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On generalized m-quasi-Einstein manifolds with constant scalar curvature
Accepted Manuscript On generalized m-quasi-Einstein manifolds with constant scalar curvature
Zejun Hu, Dehe Li, Jing Xu
PII: DOI: Reference:
S0022-247X(15)00656-3 http://dx.doi.org/10.1016/j.jmaa.2015.07.021 YJMAA 19646
To appear in:
Journal of Mathematical Analysis and Applications
Received date:
8 February 2015
Please cite this article in press as: Z. Hu et al., On generalized m-quasi-Einstein manifolds with constant scalar curvature, J. Math. Anal. Appl. (2015), http://dx.doi.org/10.1016/j.jmaa.2015.07.021
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ON GENERALIZED m-QUASI-EINSTEIN MANIFOLDS WITH CONSTANT SCALAR CURVATURE ZEJUN HU, DEHE LI, AND JING XU Abstract. We study generalized m-quasi-Einstein manifolds with constant scalar curvature. First, we establish a classification of generalized m-quasiEinstein manifolds with parallel Ricci tensor. Second, and particularly, if an mquasi-Einstein manifold possesses constant scalar curvature then we determine the scalar curvature in explicit form and show that which values are attainable by presenting examples.
1. Introduction Because of their importance in both mathematics and physics, the study of the Einstein manifolds and their various generalizations is always an attractive topic in modern Riemannian geometry. In recent years, there has been increasing interest on so-called quasi-Einstein manifolds. Recall that, for a positive integer m, a complete Riemannian manifold (M n , g, f ) with a potential function f is called m-quasi-Einstein if its associated m-Bakry-Emery Ricci tensor Ricfm := Ric + ∇2 f −
1 m df
⊗ df
is a constant multiple of the metric g (cf. [12, 17] and the references therein). The notion of m-quasi-Einstein manifolds is natural and interesting partially due to that, according to [19], an n-dimensional m-quasi-Einstein manifold is exactly the manifold which is the base of an (n + m)-dimensional Einstein warped product. Moreover, the 1-quasi-Einstein metrics that satisfies Δe−f + λe−f = 0 are more commonly called static metrics. For more details we refer to [15] and, among others, [1, 2, 3] for later development. To extend the notion of m-quasi-Einstein, Catino [13] introduced the concept of generalized quasi-Einstein manifold. More precisely, an n-dimensional (n ≥ 3) complete Riemannian manifold (M n , g) is called generalized quasi-Einstein if there exist three smooth functions f, μ, λ on M , such that the Ricci tensor Ric of (M n , g) satisfies the relation (1.1)
Ric + ∇2 f − μdf ⊗ df = λg,
where ∇2 and ⊗ denote the Hessian and the tensorial product, respectively. In 1 and m is a positive integer, (1.1) turns to be particular, when μ = m (1.2)
Ric + ∇2 f −
1 m df
⊗ df = λg.
Key words and phrases. Generalized m-quasi-Einstein manifold, m-quasi-Einstein manifold, parallel Ricci tensor, constant scalar curvature, Bakry-Emery Ricci tensor. 2010 Mathematics Subject Classification. Primary 53C21; Secondary 53C24, 53C25. This project was supported by grants of NSFC-11371330. 1
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ZEJUN HU, DEHE LI, AND JING XU
A manifold that satisfies (1.2) is called generalized m-quasi-Einstein, and is usually denoted by (M n , g, f, λ). Obviously, if λ is a constant, the generalized m-quasiEinstein manifold (M n , g, f, λ) becomes m-quasi-Einstein. It is noteworthy to point out that if m approaches infinite and λ is a constant, (1.2) becomes the equation satisfied by the gradient Ricci solitons. As is wellknown that gradient Ricci solitons play an important role in Hamilton’s Ricci flow as they correspond to the self-similar solutions. As an important aspect in the treatment of the Ricci flow, subject of gradient Ricci solitons has been treated by many authors (cf. [16, 21] etc.), we refer to Cao [9] and [10] for complete surveys up to that time. In addition, if in (1.2), λ = ρR + μ, where R is the scalar curvature of (M n , g) and ρ, μ are two real constants, then (M n , g) becomes an (m, ρ)-quasiEinstein manifold as defined in [18]. In particular, (∞, ρ)-quasi-Einstein manifold is called the gradient ρ-Einstein soliton that plays an important role in the so-called ρ-Einstein flows (cf. [14]). A generalized m-quasi-Einstein manifold (M n , g, f, λ) will be called trivial if the potential function f is constant. Otherwise, it will be called nontrivial. Obviously, the definition of triviality implies that (M n , g) is an Einstein manifold, but generally the converse is not true. In [8], there are some special examples which are Einstein, but as generalized m-quasi-Einstein manifolds they are not trivial. So far much research has been done on the characteristic functions on manifolds since they can provide much geometric information. A manifold can be well characterized if there exists a smooth function satisfying a special equation on it. For example, the well-known geometric and analytical characterizations obtained by Obata [20] and Tashiro [24], respectively. Now, a natural and interesting problem is to characterize a manifold for which (1.2) is satisfied under appropriate curvature conditions. Indeed, such characterizations of nontrivial m-quasi-Einstein manifolds (resp. generalized m-quasi-Einstein manifolds) have appeared in [12] (resp. [8]) under the condition that the manifold is Einstein. When m approaches infinite and λ is a function, there are also results about the so-called Ricci almost solitons under some curvature conditions in [5, 7, 11, 22]. The purpose of this paper is the study of generalized m-quasi-Einstein manifolds. Firstly, we assume that the Ricci tensor of the manifold is parallel. As the result we obtain a complete classification of such generalized m-quasi-Einstein manifolds. It turns out that the set of these manifolds includes not only space forms but also the Riemannian products of only two typical Einstein manifolds. Here is our first main result: Theorem 1.1. Let (M n , g, f, λ) be a complete n-dimensional (n ≥ 3) non-trivial generalized m-quasi-Einstein manifold which possesses parallel Ricci tensor. Then (M n , g) is isometric to one of the following manifolds: (1) (2) (3) (4) (5)
a space form, Dnc , R × N n−1 (b), Hp (a) × N n−p (b), b = m+p−1 p−1 a, Dpc × N n−p (b), b = (1 − m − p)c2 ,
where a, b are negative constants, N k (b) denotes a k-dimensional Einstein manifold with scalar curvature kb, here we recall that traditionally one also calls b the Einstein constant of N k (b); Hp (a) denotes the p-dimensional hyperbolic space with Einstein
3
constant a; Dkc denotes a k-dimensional Einstein warped product R ×c−1 ecr F k−1 , i.e. R × F k−1 endowed with the metric dr2 + (c−1 ecr )2 gF , c is a positive constant, F k−1 with metric gF is a Ricci flat manifold. Secondly, focusing on the m-quasi-Einstein manifolds, in section 3 we consider them under the additional condition that m is a positive integer and the manifold is of constant scalar curvature. We notice that if n = 1 and 2, a classification of m-quasi-Einstein manifolds already appeared in [4] (cf. also Remark 1.9 of [17]). In [12], one of the results states that for an n-dimensional m-quasi-Einstein manifold (M n , g, f, λ), if λ < 0 and the constant scalar curvature R is constant, then R n(n−1) is in fact bounded and satisfies nλ ≤ R ≤ m+n−1 λ. In this paper, continuing with this topic and motivated by a recent result of [16], we will show that under a suitable assumption the possible values of the constant scalar curvature are in fact quantified and they are expressed by the constant λ. The following theorem is our second main result. Theorem 1.2. Let (M n , g, f, λ) be a complete n-dimensional nontrivial m-quasiEinstein manifold with constant scalar curvature R, then λ ≤ 0. Moreover, (1) if m = 1, then R = (n − 1)λ, f (2) if m > 1, then λ < 0; and if moreover e− m attains its maximum or minimum at some point on M , then R ∈ { mn−(m−n)p−n λ | p = 1, 2, · · · , n}. m+p−1 Conversely, for each p ∈ {1, 2, · · · , n}, there exists nontrivial m-quasi-Einstein manifold with λ < 0 and constant scalar curvature, which satisfies R=
mn−(m−n)p−n λ. m+p−1
2. Generalized m-quasi-Einstein Manifolds with Parallel Ricci Tensor In this section, we deal with generalized m-quasi-Einstein manifolds with parallel Ricci tensor. For our purpose, we need the following result due to Case, Shu and Wei [12]. Proposition 2.1 (cf. Proposition 4.2 of [12]). A complete finite m-quasi-Einstein manifold (M n , g, f ) is Einstein if and only if f is constant or M is diffeomorphic f to Rn with the warped product structure (R ×c−1 ecr F n−1 , e− m = c−1 ecr ), or f
(Hn , dr2 + c−2 sinh2 (cr) gS n−1 , e− m = c−1 cosh(cr)), where F n−1 is Ricci flat, c is a positive constant. Proposition 2.1 can be extended to cover the case of generalized m-quasi-Einstein manifolds. To begin with, we assume that (M n , g, f, λ) is a nontrivial generalized m-quasi-Einstein manifold and (M n , g) is Einstein. Then, we recall that, under a suitable assumption on the critical points of e−f /m , Barros and Ribeiro [8] proved that (M n , g) is in fact a space form. Here, for a better understanding of such manifolds (by removing the assumption about the critical points of e−f /m ), we will prove the following result:
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ZEJUN HU, DEHE LI, AND JING XU
Proposition 2.2. Let (M n , g, f, λ) be a complete n-dimensional nontrivial generalized m-quasi-Einstein manifold (n ≥ 3). If (M n , g) is Einstein, then (M n , g) is either a space form or an Einstein manifold R ×c−1 ecr F n−1 , where c is a positive constant and F n−1 is Ricci flat. f
Proof. Firstly, we consider u = e− m . Then, according to Barros and Ribeiro (cf. Lemma 3 of [8]), the scalar curvature R satisfies (2.1)
R u+ ∇2 u = (− n(n−1)
c m )g,
where c is a constant. Moreover, from [8] we obtain that if R > 0 then (M n , g) is isometric to a standard sphere; and if R = 0, then (M n , g) is isometric to Rn . If R < 0, noting that (2.1) holds and u is positive, along with Tashiro’s proof of Theorem 2 in [24], we can show that (M n , g) is a hyperbolic space or a warped product R ×c−1 ecr F n−1 . In the later case, the fact that (M n , g) is Einstein implies that F is Ricci flat (see also Corollary 9.107 of [4]). Remark 2.1. Under more restricted condition than Proposition 2.2, if we assume that a nontrivial generalized m-quasi-Einstein manifold is compact and the scalar curvature is constant, then Barros and Gomes [6] showed that the manifold must be isometric to a standard Euclidean sphere. Before presenting the proof of Theorem 1.1 we first analyze in detail the examples that appear in Theorem 1.1. Example 1. As demonstrated by Examples 1, 2 and 3 in [8], a space form with an appropriate potential function f can be regarded as a nontrivial generalized mquasi-Einstein manifold, which is Einstein and is trivially of parallel Ricci tensor. Example 2. Consider the Riemannian product manifold M n = R × N n−1 (mk), here R is equipped with the metric gR = dt2 and N n−1 (mk) is an (n−1)-dimensional Einstein manifold with Einstein constant mk (k < 0). Obviously, M n is of parallel Ricci tensor. Moreover, on M n we consider the following function √ √ u(t, x) = Ce −kt or u(t, x) = C cosh( −kt), where x ∈ N n−1 and C is a positive constant. Noting that u (t) = −ku(t), by direct calculation we see that the function u(t, x) on M n satisfies Ric − mu−1 ∇2 u = mkg, where g denotes the product metric of M n = R × N n−1 (mk). This shows that, f for f defined by u = e− m , (M n , g, f, mk) is a nontrivial m-quasi-Einstein manifold with parallel Ricci tensor. Example 3. Consider the Riemannian product manifold M n = Hp ×N n−p (b), here Hp denotes the p-dimensional hyperbolic space with Einstein constant −(p − 1), N n−p (b) denotes an (n − p)-dimensional Einstein manifold with Einstein constant b = 1 − p − m. Obviously, M n is of parallel Ricci tensor. Moreover, we may as usual look Hp as a hypersurface of the Minkowski space Rp,1 : Hp = {x ∈ Rp,1 | x, x 1 = −1, x1 > 0},
5
where Rp,1 denotes the (p+1)-dimensional real vector space Rp+1 endowed with the Minkowski inner product x, x 1 = −x21 + x22 + · · · + x2p+1 . Then, given a constant element v ∈ Hp we can define a function hv : Hp → R by hv (x) := x, v 1 . Now we consider the following function u(x, y) = hv (x), x ∈ Hp , y ∈ N n−p (b) f
and define u = e− m . Then, similar to that in Example 3 of [8], we can show that on M n it holds Ric − mu−1 ∇2 u = (1 − p − m)g, where g is the standard product metric of M n = Hp × N n−p (b). Therefore, (M n , g, f, 1 − p − m) is a nontrivial m-quasi-Einstein manifold with parallel Ricci tensor. Proof of Theorem 1.1. Since the Ricci tensor of (M n , g, f, λ) is parallel, (M n , g) is locally either an Einstein manifold or the Riemannian product of a finite number of Einstein manifolds (cf. Theorem 1.100 of [4]). If (M n , g) is Einstein, from Proposition 2.2, (M n , g) is either a space form or an Einstein warped product R ×c−1 ecr F n−1 , where c is a positive constant and F n−1 is a Ricci flat manifold. If M n = M1 × M2 · · · × Ms , where s ≥ 2 and M1 , M2 , · · · , Ms are Einstein manifolds with different Einstein constants. Assume that dim Mi = mi and let {e1 , e2 , · · · , en } be a local orthonormal frame field of (M n , g) such that {ej }1≤j≤m1 are tangent to M1 , and {ej }i−1 mk +1≤j≤i−1 mk +mi k=1
k=1
are tangent to Mi for 2 ≤ i ≤ s. Let {ωi } be the dual coframe field of {ei } and {ωij } be the connection forms associated to {ei }, i.e., dωi = ωij ∧ ωj , ωij + ωji = 0. Then we have Rij := Ric(ei , ej ) = ci δij , where ci (i = 1, 2, · · · , n) are constants. For brevity, we denote [i] := {j | cj = ci }. f Considering the positive function u = e− m on M , then (1.2) can be rewritten as (2.2)
Ric −
m 2 u∇ u
Δu =
u m (R
= λg.
Taking trace of (2.2), we get (2.3)
− λn).
Here, because the Ricci tensor is parallel, the scalar curvature R is constant on M . The equation (2.2) gives muij = uRij − λuδij ,
(2.4) 2
where uij = ∇ u(ei , ej ). Using Rij = ci δij , we get 0, if i = j, (2.5) uij = ci −λ u, if i = j. m Calculating the covariant derivation of (2.4) yields muij,j = uj Rij + uRij,j − (λu)j δij .
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ZEJUN HU, DEHE LI, AND JING XU
Since uij = uji and Rij,j = 0, we have muji,j = uj Rij − (λu)j δij . The Ricci identity (see, e.g. (3.2) in [25]) then gives uk Rkjij = uj Rij − (λu)j δij , mujj,i + m k
m(Δu)i + m
(2.6)
uk Rki =
uj Rij − (λu)i .
j
k
Substituting (2.3) into (2.6), a direct calculation gives (2.7)
(λu)i =
R+(m−1)ci ui . n−1
Claim 1. For [i] = [j] and [α] = [i], it holds eα (uij ) = 0. In fact, if [i] = [j] and [α] = [i], from the product structure of M we obtain ωiα = 0, and it follows from (2.5) that uiα = 0. By definition of the covariant derivatives we get uiα,k ωk = duiα + uik ωkα + ukα ωki = 0, k
k
uij,k ωk = duij +
k
k
uik ωkj +
k∈[i]
ukj ωki .
k∈[i]
Therefore, we have uiα,j = 0 and uij,α = eα (uij ). On the other hand, the Ricci identity gives um Rmijα = uiα,j = 0. uij,α = uiα,j + m
This completes the proof of Claim 1. Claim 2. s = 2. From Claim 1, we have eα (uii ) = 0 for [α] = [i].
(2.8)
Inserting (2.5) into (2.8) we see that 0 = meα (uii ) = ci uα − (λu)α .
(2.9)
Substituting (2.7) into (2.9) we get the following relation (ci −
(2.10)
R+(m−1)cα )uα n−1
= 0 for [α] = [i].
n
Since (M , g, f, λ) is nontrivial, there exists an index j0 and a point p0 ∈ M such that uj0 |p0 = 0. Then (2.10) gives (2.11) This implies that
(ci −
R+(m−1)cj0 n−1
)uj0 |p0 = 0 for i ∈ [j0 ].
R+(m−1)c
j0 ci = , ∀i ∈ [j0 ] n−1 and therefore, we have s ≤ 2. From the assumption s ≥ 2, Claim 2 follows.
Next, without loss of generality, we may assume that j0 = 1 and M = M1 × M2 , where M1 is an m1 -dimensional Einstein manifold with Einstein constant a := c1 ; , M2 is an m2 -dimensional Einstein manifold with Einstein constant b = R+(m−1)a n−1 and a = b. In what follows we separate the discussion into two cases.
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Case I. There exists some β0 ∈ {m1 + 1, m1 + 2, · · · , m1 + m2 = n} and a point q0 ∈ M such that uβ0 (q0 ) = 0. Case II. For all β ∈ {m1 + 1, m1 + 2, · · · , m1 + m2 = n}, it holds uβ = 0. In Case I, from (2.10) and (2.11) we can solve b = R+(m−1)a , n−1 . a = R+(m−1)b n−1 It follows that
R a = b = n−m , a = b = 0,
if m = n, if m = n,
which contradicts the assumption a = b. Hence Case I does not occur. In Case II, the function u depends only on M1 . By definition we have uββ = eβ (uβ ) + uk ωkβ (eβ ). k∈[β]
It follows that uββ = 0 for all β ∈ {m1 + 1, m1 + 2, · · · , n}. Combining this with (2.5), we obtain 0 = muββ = (b − λ)u. Hence, since u is positive, we obtain that λ = b. As a function depends only on M1 , u satisfying (2.2) with λ being a constant implies that M1 is an Einstein and m-quasi-Einstein manifold with Einstein constant a. If m1 > 1, we can apply Proposition 2.1 to M1 to see that it is either a hyperbolic space Hm1 or an Einstein warped product R ×c−1 ecr F m1 −1 , where c is a positive constant and F m1 −1 is Ricci flat. From the proof of Proposition 4.2 in [12], we find m+m1 −1 m+m1 −1 that λ satisfies a = −(m1 −1) a−λ m . Thus λ = m1 −1 a < 0 and b = λ = m1 −1 a. m1 −1 , a direct calculation gives that its Einstein Moreover, if M1 = R ×c−1 ecr F constant a = −(m1 − 1)c2 , and hence b = (1 − m − m1 )c2 . If m1 = 1, then, according to Example 1 of [17], M1 is R with λ < 0 (cf. also 9.109 and 9.110 in [4]). To be specific, we assume that M1 is endowed with the metric dt2 , then (2.2) becomes u (t) = −(λ/m)u(t). Since u(t) should have√complete positivesolution, it must be that λ < 0 and the solution is u(t) = Ce −λ/mt or C cosh( −λ/mt), where C is a positive constant. Noting that u is not periodic, M 1 is not diffeomorphic to S1 . Thus M n is R × N n−1 (b), where b = λ is a negative constant. This completes the proof of Theorem 1.1. Remark 2.2. From the proof of Theorem 1.1, it is easily seen that in each case of (3), (4) and (5), M n is not Einstein, the potential function f does not depend on the factor N , and λ = b is a negative constant. Furthermore, in case (4) (resp. (5)), the scalar curvature R of (M n , g) is determined by the relation b = R+(m−1)a n−1 (resp. b =
R−(m−1)(p−1)c2 ). n−1
3. m-quasi-Einstein Manifolds with Constant Scalar Curvature In this section, we study m-quasi-Einstein manifolds with constant scalar curvature and finally we give a proof of Theorem 1.2.
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ZEJUN HU, DEHE LI, AND JING XU
First of all, related to Proposition 3.6 c) of [12] with further contributions, we prove the following Proposition 3.1. Assume that (M n , g, f, λ) is a complete n-dimensional m-quasif Einstein manifold which possesses constant scalar curvature R. If m > 1 and e− m attains its maximum or minimum at some point, then there is p ∈ {0, 1, 2, · · · , n} such that R = mn−(m−n)p−n λ. m+p−1 Remark 3.1. Proposition 3.1 is a generalization of Theorem 1 of [16] in the sense that, when m approaches infinite, an m-quasi-Einstein manifold becomes a gradient Ricci soliton. In the later situation, the assumption about u holds automatically, our result reduces to Theorem 1 of [16] that R ∈ {0, λ, · · · , (n − 1)λ, nλ}. f
Proof of Proposition 3.1. As before we assume u = e− m . From equation (2.2), together with the second Bianchi identity and the following Bochner formula div(∇2 u) = Ric(∇u) + ∇Δu, and noting that both R and λ are constants, we have the computation 2 0 = 12 ∇R = div(Ric) = div( m u ∇ u)
=
(3.1)
=
2 m m u div(∇ u) − u2 ∇∇u ∇u m m u (Ric(∇u) + ∇Δu) − u2 ∇∇u ∇u.
Substituting (2.3) into (3.1), we deduce m u Ric(∇u)
(3.2)
+ (R − nλ) ∇u u −
m u2 ∇∇u ∇u
= 0.
Using equation (2.2) again we get Ric(∇u) −
m u ∇∇u ∇u
= λ∇u.
Combining this with (3.2) we have (mλ − nλ + R) ∇u u +
m(m−1) ∇∇u ∇u u2
= 0.
This yields (3.3)
(nλ − mλ − R)∇(u2 ) = m(m − 1)∇(|∇u|2 ).
Then we find that if m > 1, the function u satisfies (3.4)
|∇u|2 =
(n−m)λ−R 2 m(m−1) u
+ C =: b(u)
where C is a constant and (3.5)
Δu =
R mu
−
n m λu
=: a(u).
Both, (3.4) and (3.5), show that, if m > 1, u is an isoparametric function (see [16] or [26] for the definition of isoparametric function) on M . The following discussion is similar to that in [16]. We denote umax = max{u(x) | x ∈ M } and umin = min{u(x) | x ∈ M }, if they exist. Recall that for the isoparametric function u, the level sets M+ (u) = {x ∈ M | u(x) = umax } and M− (u) = {x ∈ M | u(x) = umin }, if exist, are called the focal varieties of u. Now, without loss of generality we assume that u attains its maximum, thus M+ (u) is nonempty and, according to [26], it is
9
a smooth submanifold of M . As was proved by Wang [26] that the restriction of ∇2 u to M+ (u) has only two eigenvalues, 0 and 12 b (u), i.e., ∇2 u(X, Y ) = 0 for all X, Y ∈ T M+ (u), and ∇2 u(V, W ) = 12 b (u)g(V, W ) for all V, W ∈ T ⊥ M+ (u), where ⊥ T M+ (u) and T M+ (u) denote the tangent bundle and normal bundle of M+ (u), respectively. The expression of b(u) in (3.4) gives 12 b (u) = (n−m)λ−R m(m−1) u. Thus in our case 0, for X, Y ∈ T M+ (u), m 2 (3.6) u ∇ u(X, Y ) = (n−m)λ−R g(X, Y ), for X, Y ∈ T ⊥ M+ (u). m−1 Assume that dim M+ = n − p, where 0 ≤ p ≤ n. It follows from (2.2) and (3.6) that the restriction of the Ricci tensor to M+ (u) is of the form λIn−p 0 . (3.7) Ric |M+ (u) = (n−1)λ−R 0 Ip m−1 From (3.7) we deduce that the scalar curvature R satisfies R = (n−p)λ+ (n−1)λ−R p, m−1 so we have λ. R = mn−(m−n)p−n m+p−1 We have completed the proof of Proposition 3.1.
Now we give the proof of Theorem 1.2. Proof of Theorem 1.2. From Theorem 4.6 of [17] one knows that a nontrivial mquasi-Einstein manifold with λ > 0 is compact (cf. also Theorem 5 of [23]), whereas a compact m-quasi-Einstein manifold with constant scalar curvature is trivial (cf. Proposition 2.1 of [12]). Thus in our situation it must be the case λ ≤ 0. If m = 1, from (3.3) and the nontriviality, we obtain R = (n − 1)λ. If m > 1 and λ = 0, then from Proposition 3.6 b) of [12] we see that M is Ricci flat, by (2.3) we have Δu = 0 and thus u is constant, a contradiction to the nontriviality. Having proved that if m > 1 then λ < 0, we next assume that e−f /m attains its maximum or minimum at some point. It follows from Proposition 3.1 that R ∈ { mn−(m−n)p−n λ | p = 0, 1, 2, · · · , n}. m+p−1 If R = nλ (corresponding to p = 0), then by Case, Shu and Wei [12] in their proof of Proposition 3.6 b) c), (M, g) is Einstein and thus Ric = n1 Rg = λg. Hence, by (2.2) we have ∇2 u = 0, this equation has no complete nonconstant positive solution due to that the manifold is complete. Therefore, under the condition of nontriviality, R = nλ is impossible. Hence we have R ∈ { mn−(m−n)p−n λ | p = 1, 2, · · · , n}. m+p−1 Finally, we will verify the last statement of Theorem 1.2. That is, for each p ∈ {1, 2, · · · , n}, we prove that there exists nontrivial m-quasi-Einstein manifold with λ < 0 and constant scalar curvature R, which satisfies (3.8)
R=
mn−(m−n)p−n λ. m+p−1
This can be proved as follows: (i) In Example 2, (M n , g, f, mk) satisfies R = (n − 1)mk = (n − 1)λ, which corresponds to (3.8) for p = 1.
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ZEJUN HU, DEHE LI, AND JING XU
(ii) In Example 3, (M n , g, f, 1 − p − m) satisfies λ = 1 − p − m and that R = −p(p − 1) + (n − p)(−p − m − 1) = −mn + (m − n)p + n, which corresponds to (3.8) for each p ∈ {2, 3, · · · , n − 1}. (iii) The example (Hn , g, f, −m − n + 1), as considered in [8], is an m-quasiEinstein manifold with λ = −m − n + 1, where (Hn , g) is a hyperbolic space f with Einstein constant −(n−1), and f is given by e− m = hv . Then we have R = −n(n − 1), which corresponds to (3.8) for p = n. For this example, one is also referred to the discussion in Example 3 for background. We have completed the proof of Theorem 1.2.
Acknowledgements. We would like to express our sincerely thanks to the referee for his/her helpful comments and suggestions to improve our presentation.
References [1] M.T. Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. I, Geom. Funct. Anal. 9 (1999), 855-967. [2] M.T. Anderson and M. Khuri, The static extension problem in general relativity, arXiv: math.DG/0909.4550v1. [3] M.T. Anderson and M. Khuri, Uniqueness of static vacuum Einstein metrics and the Bartnik quasi-local mass, arXiv: math.DG/1003.2803v1. [4] A.L. Besse, Einstein manifolds. Springer-Verlag, Berlin, 1987. [5] A. Barros, R. Batista and E. Ribeiro Jr, Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh. Math. 174 (2014), 29-39. [6] A. Barros and J.N. Gomes, A compact gradient generalized quasi-Einstein metric with constant scalar curvature, J. Math. Anal. Appl. 401 (2013), 702-705. [7] A. Barros, J.N. Gomes and E. Ribeiro Jr. A note on rigidity of the almost Ricci soliton, Arch. Math. (Basel) 100 (2013), 481-190. [8] A. Barros and E. Ribeiro Jr, Characterizations and integral formulae for generalized m-quasi-Einstein metrics, Bull. Braz. Math. Soc. (N.S.) 45 (2014), 325-341. [9] H.D. Cao, Recent progress on Ricci solitons, Recent Advances in Geometric Analysis, 1-38, Adv. Lect. Math. 11 (2010). [10] H.D. Cao, Geometry of complete gradient shrinking Ricci solitons, Geometry and Analysis. No. 1, 227-246, Adv. Lect. Math. 17 (2011). [11] E. Calvi˜ no-Louzao, M. Fern´ andez-L´ opez, E. Garc´ıa-R´ıo and R. V´ azquez-Lorenzo, Homogeneous Ricci almost solitons, arXiv: math.DG/1501.05224v1. [12] J. Case, Y. Shu and G. Wei, Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), 93-100. [13] G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), 751-756. [14] G. Catino and L. Mazzieri, Gradient Einstein solitons, arXiv: math.DG/1201.6620. [15] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), 137-189. [16] M. Fern´ andez-L´ opez and E. Garc´ıa-R´ıo, On gradient Ricci solitons with constant scalar curvature, to appear in Proc. Amer. Math. Soc.; arXiv: math.DG/1409.3359v1. [17] C. He, P. Petersen and W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), 271-311. [18] G. Huang and Y. Wei, The classification of (m, ρ)-quasi-Einstein manifolds, Ann. Glob. Anal. Geom. 44 (2013), 269-282. [19] D.S. Kim and Y.H. Kim, Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc. 131 (2003), 2573-2576.
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[20] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan. 14 (1962), 333-340. [21] P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2009), 329-345. [22] S. Pigola, M. Rigoli, M.Rimoldi, and A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 759-799. [23] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), 235-242. [24] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275. [25] L.F. Wang, Rigid properties of quasi-almost-Einstein metrics, Chin. Ann. Math. Ser. B 33 (2012), 715-736. [26] Q.-M. Wang, Isoparametric functions on Riemannian manifolds I, Math. Ann. 277 (1987), 639-646. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China. E-mail addresses: [email protected]; [email protected]; [email protected].
Zejun Hu, Dehe Li, Jing Xu
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ON GENERALIZED m-QUASI-EINSTEIN MANIFOLDS WITH CONSTANT SCALAR CURVATURE ZEJUN HU, DEHE LI, AND JING XU Abstract. We study generalized m-quasi-Einstein manifolds with constant scalar curvature. First, we establish a classification of generalized m-quasiEinstein manifolds with parallel Ricci tensor. Second, and particularly, if an mquasi-Einstein manifold possesses constant scalar curvature then we determine the scalar curvature in explicit form and show that which values are attainable by presenting examples.
1. Introduction Because of their importance in both mathematics and physics, the study of the Einstein manifolds and their various generalizations is always an attractive topic in modern Riemannian geometry. In recent years, there has been increasing interest on so-called quasi-Einstein manifolds. Recall that, for a positive integer m, a complete Riemannian manifold (M n , g, f ) with a potential function f is called m-quasi-Einstein if its associated m-Bakry-Emery Ricci tensor Ricfm := Ric + ∇2 f −
1 m df
⊗ df
is a constant multiple of the metric g (cf. [12, 17] and the references therein). The notion of m-quasi-Einstein manifolds is natural and interesting partially due to that, according to [19], an n-dimensional m-quasi-Einstein manifold is exactly the manifold which is the base of an (n + m)-dimensional Einstein warped product. Moreover, the 1-quasi-Einstein metrics that satisfies Δe−f + λe−f = 0 are more commonly called static metrics. For more details we refer to [15] and, among others, [1, 2, 3] for later development. To extend the notion of m-quasi-Einstein, Catino [13] introduced the concept of generalized quasi-Einstein manifold. More precisely, an n-dimensional (n ≥ 3) complete Riemannian manifold (M n , g) is called generalized quasi-Einstein if there exist three smooth functions f, μ, λ on M , such that the Ricci tensor Ric of (M n , g) satisfies the relation (1.1)
Ric + ∇2 f − μdf ⊗ df = λg,
where ∇2 and ⊗ denote the Hessian and the tensorial product, respectively. In 1 and m is a positive integer, (1.1) turns to be particular, when μ = m (1.2)
Ric + ∇2 f −
1 m df
⊗ df = λg.
Key words and phrases. Generalized m-quasi-Einstein manifold, m-quasi-Einstein manifold, parallel Ricci tensor, constant scalar curvature, Bakry-Emery Ricci tensor. 2010 Mathematics Subject Classification. Primary 53C21; Secondary 53C24, 53C25. This project was supported by grants of NSFC-11371330. 1
2
ZEJUN HU, DEHE LI, AND JING XU
A manifold that satisfies (1.2) is called generalized m-quasi-Einstein, and is usually denoted by (M n , g, f, λ). Obviously, if λ is a constant, the generalized m-quasiEinstein manifold (M n , g, f, λ) becomes m-quasi-Einstein. It is noteworthy to point out that if m approaches infinite and λ is a constant, (1.2) becomes the equation satisfied by the gradient Ricci solitons. As is wellknown that gradient Ricci solitons play an important role in Hamilton’s Ricci flow as they correspond to the self-similar solutions. As an important aspect in the treatment of the Ricci flow, subject of gradient Ricci solitons has been treated by many authors (cf. [16, 21] etc.), we refer to Cao [9] and [10] for complete surveys up to that time. In addition, if in (1.2), λ = ρR + μ, where R is the scalar curvature of (M n , g) and ρ, μ are two real constants, then (M n , g) becomes an (m, ρ)-quasiEinstein manifold as defined in [18]. In particular, (∞, ρ)-quasi-Einstein manifold is called the gradient ρ-Einstein soliton that plays an important role in the so-called ρ-Einstein flows (cf. [14]). A generalized m-quasi-Einstein manifold (M n , g, f, λ) will be called trivial if the potential function f is constant. Otherwise, it will be called nontrivial. Obviously, the definition of triviality implies that (M n , g) is an Einstein manifold, but generally the converse is not true. In [8], there are some special examples which are Einstein, but as generalized m-quasi-Einstein manifolds they are not trivial. So far much research has been done on the characteristic functions on manifolds since they can provide much geometric information. A manifold can be well characterized if there exists a smooth function satisfying a special equation on it. For example, the well-known geometric and analytical characterizations obtained by Obata [20] and Tashiro [24], respectively. Now, a natural and interesting problem is to characterize a manifold for which (1.2) is satisfied under appropriate curvature conditions. Indeed, such characterizations of nontrivial m-quasi-Einstein manifolds (resp. generalized m-quasi-Einstein manifolds) have appeared in [12] (resp. [8]) under the condition that the manifold is Einstein. When m approaches infinite and λ is a function, there are also results about the so-called Ricci almost solitons under some curvature conditions in [5, 7, 11, 22]. The purpose of this paper is the study of generalized m-quasi-Einstein manifolds. Firstly, we assume that the Ricci tensor of the manifold is parallel. As the result we obtain a complete classification of such generalized m-quasi-Einstein manifolds. It turns out that the set of these manifolds includes not only space forms but also the Riemannian products of only two typical Einstein manifolds. Here is our first main result: Theorem 1.1. Let (M n , g, f, λ) be a complete n-dimensional (n ≥ 3) non-trivial generalized m-quasi-Einstein manifold which possesses parallel Ricci tensor. Then (M n , g) is isometric to one of the following manifolds: (1) (2) (3) (4) (5)
a space form, Dnc , R × N n−1 (b), Hp (a) × N n−p (b), b = m+p−1 p−1 a, Dpc × N n−p (b), b = (1 − m − p)c2 ,
where a, b are negative constants, N k (b) denotes a k-dimensional Einstein manifold with scalar curvature kb, here we recall that traditionally one also calls b the Einstein constant of N k (b); Hp (a) denotes the p-dimensional hyperbolic space with Einstein
3
constant a; Dkc denotes a k-dimensional Einstein warped product R ×c−1 ecr F k−1 , i.e. R × F k−1 endowed with the metric dr2 + (c−1 ecr )2 gF , c is a positive constant, F k−1 with metric gF is a Ricci flat manifold. Secondly, focusing on the m-quasi-Einstein manifolds, in section 3 we consider them under the additional condition that m is a positive integer and the manifold is of constant scalar curvature. We notice that if n = 1 and 2, a classification of m-quasi-Einstein manifolds already appeared in [4] (cf. also Remark 1.9 of [17]). In [12], one of the results states that for an n-dimensional m-quasi-Einstein manifold (M n , g, f, λ), if λ < 0 and the constant scalar curvature R is constant, then R n(n−1) is in fact bounded and satisfies nλ ≤ R ≤ m+n−1 λ. In this paper, continuing with this topic and motivated by a recent result of [16], we will show that under a suitable assumption the possible values of the constant scalar curvature are in fact quantified and they are expressed by the constant λ. The following theorem is our second main result. Theorem 1.2. Let (M n , g, f, λ) be a complete n-dimensional nontrivial m-quasiEinstein manifold with constant scalar curvature R, then λ ≤ 0. Moreover, (1) if m = 1, then R = (n − 1)λ, f (2) if m > 1, then λ < 0; and if moreover e− m attains its maximum or minimum at some point on M , then R ∈ { mn−(m−n)p−n λ | p = 1, 2, · · · , n}. m+p−1 Conversely, for each p ∈ {1, 2, · · · , n}, there exists nontrivial m-quasi-Einstein manifold with λ < 0 and constant scalar curvature, which satisfies R=
mn−(m−n)p−n λ. m+p−1
2. Generalized m-quasi-Einstein Manifolds with Parallel Ricci Tensor In this section, we deal with generalized m-quasi-Einstein manifolds with parallel Ricci tensor. For our purpose, we need the following result due to Case, Shu and Wei [12]. Proposition 2.1 (cf. Proposition 4.2 of [12]). A complete finite m-quasi-Einstein manifold (M n , g, f ) is Einstein if and only if f is constant or M is diffeomorphic f to Rn with the warped product structure (R ×c−1 ecr F n−1 , e− m = c−1 ecr ), or f
(Hn , dr2 + c−2 sinh2 (cr) gS n−1 , e− m = c−1 cosh(cr)), where F n−1 is Ricci flat, c is a positive constant. Proposition 2.1 can be extended to cover the case of generalized m-quasi-Einstein manifolds. To begin with, we assume that (M n , g, f, λ) is a nontrivial generalized m-quasi-Einstein manifold and (M n , g) is Einstein. Then, we recall that, under a suitable assumption on the critical points of e−f /m , Barros and Ribeiro [8] proved that (M n , g) is in fact a space form. Here, for a better understanding of such manifolds (by removing the assumption about the critical points of e−f /m ), we will prove the following result:
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ZEJUN HU, DEHE LI, AND JING XU
Proposition 2.2. Let (M n , g, f, λ) be a complete n-dimensional nontrivial generalized m-quasi-Einstein manifold (n ≥ 3). If (M n , g) is Einstein, then (M n , g) is either a space form or an Einstein manifold R ×c−1 ecr F n−1 , where c is a positive constant and F n−1 is Ricci flat. f
Proof. Firstly, we consider u = e− m . Then, according to Barros and Ribeiro (cf. Lemma 3 of [8]), the scalar curvature R satisfies (2.1)
R u+ ∇2 u = (− n(n−1)
c m )g,
where c is a constant. Moreover, from [8] we obtain that if R > 0 then (M n , g) is isometric to a standard sphere; and if R = 0, then (M n , g) is isometric to Rn . If R < 0, noting that (2.1) holds and u is positive, along with Tashiro’s proof of Theorem 2 in [24], we can show that (M n , g) is a hyperbolic space or a warped product R ×c−1 ecr F n−1 . In the later case, the fact that (M n , g) is Einstein implies that F is Ricci flat (see also Corollary 9.107 of [4]). Remark 2.1. Under more restricted condition than Proposition 2.2, if we assume that a nontrivial generalized m-quasi-Einstein manifold is compact and the scalar curvature is constant, then Barros and Gomes [6] showed that the manifold must be isometric to a standard Euclidean sphere. Before presenting the proof of Theorem 1.1 we first analyze in detail the examples that appear in Theorem 1.1. Example 1. As demonstrated by Examples 1, 2 and 3 in [8], a space form with an appropriate potential function f can be regarded as a nontrivial generalized mquasi-Einstein manifold, which is Einstein and is trivially of parallel Ricci tensor. Example 2. Consider the Riemannian product manifold M n = R × N n−1 (mk), here R is equipped with the metric gR = dt2 and N n−1 (mk) is an (n−1)-dimensional Einstein manifold with Einstein constant mk (k < 0). Obviously, M n is of parallel Ricci tensor. Moreover, on M n we consider the following function √ √ u(t, x) = Ce −kt or u(t, x) = C cosh( −kt), where x ∈ N n−1 and C is a positive constant. Noting that u (t) = −ku(t), by direct calculation we see that the function u(t, x) on M n satisfies Ric − mu−1 ∇2 u = mkg, where g denotes the product metric of M n = R × N n−1 (mk). This shows that, f for f defined by u = e− m , (M n , g, f, mk) is a nontrivial m-quasi-Einstein manifold with parallel Ricci tensor. Example 3. Consider the Riemannian product manifold M n = Hp ×N n−p (b), here Hp denotes the p-dimensional hyperbolic space with Einstein constant −(p − 1), N n−p (b) denotes an (n − p)-dimensional Einstein manifold with Einstein constant b = 1 − p − m. Obviously, M n is of parallel Ricci tensor. Moreover, we may as usual look Hp as a hypersurface of the Minkowski space Rp,1 : Hp = {x ∈ Rp,1 | x, x 1 = −1, x1 > 0},
5
where Rp,1 denotes the (p+1)-dimensional real vector space Rp+1 endowed with the Minkowski inner product x, x 1 = −x21 + x22 + · · · + x2p+1 . Then, given a constant element v ∈ Hp we can define a function hv : Hp → R by hv (x) := x, v 1 . Now we consider the following function u(x, y) = hv (x), x ∈ Hp , y ∈ N n−p (b) f
and define u = e− m . Then, similar to that in Example 3 of [8], we can show that on M n it holds Ric − mu−1 ∇2 u = (1 − p − m)g, where g is the standard product metric of M n = Hp × N n−p (b). Therefore, (M n , g, f, 1 − p − m) is a nontrivial m-quasi-Einstein manifold with parallel Ricci tensor. Proof of Theorem 1.1. Since the Ricci tensor of (M n , g, f, λ) is parallel, (M n , g) is locally either an Einstein manifold or the Riemannian product of a finite number of Einstein manifolds (cf. Theorem 1.100 of [4]). If (M n , g) is Einstein, from Proposition 2.2, (M n , g) is either a space form or an Einstein warped product R ×c−1 ecr F n−1 , where c is a positive constant and F n−1 is a Ricci flat manifold. If M n = M1 × M2 · · · × Ms , where s ≥ 2 and M1 , M2 , · · · , Ms are Einstein manifolds with different Einstein constants. Assume that dim Mi = mi and let {e1 , e2 , · · · , en } be a local orthonormal frame field of (M n , g) such that {ej }1≤j≤m1 are tangent to M1 , and {ej }i−1 mk +1≤j≤i−1 mk +mi k=1
k=1
are tangent to Mi for 2 ≤ i ≤ s. Let {ωi } be the dual coframe field of {ei } and {ωij } be the connection forms associated to {ei }, i.e., dωi = ωij ∧ ωj , ωij + ωji = 0. Then we have Rij := Ric(ei , ej ) = ci δij , where ci (i = 1, 2, · · · , n) are constants. For brevity, we denote [i] := {j | cj = ci }. f Considering the positive function u = e− m on M , then (1.2) can be rewritten as (2.2)
Ric −
m 2 u∇ u
Δu =
u m (R
= λg.
Taking trace of (2.2), we get (2.3)
− λn).
Here, because the Ricci tensor is parallel, the scalar curvature R is constant on M . The equation (2.2) gives muij = uRij − λuδij ,
(2.4) 2
where uij = ∇ u(ei , ej ). Using Rij = ci δij , we get 0, if i = j, (2.5) uij = ci −λ u, if i = j. m Calculating the covariant derivation of (2.4) yields muij,j = uj Rij + uRij,j − (λu)j δij .
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ZEJUN HU, DEHE LI, AND JING XU
Since uij = uji and Rij,j = 0, we have muji,j = uj Rij − (λu)j δij . The Ricci identity (see, e.g. (3.2) in [25]) then gives uk Rkjij = uj Rij − (λu)j δij , mujj,i + m k
m(Δu)i + m
(2.6)
uk Rki =
uj Rij − (λu)i .
j
k
Substituting (2.3) into (2.6), a direct calculation gives (2.7)
(λu)i =
R+(m−1)ci ui . n−1
Claim 1. For [i] = [j] and [α] = [i], it holds eα (uij ) = 0. In fact, if [i] = [j] and [α] = [i], from the product structure of M we obtain ωiα = 0, and it follows from (2.5) that uiα = 0. By definition of the covariant derivatives we get uiα,k ωk = duiα + uik ωkα + ukα ωki = 0, k
k
uij,k ωk = duij +
k
k
uik ωkj +
k∈[i]
ukj ωki .
k∈[i]
Therefore, we have uiα,j = 0 and uij,α = eα (uij ). On the other hand, the Ricci identity gives um Rmijα = uiα,j = 0. uij,α = uiα,j + m
This completes the proof of Claim 1. Claim 2. s = 2. From Claim 1, we have eα (uii ) = 0 for [α] = [i].
(2.8)
Inserting (2.5) into (2.8) we see that 0 = meα (uii ) = ci uα − (λu)α .
(2.9)
Substituting (2.7) into (2.9) we get the following relation (ci −
(2.10)
R+(m−1)cα )uα n−1
= 0 for [α] = [i].
n
Since (M , g, f, λ) is nontrivial, there exists an index j0 and a point p0 ∈ M such that uj0 |p0 = 0. Then (2.10) gives (2.11) This implies that
(ci −
R+(m−1)cj0 n−1
)uj0 |p0 = 0 for i ∈ [j0 ].
R+(m−1)c
j0 ci = , ∀i ∈ [j0 ] n−1 and therefore, we have s ≤ 2. From the assumption s ≥ 2, Claim 2 follows.
Next, without loss of generality, we may assume that j0 = 1 and M = M1 × M2 , where M1 is an m1 -dimensional Einstein manifold with Einstein constant a := c1 ; , M2 is an m2 -dimensional Einstein manifold with Einstein constant b = R+(m−1)a n−1 and a = b. In what follows we separate the discussion into two cases.
7
Case I. There exists some β0 ∈ {m1 + 1, m1 + 2, · · · , m1 + m2 = n} and a point q0 ∈ M such that uβ0 (q0 ) = 0. Case II. For all β ∈ {m1 + 1, m1 + 2, · · · , m1 + m2 = n}, it holds uβ = 0. In Case I, from (2.10) and (2.11) we can solve b = R+(m−1)a , n−1 . a = R+(m−1)b n−1 It follows that
R a = b = n−m , a = b = 0,
if m = n, if m = n,
which contradicts the assumption a = b. Hence Case I does not occur. In Case II, the function u depends only on M1 . By definition we have uββ = eβ (uβ ) + uk ωkβ (eβ ). k∈[β]
It follows that uββ = 0 for all β ∈ {m1 + 1, m1 + 2, · · · , n}. Combining this with (2.5), we obtain 0 = muββ = (b − λ)u. Hence, since u is positive, we obtain that λ = b. As a function depends only on M1 , u satisfying (2.2) with λ being a constant implies that M1 is an Einstein and m-quasi-Einstein manifold with Einstein constant a. If m1 > 1, we can apply Proposition 2.1 to M1 to see that it is either a hyperbolic space Hm1 or an Einstein warped product R ×c−1 ecr F m1 −1 , where c is a positive constant and F m1 −1 is Ricci flat. From the proof of Proposition 4.2 in [12], we find m+m1 −1 m+m1 −1 that λ satisfies a = −(m1 −1) a−λ m . Thus λ = m1 −1 a < 0 and b = λ = m1 −1 a. m1 −1 , a direct calculation gives that its Einstein Moreover, if M1 = R ×c−1 ecr F constant a = −(m1 − 1)c2 , and hence b = (1 − m − m1 )c2 . If m1 = 1, then, according to Example 1 of [17], M1 is R with λ < 0 (cf. also 9.109 and 9.110 in [4]). To be specific, we assume that M1 is endowed with the metric dt2 , then (2.2) becomes u (t) = −(λ/m)u(t). Since u(t) should have√complete positivesolution, it must be that λ < 0 and the solution is u(t) = Ce −λ/mt or C cosh( −λ/mt), where C is a positive constant. Noting that u is not periodic, M 1 is not diffeomorphic to S1 . Thus M n is R × N n−1 (b), where b = λ is a negative constant. This completes the proof of Theorem 1.1. Remark 2.2. From the proof of Theorem 1.1, it is easily seen that in each case of (3), (4) and (5), M n is not Einstein, the potential function f does not depend on the factor N , and λ = b is a negative constant. Furthermore, in case (4) (resp. (5)), the scalar curvature R of (M n , g) is determined by the relation b = R+(m−1)a n−1 (resp. b =
R−(m−1)(p−1)c2 ). n−1
3. m-quasi-Einstein Manifolds with Constant Scalar Curvature In this section, we study m-quasi-Einstein manifolds with constant scalar curvature and finally we give a proof of Theorem 1.2.
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ZEJUN HU, DEHE LI, AND JING XU
First of all, related to Proposition 3.6 c) of [12] with further contributions, we prove the following Proposition 3.1. Assume that (M n , g, f, λ) is a complete n-dimensional m-quasif Einstein manifold which possesses constant scalar curvature R. If m > 1 and e− m attains its maximum or minimum at some point, then there is p ∈ {0, 1, 2, · · · , n} such that R = mn−(m−n)p−n λ. m+p−1 Remark 3.1. Proposition 3.1 is a generalization of Theorem 1 of [16] in the sense that, when m approaches infinite, an m-quasi-Einstein manifold becomes a gradient Ricci soliton. In the later situation, the assumption about u holds automatically, our result reduces to Theorem 1 of [16] that R ∈ {0, λ, · · · , (n − 1)λ, nλ}. f
Proof of Proposition 3.1. As before we assume u = e− m . From equation (2.2), together with the second Bianchi identity and the following Bochner formula div(∇2 u) = Ric(∇u) + ∇Δu, and noting that both R and λ are constants, we have the computation 2 0 = 12 ∇R = div(Ric) = div( m u ∇ u)
=
(3.1)
=
2 m m u div(∇ u) − u2 ∇∇u ∇u m m u (Ric(∇u) + ∇Δu) − u2 ∇∇u ∇u.
Substituting (2.3) into (3.1), we deduce m u Ric(∇u)
(3.2)
+ (R − nλ) ∇u u −
m u2 ∇∇u ∇u
= 0.
Using equation (2.2) again we get Ric(∇u) −
m u ∇∇u ∇u
= λ∇u.
Combining this with (3.2) we have (mλ − nλ + R) ∇u u +
m(m−1) ∇∇u ∇u u2
= 0.
This yields (3.3)
(nλ − mλ − R)∇(u2 ) = m(m − 1)∇(|∇u|2 ).
Then we find that if m > 1, the function u satisfies (3.4)
|∇u|2 =
(n−m)λ−R 2 m(m−1) u
+ C =: b(u)
where C is a constant and (3.5)
Δu =
R mu
−
n m λu
=: a(u).
Both, (3.4) and (3.5), show that, if m > 1, u is an isoparametric function (see [16] or [26] for the definition of isoparametric function) on M . The following discussion is similar to that in [16]. We denote umax = max{u(x) | x ∈ M } and umin = min{u(x) | x ∈ M }, if they exist. Recall that for the isoparametric function u, the level sets M+ (u) = {x ∈ M | u(x) = umax } and M− (u) = {x ∈ M | u(x) = umin }, if exist, are called the focal varieties of u. Now, without loss of generality we assume that u attains its maximum, thus M+ (u) is nonempty and, according to [26], it is
9
a smooth submanifold of M . As was proved by Wang [26] that the restriction of ∇2 u to M+ (u) has only two eigenvalues, 0 and 12 b (u), i.e., ∇2 u(X, Y ) = 0 for all X, Y ∈ T M+ (u), and ∇2 u(V, W ) = 12 b (u)g(V, W ) for all V, W ∈ T ⊥ M+ (u), where ⊥ T M+ (u) and T M+ (u) denote the tangent bundle and normal bundle of M+ (u), respectively. The expression of b(u) in (3.4) gives 12 b (u) = (n−m)λ−R m(m−1) u. Thus in our case 0, for X, Y ∈ T M+ (u), m 2 (3.6) u ∇ u(X, Y ) = (n−m)λ−R g(X, Y ), for X, Y ∈ T ⊥ M+ (u). m−1 Assume that dim M+ = n − p, where 0 ≤ p ≤ n. It follows from (2.2) and (3.6) that the restriction of the Ricci tensor to M+ (u) is of the form λIn−p 0 . (3.7) Ric |M+ (u) = (n−1)λ−R 0 Ip m−1 From (3.7) we deduce that the scalar curvature R satisfies R = (n−p)λ+ (n−1)λ−R p, m−1 so we have λ. R = mn−(m−n)p−n m+p−1 We have completed the proof of Proposition 3.1.
Now we give the proof of Theorem 1.2. Proof of Theorem 1.2. From Theorem 4.6 of [17] one knows that a nontrivial mquasi-Einstein manifold with λ > 0 is compact (cf. also Theorem 5 of [23]), whereas a compact m-quasi-Einstein manifold with constant scalar curvature is trivial (cf. Proposition 2.1 of [12]). Thus in our situation it must be the case λ ≤ 0. If m = 1, from (3.3) and the nontriviality, we obtain R = (n − 1)λ. If m > 1 and λ = 0, then from Proposition 3.6 b) of [12] we see that M is Ricci flat, by (2.3) we have Δu = 0 and thus u is constant, a contradiction to the nontriviality. Having proved that if m > 1 then λ < 0, we next assume that e−f /m attains its maximum or minimum at some point. It follows from Proposition 3.1 that R ∈ { mn−(m−n)p−n λ | p = 0, 1, 2, · · · , n}. m+p−1 If R = nλ (corresponding to p = 0), then by Case, Shu and Wei [12] in their proof of Proposition 3.6 b) c), (M, g) is Einstein and thus Ric = n1 Rg = λg. Hence, by (2.2) we have ∇2 u = 0, this equation has no complete nonconstant positive solution due to that the manifold is complete. Therefore, under the condition of nontriviality, R = nλ is impossible. Hence we have R ∈ { mn−(m−n)p−n λ | p = 1, 2, · · · , n}. m+p−1 Finally, we will verify the last statement of Theorem 1.2. That is, for each p ∈ {1, 2, · · · , n}, we prove that there exists nontrivial m-quasi-Einstein manifold with λ < 0 and constant scalar curvature R, which satisfies (3.8)
R=
mn−(m−n)p−n λ. m+p−1
This can be proved as follows: (i) In Example 2, (M n , g, f, mk) satisfies R = (n − 1)mk = (n − 1)λ, which corresponds to (3.8) for p = 1.
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ZEJUN HU, DEHE LI, AND JING XU
(ii) In Example 3, (M n , g, f, 1 − p − m) satisfies λ = 1 − p − m and that R = −p(p − 1) + (n − p)(−p − m − 1) = −mn + (m − n)p + n, which corresponds to (3.8) for each p ∈ {2, 3, · · · , n − 1}. (iii) The example (Hn , g, f, −m − n + 1), as considered in [8], is an m-quasiEinstein manifold with λ = −m − n + 1, where (Hn , g) is a hyperbolic space f with Einstein constant −(n−1), and f is given by e− m = hv . Then we have R = −n(n − 1), which corresponds to (3.8) for p = n. For this example, one is also referred to the discussion in Example 3 for background. We have completed the proof of Theorem 1.2.
Acknowledgements. We would like to express our sincerely thanks to the referee for his/her helpful comments and suggestions to improve our presentation.
References [1] M.T. Anderson, Scalar curvature, metric degenerations and the static vacuum Einstein equations on 3-manifolds. I, Geom. Funct. Anal. 9 (1999), 855-967. [2] M.T. Anderson and M. Khuri, The static extension problem in general relativity, arXiv: math.DG/0909.4550v1. [3] M.T. Anderson and M. Khuri, Uniqueness of static vacuum Einstein metrics and the Bartnik quasi-local mass, arXiv: math.DG/1003.2803v1. [4] A.L. Besse, Einstein manifolds. Springer-Verlag, Berlin, 1987. [5] A. Barros, R. Batista and E. Ribeiro Jr, Compact almost Ricci solitons with constant scalar curvature are gradient, Monatsh. Math. 174 (2014), 29-39. [6] A. Barros and J.N. Gomes, A compact gradient generalized quasi-Einstein metric with constant scalar curvature, J. Math. Anal. Appl. 401 (2013), 702-705. [7] A. Barros, J.N. Gomes and E. Ribeiro Jr. A note on rigidity of the almost Ricci soliton, Arch. Math. (Basel) 100 (2013), 481-190. [8] A. Barros and E. Ribeiro Jr, Characterizations and integral formulae for generalized m-quasi-Einstein metrics, Bull. Braz. Math. Soc. (N.S.) 45 (2014), 325-341. [9] H.D. Cao, Recent progress on Ricci solitons, Recent Advances in Geometric Analysis, 1-38, Adv. Lect. Math. 11 (2010). [10] H.D. Cao, Geometry of complete gradient shrinking Ricci solitons, Geometry and Analysis. No. 1, 227-246, Adv. Lect. Math. 17 (2011). [11] E. Calvi˜ no-Louzao, M. Fern´ andez-L´ opez, E. Garc´ıa-R´ıo and R. V´ azquez-Lorenzo, Homogeneous Ricci almost solitons, arXiv: math.DG/1501.05224v1. [12] J. Case, Y. Shu and G. Wei, Rigidity of quasi-Einstein metrics, Differential Geom. Appl. 29 (2011), 93-100. [13] G. Catino, Generalized quasi-Einstein manifolds with harmonic Weyl tensor, Math. Z. 271 (2012), 751-756. [14] G. Catino and L. Mazzieri, Gradient Einstein solitons, arXiv: math.DG/1201.6620. [15] J. Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), 137-189. [16] M. Fern´ andez-L´ opez and E. Garc´ıa-R´ıo, On gradient Ricci solitons with constant scalar curvature, to appear in Proc. Amer. Math. Soc.; arXiv: math.DG/1409.3359v1. [17] C. He, P. Petersen and W. Wylie, On the classification of warped product Einstein metrics, Comm. Anal. Geom. 20 (2012), 271-311. [18] G. Huang and Y. Wei, The classification of (m, ρ)-quasi-Einstein manifolds, Ann. Glob. Anal. Geom. 44 (2013), 269-282. [19] D.S. Kim and Y.H. Kim, Compact Einstein warped product spaces with nonpositive scalar curvature, Proc. Amer. Math. Soc. 131 (2003), 2573-2576.
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[20] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan. 14 (1962), 333-340. [21] P. Petersen and W. Wylie, Rigidity of gradient Ricci solitons, Pacific J. Math. 241 (2009), 329-345. [22] S. Pigola, M. Rigoli, M.Rimoldi, and A. Setti, Ricci almost solitons, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 10 (2011), 759-799. [23] Z. Qian, Estimates for weighted volumes and applications, Quart. J. Math. Oxford Ser. (2) 48 (1997), 235-242. [24] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251-275. [25] L.F. Wang, Rigid properties of quasi-almost-Einstein metrics, Chin. Ann. Math. Ser. B 33 (2012), 715-736. [26] Q.-M. Wang, Isoparametric functions on Riemannian manifolds I, Math. Ann. 277 (1987), 639-646. School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, People’s Republic of China. E-mail addresses: [email protected]; [email protected]; [email protected].