In [4], Miao-Tam studied the volume functional on the space of constant scalar curvature metrics with a given boundary metric. They derived a necessary and sufficient condition for a metric to be a critical point as follows:

On a compact Riemannian manifold (Mⁿ, g), n ≥ 3, with smooth boundary, if there exists a non-zero smooth function λ : Mⁿ → ℝ (called potential function) such that

and λ = 0 on ∂Mⁿ, where Δ is the Laplacian operator, Hess is the Hessian operator and S is the Ricci tensor with respect to the metric g, then g is said to satisfy the Miao-Tam critical condition.

In particular, if the potential function λ is a non-zero constant, then (1.1) is just an Einstein metric. Recently, Hwang [2] proved that the CPE conjecture is also true under certain condition on the bounds of the potential function λ. In 2017, Wang [8] proved that if the metric of a 3-dimensional (k, μ)′-almost Kenmotsu manifold satisfies the Miao-Tam critical condition, then the manifold is locally isometric either to the hyperbolic space ℍ³(−1) or to the Riemannian product ℍ²(−4)×ℝ. In [7], Ghosh and Patra considered the CPE in the framework of K-contact manifolds and (k, μ)-contact manifolds.

Motivated by the above studies, in the present paper we study 3-dimensional α-cosymplectic manifolds admitting CPE, i.e. satisfying the relation (1.1). The paper is organized as follows. In section 2, we recall the definition of α-cosymplectic manifolds and some basic formulas and section 3 is devoted to prove our main result, precisely:

Theorem 1.1

If a 3-dimensional connected α-cosymplectic manifold (M,φ, ξ, η, g, α) satisfies CPE, then the manifold is of constant sectional curvature −α², provided Dλ ≠ (ξλ)ξ, where D denotes the gradient operator with respect to g.

An almost contact metric structure on a (2n+1)-dimensional smooth manifold M consists of a 1-form η, a vector field ξ (called the Reeb field), a (1, 1)-tensor field φ and a Riemannian metric g satisfying the following conditions:

and

for any X, Y ∈ χ(M). The above relations imply

for any X, Y ∈ χ(M).

An almost contact metric structure is said to be normal if the induced almost complex structure J on the product manifold M × ℝ defined by

is integrable, where X ∈ χ(M), t is the coordinate on ℝ and f is a smooth function on M × ℝ.

An almost contact metric structure is said to be a contact metric structure if

for any X, Y ∈ χ(M). In this case, the 1-form η is called the contact metric form. We define a (1, 1)-tensor field h by h:=12£ξφ, where £ denotes the Lie derivative in the direction of the vector field ξ. It is symmetric and satisfies hφ = −φh. Also, we have Tr.h = Tr.φh = 0, hξ = 0 and

for any X ∈ χ(M), where ∇ is the Levi-Civita connection of g.

An almost contact metric manifold is called Kenmotsu if

for any X, Y ∈ χ(M).

In 2005, Kim and Pak [3] introduced the notion of almost α-cosymplectic manifold, which is an almost contact metric manifold that satisfies dη = 0 and dΦ = 2αη ∧ Φ, for α a real number. Recently, Erken [1] and Öztürk et. al [5, 6] obtained some fundamental properties of almost α-cosymplectic manifolds. An α-cosymplectic manifold is a normal almost α-cosymplectic manifold. An α-cosymplectic manifold with α = 0 is a cosymplectic manifold and with α = 1, it is a Kenmotsu manifold.

On a (2n + 1)-dimensional α-cosymplectic manifold M, for any X, Y ∈ χ(M), the following relations hold:

Before proving our main result, we recall the following lemma, given by Miao-Tam.

Lemma 3.1. ([4, Theorem 7])

If the metric of a connected Riemannian manifold satisfies the Miao-Tam critical condition, then the scalar curvature is constant.

It is known that the Riemannian curvature tensor of a 3-dimensional Riemannian manifold (M,g) is given by:

for any X, Y, Z ∈ χ(M), where S is the Ricci tensor, Q is the Ricci operator and r is the scalar curvature.

Assume that (M,φ, ξ, η, g,α) is a 3-dimensional connected α-cosymplectic manifold which satisfies the Miao-Tam critical condition, i.e. which satisfies (1.1). Putting Y = Z = ξ in (3.1) and using (2.9), (2.10) and (2.11), the Ricci operator can be written as

for any X ∈ χ(M).

Taking covariant derivative of the above equation with respect to Y and using (2.7) and Lemma 3.1, we obtain

for any X, Y ∈ χ(M).

Taking trace of the equation (1.1), we have

Using (3.4) in (1.1), we obtain

for any X ∈ χ(M), where D denotes the gradient operator with respect to g.

Taking the covariant derivative of (3.5) with respect to Y, we get

for any X, Y ∈ χ(M).

Similarly, we get

Also

and using (3.6), (3.7) and (3.8) we have

In view of (3.3) and (3.9) yields

for any X, Y ∈ χ(M).

By setting X = ξ in the above equation and using (2.10) and (3.2) we get

Taking inner product with ξ in the above equation, we easily compute

On the other hand, from (2.9) we have

Making use of (3.12) and (3.13) we get

Removing Y from both sides in the above equation, we obtain

From f=-12(rλ+1), we get

Using the above relations in (3.15) we easily have

If Dλ = (ξλ)ξ, then taking the covariant derivative with respect to X and using (3.5) we obtain

Then taking trace we get

From (1.1) we get

hence, for X = ξ, together with (3.4) and (2.11) imply

Since ∇_ξDλ = ξ(ξλ)ξ, we deduce that

which together with (3.18) imply

Then we can state:

Proposition 3.1

If a 3-dimensional connected α-cosymplectic manifold (M,φ, ξ, η, g,α) satisfies CPE and r ≠ −6α², then the gradient of λ is collinear with ξ. Moreover

Corollary 3.1

If a 3-dimensional connected α-cosymplectic manifold (M,φ, ξ, η, g, α) satisfies CPE and r ≠ −6α², then it can not be a cosymplectic manifold.

If 3α2+r2=0, then r = −6α². Putting the value of r = −6α² in (3.1) and in view of (3.2), we find that manifold is of constant sectional curvature −α².

Hence we can state the following:

Theorem 3.1

If a 3-dimensional connected α-cosymplectic manifold (M,φ, ξ, η, g, α) satisfies CPE, then the manifold is of constant sectional curvature −α², provided Dλ ≠ (ξλ)ξ, where D denotes the gradient operator with respect to g.

If α = 0, then the manifold is a cosymplectic manifold and we have the following:

Corollary 3.2

If a 3-dimensional connected cosymplectic manifold (M,φ, ξ, η, g) satisfies CPE, then the manifold is flat, provided Dλ ≠ (ξλ)ξ.

If α = 1, then the manifold is a Kenmotsu manifold and we have the following:

Corollary 3.3

If a 3-dimensional connected Kenmotsu manifold (M,φ, ξ, η, g) satisfies CPE, then the manifold is locally isometric to the hyperbolic space H³(−1), provided Dλ ≠ (ξλ)ξ.