# Paracontact Metricκ,μ-Manifold Satisfying the Miao-Tam Equation

Authors: Dehe Li; Jiabin Yin
Journal: Advances in Mathematical Physics
 (2021)

---

## Abstract

In this paper, we classified the paracontact metricκ,μ-manifold satisfying the Miao-Tam critical equation with κ>−1. We proved that it is locally isometric to the product of a flat n+1-dimensional manifold and an n-dimensional manifold of negative constant curvature −4.


---

## Body

## 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]).
LetΩ be a compact n-dimensional Riemannian manifold with smooth boundary Σ, γ be a given metric on Σ, and MγK be the space of metrics on Ω which have constant scalar curvature K and have induced metric on Σ given by γ. Let g∈MγK be a smooth metric such that the first Dirichlet eigenvalue of n−1Δg+K is positive. Then, g is a critical point of the volume functional in MγK if and only if there is a smooth function λ on Ω such that λ=0 on Σ and
(1)−Δgλg+∇g2λ−λRicg=g,where Δg and ∇g2 are the Laplacian and Hessian operators with respect to g, and Ric(g) is the Ricci curvature of g.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 [6–10], 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 K-contact metric satisfying the Miao-Tam critical condition is isometric to a unit sphere S2n+1. Furthermore, they studied k,μ-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 K-paracontact manifolds; especially, they proved that any K-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 κ,μ-manifold (see [14]). In this direction, it is natural to study paracontact metric κ,μ-manifold satisfying the Miao-Tam equation. In this paper, we will prove the following main result:Theorem 2.
LetM2n+1φ,ξ,η,g be a paracontact metric κ,μ-manifold of dimensional 2n+1 with κ>−1. If g,λ is a nonconstant solution of the Miao-Tam equation, then M2n+1 is locally flat in dimension 3, and in higher dimensions n>1, it is locally isometric to the product of a flat n+1-dimensional manifold and an n-dimensional manifold of negative constant curvature equal to −4.


## 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 [15–26].A2n+1-dimensional smooth manifold M2n+1 is said to have an almost paracontact structureφ,ξ,η, if it admits a 1,1-tensor field φ, a vector field ξ, and a 1-form η satisfying the following conditions:
(i)
ηξ=1, φ2=id−η⊗ξ(ii)
The tensor fieldφ induces an almost paracomplex structure on each fiber of D=Kerη, i.e., the eigendistributions D+ and D− of φ corresponding to the eigenvalues 1 and −1, respectively, have same dimension nFrom the definition, it is easy to see thatφξ=0, η∘φ=0, and the endomorphism φ have rank 2n. An almost paracontact structure is said to be normal if and only if the tensor field Nφ≔φ,φ−2dη⊗ξ vanishes identically. If an almost paracontact manifold admits a pseudo-Riemannian metric g such that
(2)gφX,φY=−gX,Y+ηXηY,for all X,Y∈ΓTM, then we say that M has an almost paracontact metric structure, and g is called compatible metric. It follows that η=g·,ξ and g·,φ·=−gφ·,·. Notice that any such a pseudo-Riemannian metric is necessarily of signature n+1,n.If in additiondηX,Y=gX,φY for all vector fields X,Y on M, then the manifold M2n+1φ,ξ,η,g is said to be a paracontact metric manifold. In this case, η becomes a contact form, i.e., η∧dηn≠0, with ξ its Reeb vector field. In a paracontact metric manifold, one defines two self-adjoint operators h and l by h=1/2Lξφ and l=R·,ξξ, where Lξ is the Lie derivative along ξ, and R is the curvature tensor of g. It is known in [25] that the two operators h and l satisfy
(3)Trh=0,hξ=0,lξ=0,hφ=−φh.And there also holds(4)∇Xξ=−φX+φhX,(5)∇ξh=φh2−φ−φl,where ∇ is the Levi-Civita connection of the pseudo-Riemannian manifold M,g. Moreover, h=0 if and only if ξ is a Killing vector field, and in this case, the paracontact metric manifold M is said to be a K-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 manifoldM2n+1φ,ξ,η,g is said to be a paracontact metric κ,μ-manifold, if its curvature tensor R satisfies
(6)RX,Yξ=κηYX−ηXY+μηYhX−ηXhY,for all tangent vector fields X,Y on M, where κ,μ are real constants.On a paracontact metricκ,μ-manifold M2n+1φ,η,ξ,gn≥1, the following formulas are valid [18]:
(7)h2=1+κφ2,(8)Qξ=2nκξ,where Q is the Ricci operator associated with the Ricci tensor Ric.Paracontact metricκ,μ-spaces satisfy (7) but this condition does not give any type of restriction over the value of κ, 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 κ,μ-manifold is different according κ<−1,κ=1 and κ>−1. In particular, for the case κ<−1 and κ>−1, κ,μ-nullity condition (7) determines the whole curvature tensor field completely. The case κ=−1 is equivalent to h2=0 but not to h=0, which is different from contact κ,μ-space. Indeed, there are examples of paracontact metric κ,μ-spaces with h2=0 but h=0, as was first shown in [18, 27, 28]. In this paper, we consider the paracontact metric κ,μ-manifolds with the condition κ>−1.


## 3. The Proof of Theorem2

Before giving the proof of Theorem2, we introduce some important lemmas which will be used later. First of all, we recall a basic fact about paracontact metric κ,μ-manifold.Lemma 4 (Corollary 4.14 in [18]).
In any2n+1-dimensional paracontact metric κ,μ-manifold M2n+1φ,ξ,η,g such that κ>−1, the Ricci operator Q of M is given by
(9)QX=21−n+nμX+2n−1+μhX+2n−1+n2κ−μηXξ,for any vector field X. In particular, M,g is η-Einstein if and only if μ=21−n, Einstein if and only if κ=μ=0 and n=1 (in this case, the manifold is Ricci-flat). Further, the scalar curvature of M is 2n21−n+κ+nμ.In the following, we consider paracontact metricκ,μ-manifold satisfying the Miao-Tam equation.Lemma 5.
Letg,λ be a nonconstant solution of the Miao-Tam equation on the k-dimensional semi-Riemannian manifold Mk with scalar curvature S. Then, the curvature tensor R can be expressed as
(10)RX,YDλ=XλQY−YλQX+λ∇XQY−λ∇YQX+XfY−YfX,for any vector field X,Y on M, where f=−λS+1/k−1.Proof.
Tracing (1), we obtain
(11)Δgλ=−λS+kk−1.Then, the Miao-Tam equation (1) can be exhibited as
(12)∇XDλ=λQX+fX,for any vector field X on M, where f=−λS+1/k−1. Taking the covariant derivative of (12) along an arbitrary vector field Y on M, we obtain
(13)∇Y∇XDλ=YλQX+λ∇YQX+λQ∇YX+YfX+f∇YX.Similarly, we have(14)∇X∇YDλ=XλQY+λ∇XQY+λQ∇XY+XfY+f∇XY,for any vector field X,Y on M. Comparing the preceding two equations and using (12) in the well-known expression of the curvature tensor RX,Y=∇X,∇Y−∇X,Y, we obtain the result.Lemma 6.
LetM2n+1φ,ξ,η,g be a paracontact metric κ,μ-manifold of dimensional 2n+1 with κ>−1, and g,λ be a nonconstant solution of the Miao-Tam equation on M2n+1. Then, we have
(15)μn+κ+1=2κ.Proof.
Firstly, taking covariant derivative of (8) along any vector field X, and using (4), we can obtain
(16)∇XQξ=QφX−φhX−2nκφX−φhX.Taking the inner product of (10) with ξ and using (8) and (16), we have
(17)gRX,YDλ,ξ=2nκXληY−YληX+λgQφX−φhX,Y−λgQφY−φhY,X+4λnκgφY,X+XfηY−YfηX,where f=−λS+1/2n (noting that the dimension of M is 2n+1).It follows from (6) that RφX,φYξ=0. Then, replacing X by φX and Y by φY in (17), respectively, we obtain
(18)λQφ+φQ+φQh+hQφ−4nκφX=0.Sinceλ is nonconstant on M, it is easy to see that
(19)Qφ+φQ+φQh+hQφ−4nκφX=0.ReplacingX by φX in (9), we have
(20)QφX=21−n+nμφX+2n−1+μhφX.Then, the action ofh on the (20) gives
(21)hQφX=21−n+nμhφX+1+κ2n−1+μφX,where we have used (7).Operating (9) by φ, we have
(22)φQX=21−n+nμφX+2n−1+μφhX.ReplacingX by hX in (22) and using (7) again, we get
(23)φQhX=21−n+nμφhX+1+κ2n−1+μφX.Substituting equations (20)-(23) into (19) yields
(24)μn+κ+1=2κ,which completes the proof of Lemma 6.Next, we will give the complete proof of Theorem2.Proof.
Firstly, takingX=ξ in (17) gives
(25)gRξ,Yξ,Dλ=gκηYξ−Y−μhY,Dλ=κξληY−κYλ−μhYλ.PuttingX=ξ in (6) and comparing with the forgoing equation, we obtain
(26)κDλ+μhDλ−2nκξλξ−Dλ−κξλ+ξfξ+Df=0.Noting that the scalar curvatureS is a constant, it follows from f=−λS+1/2n that
(27)2nDf=−SDλ.Then, we can obtain from (26) and (27) that
(28)2nκDλ+2nμhDλ−4n2κξλξ−Dλ−2nκξλ+ξfξ−SDλ=0.On the one hand, takingY=ξ in (6), since hξ=0, it follows that
(29)RX,ξξ=κX−ηXξ+μhX−ηXhξ=κφ2X+μhX,which gives
(30)l=κφ2+μh.Substituting (7) and (30) in (5), we get
(31)∇ξh=−μφh=μhφ.On the other hand, we obtain from (12) and (8) that
(32)∇ξDλ=2nκλ+fξ.Next, taking covariant derivative of (28) along ξ and making use of (31) and (32), we have
(33)2nκ+4n2κ−S2nκλ+fξ+2nμ2hφDλ−4n2κξξλξ−2nκξξλξ−2nξξfξ=0.Operating this equation byφ shows
(34)2nμ2hDλ=0.By the action ofh in (34), it follows from (7) that
(35)μ2κ+1φ2Dλ=0.Since we assume thatκ>−1, we divide it into two cases:Case (i):μ=0; case (ii): φ2Dλ=0.If case (i) occurs, it follows from Lemma6 that κ=0. Hence, the definition of paracontact metric κ,μ-manifold gives that RX,Yξ=0 for any vector field X,Y. From Theorem 3.3 of [26], M2n+1 is locally flat in dimension 3, and in higher dimensions (n>1), it is locally isometric to the product of a flat n+1-dimensional manifold and an n-dimensional manifold of negative constant curvature −4.If case (ii) occurs, thenφ2Dλ=Dλ−ξλξ=0, i.e., Dλ=ξλξ. Differentiating this along an arbitrary vector field X together with (4) implies that
(36)∇XDλ=Xξλξ−ξλφX−φhX.It follows from (12) that g∇XDλ,Y=g∇YDλ,X, and then the foregoing equation shows that
(37)XξληY−YξληX−ξλdηX,Y=0.ReplacingX by φX, Y by φY, and noting that dη is nonzero for any paracontact metric manifolds, it follows that ξλ=0. Hence, Dλ=0, λ, is a constant, which gives a contradiction.This completes the proof of Theorem2.



---

*Source: 6687223-2021-04-20.xml*