- Email: [email protected]
A compact gradient generalized quasi-Einstein metric with constant scalar curvature
J. Math. Anal. Appl. 401 (2013) 702–705
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
A compact gradient generalized quasi-Einstein metric with constant scalar curvature A. Barros a,∗ , J.N. Gomes a,b a
Departamento de Matemática-UFC, 60455-760-Fortaleza-CE-BR, Brazil
b
Departamento de Matemática-UFAM, 69077-070-Manaus-AM-BR, Brazil1
article
info
Article history: Received 27 June 2012 Available online 3 January 2013 Submitted by Jie Xiao
abstract In this paper we shall show that a compact gradient generalized m-quasi-Einstein metric (M n , g , ∇ f , λ) with constant scalar curvature must be isometric to a standard Euclidean sphere Sn with the potential f well determined. © 2013 Elsevier Inc. All rights reserved.
Keywords: Ricci soliton Quasi-Einstein metrics Bakry–Emery Ricci tensor Scalar curvature
1. Introduction and statement of the main theorem A gradient generalized m-quasi-Einstein metric on a complete Riemannian manifold (M n , g ) is a choice of a potential function f : M n → R as well as a function λ : M n → R such that Ric + ∇ 2 f −
1 m
df ⊗ df = λg ,
(1.1)
where Ric denotes the Ricci tensor of (M n , g ), while 0 < m ≤ ∞ is an integer, and ∇ 2 and ⊗ stand for the Hessian and the tensorial product, respectively. 1 The tensor Ricf = Ric +∇ 2 f − m df ⊗ df is called the Bakry–Emery Ricci tensor. Therefore, such structure on a Riemannian manifold has some similarity with the Einstein metric, since we must have Ricf = λg. But in general we do not have a Schurtype lemma, which gives that λ must be constant provided n ≥ 3. Moreover, if R stands for the scalar curvature of (M n , g ), then, taking the traces of the two members of Eq. (1.1), we deduce R + ∆f −
1 m
|∇ f |2 = nλ.
(1.2)
It is important to point out that if m = ∞ and λ is constant, Eq. (1.1) reduces to a form associated with a gradient Ricci soliton; for a good survey of this subject we recommend the work due to Cao in [5]. Also, considering m = ∞ and λ not constant, we obtain the almost Ricci soliton equation; for more details see [11] and [1]. In addition, if λ is constant and m
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (A. Barros), [email protected] (J.N. Gomes). URL: http://www.mat.ufc.br/pgmat (J.N. Gomes).
1 Permanent address. 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.12.068
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
703
is a positive integer, it corresponds to m-quasi-Einstein metrics that are exactly those n-dimensional manifolds which are the base of an (n + m)-dimensional Einstein warped product; for more details see [6,7,9,2]. In [9], some classifications were given for m-quasi-Einstein metrics where the base has non-empty boundary. Moreover, they proved a characterization for m-quasi-Einstein metrics when the base is locally conformally flat. We also pointout that Catino [8] have proved that around any regular point of f , a generalized m-quasi-Einstein metric M n , g , ∇ f , λ with harmonic Weyl tensor and W (∇ f , . . . , ∇ f ) = 0 is locally a warped product with (n − 1)-dimensional Einstein fibers. A generalized m-quasi-Einstein metric M n , g , ∇ f , λ will be called trivial if the potential function f is constant. Otherwise, it will be called nontrivial. We observe that the triviality definition implies that M n is an Einstein manifold, but the converse is not true. Introducing f
u ∇ f ; moreover the following relations, which can be found the function u = e− m on M n , we immediately have ∇ u = − m in [6], are also true:
m u
∇ 2 u = −∇ 2 f +
1 m
df ⊗ df
(1.3)
and u (R − nλ). m In particular, Eq. (1.1) becomes
∆u =
(1.4)
m
∇ 2 u = λg . u Now we recall that the traceless tensor associated with a tensor T is defined by Ric −
tr T
g. n Therefore, from (1.4) and (1.5) we can easily check the following identity: T˚ = T −
˚ = Ric
m u
∇˚ 2 u.
(1.5)
(1.6)
(1.7)
Hence, we deduce that ∇ u is a conformal vector field, i.e. 12 L∇ u g = ρ g, for some smooth function ρ defined on M n , if and only if M n is an Einstein manifold, where L stands for the Lie derivative. On the other hand, in [3] a complete description of gradient generalized m-quasi-Einstein metrics which are also Einstein metrics was given. Before presenting our main result we recall that a height function hv on Sn is defined by hv (x) = ⟨x, v⟩, where v is some fixed unit vector v ∈ Sn ⊂ Rn+1 ; here we are considering Sn as a hypersurface in Rn+1 . Now we are in position to announce our main theorem. Theorem 1. Let M n , g , ∇ f , λ be a nontrivial compact generalized m-quasi-Einstein metric with n ≥ 3. In addition, suppose n n that L∇ u R ≥ 0. Then R is constant and M is isometric to a standard sphere S (r ). Moreover, up tonrescaling and a constant, hv f = −m ln τ − n , where τ is a real parameter lying in (1/n, +∞) and hv is a height function on S .
2. Preliminaries In this section we shall show a main lemma which will be crucial for our goal. Firstly, we recall that the divergence of a
(1, r )-tensor T on M n is the (0, r )-tensor given by (div T )(v1 , . . . , vr )(p) = tr w → (∇w T )(v1 , . . . , vr )(p) , where p ∈ M n and (v1 , . . . , vr ) ∈ Tp M × · · · × Tp M . On the other hand, if T is a (0, 2)-tensor on a Riemannian manifold (M n , g ), then we can associate with T a unique (1, 1)-tensor, which will also be indicated by T , according to g (T (Z ), Y ) := T (Z , Y ), (2.1) for all Y , Z ∈ X(M ). In addition, if T is symmetric, choosing an orthonormal frame {e1 , . . . , en } on (M n , g ), we have for each Z ∈ X(M ), (div T )(Z ) = g ( (∇ei T )(Z ), ei ) i
=
g ( ∇ei T (Z ) − T (∇ei Z ), ei ),
i
i.e.,
(div T )(Z ) = div(T (Z )) −
g (∇ei Z , Tei ).
i
Before deriving our main lemma, we present the following one.
(2.2)
704
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
Lemma 1. Let T be a symmetric (0, 2)-tensor on a Riemannian manifold (M n , g ) and ϕ a smooth function on M n . Then we have div(T (ϕ Z )) = ϕ(div T )(Z ) + ϕ⟨∇ Z , T ⟩ + T (∇ϕ, Z ), for each Z ∈ X(M ). Proof. Let {e1 , . . . , en } be an orthonormal frame on (M n , g ). Since T is symmetric we can use (2.1) and (2.2) to deduce div(T (ϕ Z )) = div(ϕ T (Z )) = ϕ div(T (Z )) + g ( ∇ϕ, T (Z ) )
= ϕ(div T )(Z ) + ϕ
g ( ∇ei Z , T (ei ) ) + T (∇ϕ, Z )
i
= ϕ(div T )(Z ) + ϕ⟨∇ Z , T ⟩ + T (∇ϕ, Z ), for each Z ∈ X(M ) and any smooth function ϕ on M n , which was what we wanted to prove.
We obtain our main lemma as a consequence of the previous lemma: Lemma 2. Let M n , g , ∇ f , λ be a generalized m-quasi-Einstein metric. Then we have:
n −2 ˚ (1) div Ric L∇ u R + (∇ u) = n2n ˚ (∇ u) = −2 L∇ u R + (2) div Ric 2n
u m m u
2 Ric ˚ 2 .2 ∇˚ u| .
˚ ϕ = 1 and Z = ∇ u and we use Lemma 1 to infer Proof. We choose T = Ric,
R 2 div Ric − g (∇ u) + ∇ u, Ric − g .
˚ (∇ u) = div Ric
R n
(2.3)
n
Next we use the second contracted Bianchi identity to obtain div
Ric −
R
g
n
(∇ u) =
n−2 2n
⟨∇ R, ∇ u⟩.
(2.4)
On the other hand, since ⟨g , Ric − Rn g ⟩ = 0 we use (1.5) to write
R
u
∇ u, Ric − g = 2
n
m
˚ |2 . |Ric
(2.5)
Now we compare identities (2.3)–(2.5) to deduce ˚ (∇ u) = div Ric
n−2 2n
⟨∇ R, ∇ u⟩ +
u m
˚ |2 , |Ric
(2.6)
which gives the first statement. Finally, it suffices to use (1.7) and the first statement in order to complete the proof of the lemma. We deduce the following proposition as a consequence of this lemma. Proposition 1. Let M n , g , ∇ f , λ be a generalized m-quasi-Einstein metric with n ≥ 3. Then we have:
˚ (∇ u) lies in L1 (M n ), then ∇ u is a conformal vector field. In particular, (M n , g ) is Einstein. (1) If L∇ u R ≥ 0 and Ric 2 n ˚ dν = − n−2 (2) If M is compact, then n u Ric n L∇ u R dν. M
m
2n
M
˚ (∇ u) ≥ 0. Now we use Proposition Proof. First we suppose that L∇ u R ≥ 0. Then the first item of Lemma 2 gives that div Ric ˚ (∇ u) = 0. Hence, going back to the first item of Lemma 2 we conclude that L∇ u R = 0 and 1 of [4] to deduce that div Ric ˚ = 0. Hence we use (1.7) to deduce that ∇ u is a conformal vector field, which gives the first statement. On integrating Ric the first item of Lemma 2 we use Stokes’s formula to arrive at the desired result. Hence we complete the proof of the proposition.
2.1. Proof of Theorem 1 Proof. Since L∇ u R ≥ 0 and M n is compact we can use one of the statements of Proposition 1 to deduce that ∇ u is a nontrivial conformal vector field, which gives that (M n , g ) is Einstein. Now we are in position to apply the result obtained in Theorem 1 of [3] to conclude that (M n , g ) is isometric to a standard sphere Sn (r ); moreover the potential f is given in accordance with the theorem, so we finish the proof of our theorem. For the sake of completeness we shall present a brief sketch of the R ∆u = 0; see e.g. Eq. (1.25) of [10]. Rescaling the metric we last claim. Indeed, since ∇ 2 u = ∆nu g we must have ∆(∆u) + n− 1 can assume that R = n(n − 1). Then we conclude that ∆u is a first eigenvalue of Sn . Hence, there exists a fixed vector v ∈ Sn such that ∆u = hv = − 1n ∆hv . Thus we have ∆(u + 1n hv ) = 0, which gives u = τ − 1n hv and this completes our sketch.
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
705
3. Concluding remarks Regarding the hyperbolic space Hn (−1) ⊂ Rn,1 : ⟨x, x⟩0 = −1, x1 > 0, where Rn,1 is the Euclidean space Rn+1 endowed with the inner product ⟨x, x⟩0 = −x21 + x22 + · · · + x2n+1 , we fixed a vector v ∈ Hn (−1) ⊂ Rn,1 and we consider a height function hv : Hn (−1) → R which is given by hv (x) = ⟨x, v⟩0 . Then we can use the same technique as was utilized in [3] to obtain the following result. Theorem 2. Let M n , g , ∇ f , λ be a nontrivial noncompact generalized m-quasi-Einstein metric with n ≥ 3. In addition, suppose ˚ (∇ u) lies in L1 (M n ). Then we have that R is constant. Moreover: that L∇ u R ≥ 0 and div Ric
(1) If R = 0, then M n is isometric to a Euclidean space (Rn , g0 ), and f = −m ln τ + |x|2 , where τ is a positive real parameter and |x| is the Euclidean norm. n (2) If R < 0, then M n is isometric to a hyperbolic space H , provided u has only one critical point. Moreover, f is, up to a constant, given by f = −m ln τ + hv , τ > −1.
Acknowledgment The first and second authors were partially supported by CNPq-BR. References [1] A. Barros, E. Ribeiro Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (2012) 1033–1040. [2] A. Barros, E. Ribeiro Jr., Integral formulae on quasi-Einstein manifolds and applications, Glasg. Math. J. 54 (2012) 213–223. [3] A. Barros, E. Ribeiro Jr., Characterizations and integral formulae for generalized quasi-Einstein metrics, Bull. Braz. Math. Soc., 2012 (in press) arXiv:1206.4980v1 [math.DG]. [4] A. Caminha, F. Camargo, P. Souza, Complete foliations of space forms by hypersurfaces, Bull. Braz. Math. Soc. 41 (2010) 339–353. [5] H.D. Cao, Recent progress on Ricci soliton, Adv. Lect. Math. (ALM) 11 (2009) 1–38. [6] J. Case, Y. Shu, G. Wei, Rigidity of quasi-Einstein metrics, Differ. Geom. Appl. 29 (2011) 93–100. [7] J. Case, On the nonexistence of quasi-Einstein metrics, Pacific J. Math. 248 (2010) 227–284. [8] G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012) 751–756. http://dx.doi.org/10.1007/s00209-0110888-5. [9] C. He, P. Petersen, W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012) 271–312. arXiv:1010.5488v2 [math.DG]. [10] M. Obata, K. Yano, Conformal changes of Riemannian metrics, J. Diff. Geom. 4 (1970) 53–72. [11] S. Pigola, M. Rigoli, M. Rimoldi, A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10 (2011) 757–799.
Contents lists available at SciVerse ScienceDirect
Journal of Mathematical Analysis and Applications journal homepage: www.elsevier.com/locate/jmaa
A compact gradient generalized quasi-Einstein metric with constant scalar curvature A. Barros a,∗ , J.N. Gomes a,b a
Departamento de Matemática-UFC, 60455-760-Fortaleza-CE-BR, Brazil
b
Departamento de Matemática-UFAM, 69077-070-Manaus-AM-BR, Brazil1
article
info
Article history: Received 27 June 2012 Available online 3 January 2013 Submitted by Jie Xiao
abstract In this paper we shall show that a compact gradient generalized m-quasi-Einstein metric (M n , g , ∇ f , λ) with constant scalar curvature must be isometric to a standard Euclidean sphere Sn with the potential f well determined. © 2013 Elsevier Inc. All rights reserved.
Keywords: Ricci soliton Quasi-Einstein metrics Bakry–Emery Ricci tensor Scalar curvature
1. Introduction and statement of the main theorem A gradient generalized m-quasi-Einstein metric on a complete Riemannian manifold (M n , g ) is a choice of a potential function f : M n → R as well as a function λ : M n → R such that Ric + ∇ 2 f −
1 m
df ⊗ df = λg ,
(1.1)
where Ric denotes the Ricci tensor of (M n , g ), while 0 < m ≤ ∞ is an integer, and ∇ 2 and ⊗ stand for the Hessian and the tensorial product, respectively. 1 The tensor Ricf = Ric +∇ 2 f − m df ⊗ df is called the Bakry–Emery Ricci tensor. Therefore, such structure on a Riemannian manifold has some similarity with the Einstein metric, since we must have Ricf = λg. But in general we do not have a Schurtype lemma, which gives that λ must be constant provided n ≥ 3. Moreover, if R stands for the scalar curvature of (M n , g ), then, taking the traces of the two members of Eq. (1.1), we deduce R + ∆f −
1 m
|∇ f |2 = nλ.
(1.2)
It is important to point out that if m = ∞ and λ is constant, Eq. (1.1) reduces to a form associated with a gradient Ricci soliton; for a good survey of this subject we recommend the work due to Cao in [5]. Also, considering m = ∞ and λ not constant, we obtain the almost Ricci soliton equation; for more details see [11] and [1]. In addition, if λ is constant and m
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (A. Barros), [email protected] (J.N. Gomes). URL: http://www.mat.ufc.br/pgmat (J.N. Gomes).
1 Permanent address. 0022-247X/$ – see front matter © 2013 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.12.068
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
703
is a positive integer, it corresponds to m-quasi-Einstein metrics that are exactly those n-dimensional manifolds which are the base of an (n + m)-dimensional Einstein warped product; for more details see [6,7,9,2]. In [9], some classifications were given for m-quasi-Einstein metrics where the base has non-empty boundary. Moreover, they proved a characterization for m-quasi-Einstein metrics when the base is locally conformally flat. We also pointout that Catino [8] have proved that around any regular point of f , a generalized m-quasi-Einstein metric M n , g , ∇ f , λ with harmonic Weyl tensor and W (∇ f , . . . , ∇ f ) = 0 is locally a warped product with (n − 1)-dimensional Einstein fibers. A generalized m-quasi-Einstein metric M n , g , ∇ f , λ will be called trivial if the potential function f is constant. Otherwise, it will be called nontrivial. We observe that the triviality definition implies that M n is an Einstein manifold, but the converse is not true. Introducing f
u ∇ f ; moreover the following relations, which can be found the function u = e− m on M n , we immediately have ∇ u = − m in [6], are also true:
m u
∇ 2 u = −∇ 2 f +
1 m
df ⊗ df
(1.3)
and u (R − nλ). m In particular, Eq. (1.1) becomes
∆u =
(1.4)
m
∇ 2 u = λg . u Now we recall that the traceless tensor associated with a tensor T is defined by Ric −
tr T
g. n Therefore, from (1.4) and (1.5) we can easily check the following identity: T˚ = T −
˚ = Ric
m u
∇˚ 2 u.
(1.5)
(1.6)
(1.7)
Hence, we deduce that ∇ u is a conformal vector field, i.e. 12 L∇ u g = ρ g, for some smooth function ρ defined on M n , if and only if M n is an Einstein manifold, where L stands for the Lie derivative. On the other hand, in [3] a complete description of gradient generalized m-quasi-Einstein metrics which are also Einstein metrics was given. Before presenting our main result we recall that a height function hv on Sn is defined by hv (x) = ⟨x, v⟩, where v is some fixed unit vector v ∈ Sn ⊂ Rn+1 ; here we are considering Sn as a hypersurface in Rn+1 . Now we are in position to announce our main theorem. Theorem 1. Let M n , g , ∇ f , λ be a nontrivial compact generalized m-quasi-Einstein metric with n ≥ 3. In addition, suppose n n that L∇ u R ≥ 0. Then R is constant and M is isometric to a standard sphere S (r ). Moreover, up tonrescaling and a constant, hv f = −m ln τ − n , where τ is a real parameter lying in (1/n, +∞) and hv is a height function on S .
2. Preliminaries In this section we shall show a main lemma which will be crucial for our goal. Firstly, we recall that the divergence of a
(1, r )-tensor T on M n is the (0, r )-tensor given by (div T )(v1 , . . . , vr )(p) = tr w → (∇w T )(v1 , . . . , vr )(p) , where p ∈ M n and (v1 , . . . , vr ) ∈ Tp M × · · · × Tp M . On the other hand, if T is a (0, 2)-tensor on a Riemannian manifold (M n , g ), then we can associate with T a unique (1, 1)-tensor, which will also be indicated by T , according to g (T (Z ), Y ) := T (Z , Y ), (2.1) for all Y , Z ∈ X(M ). In addition, if T is symmetric, choosing an orthonormal frame {e1 , . . . , en } on (M n , g ), we have for each Z ∈ X(M ), (div T )(Z ) = g ( (∇ei T )(Z ), ei ) i
=
g ( ∇ei T (Z ) − T (∇ei Z ), ei ),
i
i.e.,
(div T )(Z ) = div(T (Z )) −
g (∇ei Z , Tei ).
i
Before deriving our main lemma, we present the following one.
(2.2)
704
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
Lemma 1. Let T be a symmetric (0, 2)-tensor on a Riemannian manifold (M n , g ) and ϕ a smooth function on M n . Then we have div(T (ϕ Z )) = ϕ(div T )(Z ) + ϕ⟨∇ Z , T ⟩ + T (∇ϕ, Z ), for each Z ∈ X(M ). Proof. Let {e1 , . . . , en } be an orthonormal frame on (M n , g ). Since T is symmetric we can use (2.1) and (2.2) to deduce div(T (ϕ Z )) = div(ϕ T (Z )) = ϕ div(T (Z )) + g ( ∇ϕ, T (Z ) )
= ϕ(div T )(Z ) + ϕ
g ( ∇ei Z , T (ei ) ) + T (∇ϕ, Z )
i
= ϕ(div T )(Z ) + ϕ⟨∇ Z , T ⟩ + T (∇ϕ, Z ), for each Z ∈ X(M ) and any smooth function ϕ on M n , which was what we wanted to prove.
We obtain our main lemma as a consequence of the previous lemma: Lemma 2. Let M n , g , ∇ f , λ be a generalized m-quasi-Einstein metric. Then we have:
n −2 ˚ (1) div Ric L∇ u R + (∇ u) = n2n ˚ (∇ u) = −2 L∇ u R + (2) div Ric 2n
u m m u
2 Ric ˚ 2 .2 ∇˚ u| .
˚ ϕ = 1 and Z = ∇ u and we use Lemma 1 to infer Proof. We choose T = Ric,
R 2 div Ric − g (∇ u) + ∇ u, Ric − g .
˚ (∇ u) = div Ric
R n
(2.3)
n
Next we use the second contracted Bianchi identity to obtain div
Ric −
R
g
n
(∇ u) =
n−2 2n
⟨∇ R, ∇ u⟩.
(2.4)
On the other hand, since ⟨g , Ric − Rn g ⟩ = 0 we use (1.5) to write
R
u
∇ u, Ric − g = 2
n
m
˚ |2 . |Ric
(2.5)
Now we compare identities (2.3)–(2.5) to deduce ˚ (∇ u) = div Ric
n−2 2n
⟨∇ R, ∇ u⟩ +
u m
˚ |2 , |Ric
(2.6)
which gives the first statement. Finally, it suffices to use (1.7) and the first statement in order to complete the proof of the lemma. We deduce the following proposition as a consequence of this lemma. Proposition 1. Let M n , g , ∇ f , λ be a generalized m-quasi-Einstein metric with n ≥ 3. Then we have:
˚ (∇ u) lies in L1 (M n ), then ∇ u is a conformal vector field. In particular, (M n , g ) is Einstein. (1) If L∇ u R ≥ 0 and Ric 2 n ˚ dν = − n−2 (2) If M is compact, then n u Ric n L∇ u R dν. M
m
2n
M
˚ (∇ u) ≥ 0. Now we use Proposition Proof. First we suppose that L∇ u R ≥ 0. Then the first item of Lemma 2 gives that div Ric ˚ (∇ u) = 0. Hence, going back to the first item of Lemma 2 we conclude that L∇ u R = 0 and 1 of [4] to deduce that div Ric ˚ = 0. Hence we use (1.7) to deduce that ∇ u is a conformal vector field, which gives the first statement. On integrating Ric the first item of Lemma 2 we use Stokes’s formula to arrive at the desired result. Hence we complete the proof of the proposition.
2.1. Proof of Theorem 1 Proof. Since L∇ u R ≥ 0 and M n is compact we can use one of the statements of Proposition 1 to deduce that ∇ u is a nontrivial conformal vector field, which gives that (M n , g ) is Einstein. Now we are in position to apply the result obtained in Theorem 1 of [3] to conclude that (M n , g ) is isometric to a standard sphere Sn (r ); moreover the potential f is given in accordance with the theorem, so we finish the proof of our theorem. For the sake of completeness we shall present a brief sketch of the R ∆u = 0; see e.g. Eq. (1.25) of [10]. Rescaling the metric we last claim. Indeed, since ∇ 2 u = ∆nu g we must have ∆(∆u) + n− 1 can assume that R = n(n − 1). Then we conclude that ∆u is a first eigenvalue of Sn . Hence, there exists a fixed vector v ∈ Sn such that ∆u = hv = − 1n ∆hv . Thus we have ∆(u + 1n hv ) = 0, which gives u = τ − 1n hv and this completes our sketch.
A. Barros, J.N. Gomes / J. Math. Anal. Appl. 401 (2013) 702–705
705
3. Concluding remarks Regarding the hyperbolic space Hn (−1) ⊂ Rn,1 : ⟨x, x⟩0 = −1, x1 > 0, where Rn,1 is the Euclidean space Rn+1 endowed with the inner product ⟨x, x⟩0 = −x21 + x22 + · · · + x2n+1 , we fixed a vector v ∈ Hn (−1) ⊂ Rn,1 and we consider a height function hv : Hn (−1) → R which is given by hv (x) = ⟨x, v⟩0 . Then we can use the same technique as was utilized in [3] to obtain the following result. Theorem 2. Let M n , g , ∇ f , λ be a nontrivial noncompact generalized m-quasi-Einstein metric with n ≥ 3. In addition, suppose ˚ (∇ u) lies in L1 (M n ). Then we have that R is constant. Moreover: that L∇ u R ≥ 0 and div Ric
(1) If R = 0, then M n is isometric to a Euclidean space (Rn , g0 ), and f = −m ln τ + |x|2 , where τ is a positive real parameter and |x| is the Euclidean norm. n (2) If R < 0, then M n is isometric to a hyperbolic space H , provided u has only one critical point. Moreover, f is, up to a constant, given by f = −m ln τ + hv , τ > −1.
Acknowledgment The first and second authors were partially supported by CNPq-BR. References [1] A. Barros, E. Ribeiro Jr., Some characterizations for compact almost Ricci solitons, Proc. Amer. Math. Soc. 140 (2012) 1033–1040. [2] A. Barros, E. Ribeiro Jr., Integral formulae on quasi-Einstein manifolds and applications, Glasg. Math. J. 54 (2012) 213–223. [3] A. Barros, E. Ribeiro Jr., Characterizations and integral formulae for generalized quasi-Einstein metrics, Bull. Braz. Math. Soc., 2012 (in press) arXiv:1206.4980v1 [math.DG]. [4] A. Caminha, F. Camargo, P. Souza, Complete foliations of space forms by hypersurfaces, Bull. Braz. Math. Soc. 41 (2010) 339–353. [5] H.D. Cao, Recent progress on Ricci soliton, Adv. Lect. Math. (ALM) 11 (2009) 1–38. [6] J. Case, Y. Shu, G. Wei, Rigidity of quasi-Einstein metrics, Differ. Geom. Appl. 29 (2011) 93–100. [7] J. Case, On the nonexistence of quasi-Einstein metrics, Pacific J. Math. 248 (2010) 227–284. [8] G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012) 751–756. http://dx.doi.org/10.1007/s00209-0110888-5. [9] C. He, P. Petersen, W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012) 271–312. arXiv:1010.5488v2 [math.DG]. [10] M. Obata, K. Yano, Conformal changes of Riemannian metrics, J. Diff. Geom. 4 (1970) 53–72. [11] S. Pigola, M. Rigoli, M. Rimoldi, A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. 10 (2011) 757–799.