#### Abstract

In this paper, we classified the paracontact metric -manifold satisfying the Miao-Tam critical equation with . We proved that it is locally isometric to the product of a flat -dimensional manifold and an -dimensional manifold of negative constant curvature .

#### 1. Introduction

Inspired by the positive mass theorem and the variational characterization of Einstein metrics on a closed manifold, with an aim to find a proper concept of metrics that would sit between constant scalar curvature metrics and Einstein metrics, in [1], Miao and Tam studied the variational properties of the volume functional on the space of constant scalar curvature metrics with a prescribed boundary metric. Specifically, they derived the following sufficient and necessary condition for a metric to be a critical point:

Theorem 1 (Theorem 5 in [1]). *Let be a compact -dimensional Riemannian manifold with smooth boundary , be a given metric on , and be the space of metrics on which have constant scalar curvature and have induced metric on given by . Let be a smooth metric such that the first Dirichlet eigenvalue of is positive. Then, is a critical point of the volume functional in if and only if there is a smooth function on such that on and
where and are the Laplacian and Hessian operators with respect to , and Ric() is the Ricci curvature of .*

For brevity, we call such critical metric as Miao-Tam critical metric and refer to equation (1) as the Miao-Tam equation. A fundamental property of a Miao-Tam critical metric is that its scalar curvature is a constant (see Theorem 7 in [1]). Some explicit examples of Miao-Tam critical metrics can be found in [1, 2], including not only the standard metrics on geodesic balls in space forms but the spatial Schwarzschild metrics and AdS-Schwarzschild metrics restricted to certain domains containing their horizon and bounded by two spherically symmetric spheres. In [2], the authors classified all Einstein and conformally flat Miao-Tam critical metrics. In fact, they proved that any connected, compact, Einstein manifold with smooth boundary satisfying Miao-Tam critical condition is isometric to a geodesic ball in a simply connected space form. And then several generalizations of this rigidity result were found by different authors, replacing the Einstein assumption by a weaker condition such as harmonic Weyl tensor [3], parallel Ricci tensor [4], or cyclic parallel Ricci tensor [5]. For Some other generalizations or rigidity results, we can refer to [6â€“10], etc.

Recently, some geometricians focus on the study of Miao-Tam equation within the framework of contact metric manifolds. In [11], the authors proved that a complete -contact metric satisfying the Miao-Tam critical condition is isometric to a unit sphere . Furthermore, they studied -contact metrics satisfying the Miao-Tam equation. Moreover, the Miao-Tam equation within the framework of Kenmotsu and almost Kenmotsu manifolds was studied in [12], and it was proved that a Kenmotsu metric satisfying the Miao-Tam equation is Einstein. In addition, in [13], the authors studied the critical point equation on -paracontact manifolds; especially, they proved that any -paracontact manifolds satisfying the Miao-Tam equation must be Einstein. We also note that some geometric structures such as Ricci soliton were studied within the framework of paracontact metric -manifold (see [14]). In this direction, it is natural to study paracontact metric -manifold satisfying the Miao-Tam equation. In this paper, we will prove the following main result:

Theorem 2. *Let be a paracontact metric -manifold of dimensional with . If is a nonconstant solution of the Miao-Tam equation, then is locally flat in dimension , and in higher dimensions , it is locally isometric to the product of a flat -dimensional manifold and an -dimensional manifold of negative constant curvature equal to .*

#### 2. Preliminaries

In this section, we recall some basic definitions and facts on paracontact metric manifolds which we will use later. For more details and some examples, we refer to [15â€“26].

A -dimensional smooth manifold is said to have an *almost paracontact structure*, if it admits a -tensor field , a vector field , and a 1-form satisfying the following conditions:
(i), (ii)The tensor field induces an almost paracomplex structure on each fiber of , i.e., the eigendistributions and of corresponding to the eigenvalues 1 and , respectively, have same dimension

From the definition, it is easy to see that , , and the endomorphism have rank . An almost paracontact structure is said to be *normal* if and only if the tensor field vanishes identically. If an almost paracontact manifold admits a pseudo-Riemannian metric such that
for all , then we say that has an almost paracontact metric structure, and is called compatible metric. It follows that and . Notice that any such a pseudo-Riemannian metric is necessarily of signature .

If in addition for all vector fields on , then the manifold is said to be a *paracontact metric manifold*. In this case, becomes a contact form, i.e., , with its Reeb vector field. In a paracontact metric manifold, one defines two self-adjoint operators and by and , where is the Lie derivative along , and is the curvature tensor of . It is known in [25] that the two operators and satisfy

And there also holds
where is the Levi-Civita connection of the pseudo-Riemannian manifold . Moreover, if and only if is a Killing vector field, and in this case, the paracontact metric manifold is said to be a *-paracontact manifold*. A normal paracontact metric manifold is said to be a *paraSasakian manifold*.

The study of nullity conditions on paracontact geometry is the most interesting topics in paracontact geometry. Motivated by the relationship between contact metric and paracontact geometry, in [18],. Cappelletti Montano et al. introduced the following.

*Definition 3. *A paracontact metric manifold is said to be a paracontact metric *-*manifold, if its curvature tensor satisfies
for all tangent vector fields on , where are real constants.

On a paracontact metric -manifold , the following formulas are valid [18]: where is the Ricci operator associated with the Ricci tensor .

Paracontact metric -spaces satisfy (7) but this condition does not give any type of restriction over the value of , unlike in contact metric geometry, because the metric of a paracontact metric manifold is not positive definite. However, The geometric behavior of the paracontact metric -manifold is different according and . In particular, for the case and , -nullity condition (7) determines the whole curvature tensor field completely. The case is equivalent to but not to , which is different from contact -space. Indeed, there are examples of paracontact metric -spaces with but , as was first shown in [18, 27, 28]. In this paper, we consider the paracontact metric -manifolds with the condition .

#### 3. The Proof of Theorem 2

Before giving the proof of Theorem 2, we introduce some important lemmas which will be used later. First of all, we recall a basic fact about paracontact metric -manifold.

Lemma 4 (Corollary 4.14 in [18]). *In any -dimensional paracontact metric -manifold such that , the Ricci operator of is given by
for any vector field . In particular, is -Einstein if and only if , Einstein if and only if and (in this case, the manifold is Ricci-flat). Further, the scalar curvature of is .*

In the following, we consider paracontact metric -manifold satisfying the Miao-Tam equation.

Lemma 5. *Let be a nonconstant solution of the Miao-Tam equation on the -dimensional semi-Riemannian manifold with scalar curvature . Then, the curvature tensor can be expressed as
for any vector field on , where .*

*Proof. *Tracing (1), we obtain

Then, the Miao-Tam equation (1) can be exhibited as for any vector field on , where . Taking the covariant derivative of (12) along an arbitrary vector field on , we obtain

Similarly, we have for any vector field on . Comparing the preceding two equations and using (12) in the well-known expression of the curvature tensor , we obtain the result.

Lemma 6. *Let be a paracontact metric -manifold of dimensional with , and be a nonconstant solution of the Miao-Tam equation on . Then, we have
*

*Proof. *Firstly, taking covariant derivative of (8) along any vector field and using (4), we can obtain

Taking the inner product of (10) with and using (8) and (16), we have where (noting that the dimension of is ).

It follows from (6) that . Then, replacing by and by in (17), respectively, we obtain

Since is nonconstant on , it is easy to see that

Replacing by in (9), we have

Then, the action of on the (20) gives where we have used (7).

Operating (9) by , we have

Replacing by in (22) and using (7) again, we get

Substituting equations (20)-(23) into (19) yields which completes the proof of Lemma 6.

Next, we will give the complete proof of Theorem 2.

*Proof. *Firstly, taking in (17) gives

Putting in (6) and comparing with the forgoing equation, we obtain

Noting that the scalar curvature is a constant, it follows from that

Then, we can obtain from (26) and (27) that

On the one hand, taking in (6), since , it follows that which gives

Substituting (7) and (30) in (5), we get

On the other hand, we obtain from (12) and (8) that

Next, taking covariant derivative of (28) along and making use of (31) and (32), we have

Operating this equation by shows

By the action of in (34), it follows from (7) that

Since we assume that , we divide it into two cases:

Case (i): ; case (ii):

If case (i) occurs, it follows from Lemma 6 that . Hence, the definition of paracontact metric -manifold gives that for any vector field ,. From Theorem 3.3 of [26], is locally flat in dimension , and in higher dimensions (), it is locally isometric to the product of a flat -dimensional manifold and an -dimensional manifold of negative constant curvature .

If case (ii) occurs, then , i.e., . Differentiating this along an arbitrary vector field together with (4) implies that

It follows from (12) that , and then the foregoing equation shows that

Replacing by , by , and noting that is nonzero for any paracontact metric manifolds, it follows that . Hence, , , is a constant, which gives a contradiction.

This completes the proof of Theorem 2.

#### Data Availability

No data were used to support this study.

#### Conflicts of Interest

The author(s) declare(s) that they have no conflicts of interest.

#### Acknowledgments

The first author was supported by grant of NSFC (No. 11801011) and the Key Scientific Research Projects of Colleges and Universities of Henan Province (No. 19B110001).