (ϕeλ), then from Eqn (4.8) we obtain = (ξλ)ξ. From this it is not hard to see that Eqn (4.12) holds and Eqn (4.13) implies Δλ = ξ(ξλ). Thus Eqn (4.12) transforms into (λ+1)ar3=0, which means λ is constant on O.

If () = 0 and (ϕeλ) ≠ 0, then from Eqns (4.3) and (4.8) we compute


Utilization of above relation in ξDλ=(Δλr3)ξ+(λ+1)aξ, we find b(ϕeλ) = 0. Because of b ≠ 0, we have (ϕeλ) = 0, which is a contradiction. In a similar manner, we also come to contradiction if we consider () ≠ 0 and (ϕeλ) = 0.

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.


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H.A. Kumara and V. Venkatesha are thankful to Department of Science and Technology, New Delhi for financial assistance to the Department of Mathematics, Kuvempu University under the FISt program (Ref. No. SR/FST/MS-I/2018-23(C)).

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V. Venkatesha can be contacted at:

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