aa r X i v : . [ m a t h . DG ] F e b VACUUM STATIC SPACES WITH HARMONIC CURVATURE
FENGJIANG LI
Abstract.
In this paper, we classify n -dimensional ( n ≥
5) vacuum staticspaces with harmonic curvature, thus extending the 4-dimensional work byKim-Shin [19]. As a consequence, we provide new counterexamples to theFischer-Marsden conjecture on compact vacuum static spaces. Introduction An n -dimensional Riemannian manifold ( M n , g ) is said to be a vacuum staticspace if there exists a nonzero smooth (lapse) function f on M such that(1.1) Hessf = f ( Rc − Rn − g ) . Vacuum static metrics arise naturally in the study of static space-times in generalrelativity. Actually, let (
M, g ) be an n -dimensional Riemannian manifold and f bea nonzero smooth positive function on M . It can be shown that the Lorentzianmanifold ( R × M, − f dt + g ) satisfies the Einstein equation for the energy mo-mentum tensor T of a perfect fluid if and only if ( M, g, f ) is a vacuum static spacesatisfying (1.1), (cf. [21, 15, 8, 25]).It is very interesting to notice that the vacuum static equation (1.1) is also con-sidered by Fischer and Marsden [14] in their study of the surjectivity of scalarcurvature function on the space of Riemannian metrics, originally derived fromthe linearization of the scalar curvature equation (cf. [3, 14, 19]). Moreover, Bour-guignon [3] and Fischer-Marsden [14] independently proved that a complete vacuumstatic space has constant scalar curvature, which is necessarily non-negative if fur-ther M is compact. Based on this fact, Fischer-Marsden [14] made the followingconjecture. Fischer-Marsden Conjecture.
Any compact vacuum static space is an Einsteinspace.
If it is true, by Obata’s theorem [22], such a space must be a standard sphereor a Ricci flat space. However, it turns out the conjecture is not true. The firstcounterexample was provided by Kobayashi [20] and Lafontaine [16] independently.They proved that compact locally conformally flat vacuum static spaces are S n , S × S n − and certain warped product S × h S n − . Moreover, Kobayashi-Obata [21]showed that if a complete vacuum static metric ( M n , g, f ) is locally conformally flat,then it is isometric to a warped product I × h N n − of an open interval I ⊂ R with N of constant sectional curvature. Furthermore, Kobayashi [20] constructed fiveimportant examples of such warped products and ultimately gave the classificationof complete locally conformally flat vacuum static spaces. Later, Qing-Yuan [23] Mathematics Subject Classification.
Primary 53C21; Secondary 53C25, 83C20.
Key words and phrases. vacuum static space, harmonic curvature, Codazzi tensor, D -fat. classified complete Bach-flat vacuum static spaces with compact level sets. Inspiredby the work of Cao-Chen [4, 5], they defined a covariant 3-tensor D for vacuumstatic spaces. In fact, what they obtained is essentially the classification result ofD-flat vacuum static spaces; see Section 2 for more details. For more results, werefer to [2, 8, 9, 10, 24, 26] and the references therein.Recently, Kim-Shin [19] studied the 4-dimensional vacuum static space of har-monic curvature and obtained a local description of the metric and potential func-tion. Their method of proof was motivated by Kim’s work [18] on the classificationof 4-dimensional gradient Ricci solitons with harmonic Weyl curvature. In a veryrecent paper [17], the author has succeeded in extending the work of Kim [18] andclassified n -dimensional gradient Ricci solitons with harmonic Weyl curvature forall n ≥ n -dimensional ( n ≥
5) vacuum static space with harmonic curvature satisfying(1.1), and extend the work of Kim-Shin [19] to all dimensions n ≥
5. Our mainresult is the following
Theorem 1.1.
Let ( M n , g, f ) be an n -dimensional, n ≥ , complete vacuum staticspace with harmonic curvature satisfying (1.1) . Then it is one of the followingtypes: (i) ( M, g ) is D -flat. Consequently, by Qing-Yuan [23] , ( M, g ) is either Einsteinor isometric to one of the spaces in Examples 1-5 as described in Section 2. (ii) ( M, g ) is isometric to a quotient of S (cid:16) R n − (cid:17) × N n − with R > , where (cid:0) N n − , g (cid:1) is Einstein with positive Einstein constant Rn − . f = c cos (cid:16)q R n − s (cid:17) for some constant c = 0 , where s is the distance on S ( R n − ) from a point. (iii) ( M, g ) is isometric to a quotient of H (cid:16) R n − (cid:17) × N n − with R < , where (cid:0) N n − , g (cid:1) is Einstein with negative Einstein constant Rn − . f = c cosh (cid:16)q − R n − s (cid:17) for some constant c = 0 , where s is the distance function on H (cid:16) R n − (cid:17) from apoint. (iv) ( M, g ) is isometric to a quotient of the Riemannian product ( W r × N n − r , g =¯ g + g ) , where ≤ r ≤ n − , ( W r , ¯ g = ds + h ( s )˜ g ) is an r -dimensional D -flatvacuum static space and ( N n − r , g ) is an Einstein manifold of Einstein constant Rn − . f = ch ′ for some constant c . Particularly, R = 0 for r = n − . Finally, we pick up compact spaces in Theorem 1.1.
Theorem 1.2.
Let ( M n , g, f ) be an n -dimensional, n ≥ , compact vacuum staticspace with harmonic curvature satisfying (1.1) . Then it is one of the followingtypes: (i) ( M, g ) is D -flat. Hence, ( M, g ) is either isometric to the Euclidean sphere S n , or a quotient of ( S , ds ) × ( N n − , g ) , or a quotient of a product torus ( S × N n − , ds + h ( s ) g ) , where ( N n − , g ) is Einstein. (ii) ( M, g ) is isometric to a quotient of S (cid:16) R n − (cid:17) × N n − with R > , where (cid:0) N n − , g (cid:1) is Einstein with positive Einstein constant Rn − . f = c cos (cid:16)q R n − s (cid:17) for some constant c = 0 , where s is the distance on S ( R n − ) from a point. SS WITH HARMONIC CURVATURE 3 (iii) (
M, g ) is isometric to a quotient of the Riemannian product ( W r × N n − r , g =¯ g + g ) , where ≤ r ≤ n − , ( W r , ¯ g = ds + h ( s )˜ g ) is an r -dimensional D -flatvacuum static space and ( N n − r , g ) is an Einstein manifold of the Einstein constant Rn − . f = ch ′ for some constant c . Particularly, R = 0 for r = n − .Remark . It is worth noting that types (ii) and (iii) will provide new coun-terexamples, which are not D -flat, to the Fischer-Marsden conjecture. Moreover,Examples in type (iii) are the Riemannian products of D -flat vacuum static spacesand Einstein manifolds, which are similar to the rigid gradient Ricci solitons thatare the Riemannian products of Gaussian solitons and Einstein manifolds. There-fore, we see that the D -flatness property plays an important role in vacuum staticspaces.This paper is organized as follows. In Section 2, we give some formulae andnotations for Riemannian manifolds and vacuum static spaces by using the methodof moving frames. In Section 3, we derive the integrability conditions (ODEs) fora vacuum static space with harmonic curvature and show that, locally, the metricis a multiply warped product. In Section 4-5, in order to complete the proof ofTheorem 1.1, we divide our discussion into three cases according to the numbersand multiplicities of distinct Ricci-eigenvalues, excluding the one with respect tothe gradient vector of the lapse function.2. Preliminaries
In this section, we first recall some formulae and notations for Riemannian man-ifolds by using the method of moving frames. Then we give some facts on vacuumstatic spaces.2.1.
Some notations for Riemannian manifolds.
Let M n ( n ≥
3) be an n -dimensional Riemannian manifold, E , · · · , E n be a local orthonormal frame fieldson M n , and ω , · · · , ω n be their dual 1-forms. In this paper we make the followingconventions on the range of indices:1 ≤ i, j, k, · · · ≤ n and agree that repeated indices are summed over the respective ranges. Then wecan write the structure equations of M n as follows:(2.1) dω i = ω j ∧ ω ji and ω ij + ω ji = 0;(2.2) − R ijkl ω k ∧ ω l = dω ij − ω ik ∧ ω kj and R ijkl = − R jikl , where d is the exterior differential operator on M , ω ij is the Levi-Civita connectionform and R ijkl is the Riemannian curvature tensor of M . It is known that theRiemannian curvature tensor satisfies the following identities:(2.3) R ijkl = − R ijlk , R ijkl = R klij and R ijkl + R iklj + R iljk = 0 . The Ricci tensor R ij and scalar curvature R are defined respectively by(2.4) R ij := X k R ikjk and R = X i R ii . FENGJIANG LI
Let f be a smooth function on M n , we define the covariant derivatives f i , f i,j and f i,jk as follows:(2.5) f i ω i := df, f i,j ω j := df i + f j ω ji , and(2.6) f i,jk ω k := df i,j + f k,j ω ki + f i,k ω kj . We know that(2.7) f i,j = f j,i and f i,jk − f i,kj = f l R lijk . The gradient, Hessian and Laplacian of f are defined by the following formulae:(2.8) ∇ f := f i E i , Hess ( f ) := f i,j ω i ⊗ ω j and ∆ f := X i f i,i . The covariant derivatives of tensors R ij and R ijkl are defined by the followingformulae:(2.9) R ij,k ω k := dR ij + R kj ω ki + R ik ω kj and(2.10) R ijkl,m ω m := dR ijkl + R mjkl ω mi + R imkl ω mj + R ijml ω mk + R ijkm ω ml . By exterior differentiation of (2.2), one can get the second Bianchi identity(2.11) R ijkl,m + R ijlm,k + R ijmk,l = 0 . From (2.4), (2.10) and (2.11), we have(2.12) R ij,k − R ik,j = − X l R lijk,l , and so(2.13) X j R ji,j = 12 R i . We define the Schouten tensor as A = A ij ω i ⊗ ω j , where(2.14) A ij := R ij − n − Rδ ij , then A ij = A ji . The tensor(2.15) W ijkl := R ijkl − n − A ik δ jl + A jl δ ik − A il δ jk − A jk δ il )is called the Weyl conformal curvature tensor.In dimension three, W is identicallyzero on every Riemannian manifold, whereas, when n ≥
4, the vanishing of theWeyl tensor is equivalent to the locally conformal flatness of ( M n , g ). We alsorecall that in dimension n = 3, ( M, g ) is locally conformally flat iff the Cottontensor C , defined as follows, vanishes(2.16) C ijk := A ij,k − A ik,j . We recall that, for n ≥
4, using the second Bianchi identity the Cotton tensor canalso be defined as one of the possible divergences of the Weyl tensor:(2.17) − n − n − X l W lijk,l = C ijk . SS WITH HARMONIC CURVATURE 5
On any n -dimensional manifold ( M, g ) ( n ≥ B ij := 1 n − W ikjl,kl + 1 n − R kl W ikjl and by (2.17), we have an equivalent expression of the Bach tensor:(2.19) B ij = 1 n − C ijk,k + R kl W ikjl ) . The D -tensor and D -flat vacuum static spaces. We will recall the D -tensor defined in [23] for vacuum static spaces and the classification result for D -flatvacuum static spaces by Qing-Yuan [23].The covariant 3-tensor D ijk (see [23]) is defined by(2.20) D ijk = n − n − R ik f j − R ij f k ) + Rn − f k δ ij − f j δ ik )+ 1 n − f l ( R lj δ ik − R lk δ ij ) , which is the analog of the D -tensor defined by Cao-Chen [4, 5] for Ricci solitons.First of all, we have the following lemma. Lemma 2.1 ( Qing-Yuan [23] ). Let ( M n , g, f ) be a vacuum static space satisfying (1.1) . Then the following formulae hold: (2.21) f C ijk = f l W lijk + D ijk . (2.22) ( n − f B ij = D ijk,k − f k (cid:18) C ijk + n − n − C jik (cid:19) . Next, we recall the following classification of D -flat vacuum static spaces, whichwas obtained by Qing-Yuan [23] (even though it was not explicitly stated). Theorem 2.2. (Qing-Yuan [23] ) Suppose that ( M n , g, f ) is an n -dimensional D -flat vacuum static space satisfying (1.1) . Then the metric g is a locally warpedproduct, g = ds + h ( s )˜ g for a positive function h , where the Riemannian metric ˜ g is Einstein with the Ein-stein constant ( n − k . ( M, g ) is either Einstein or isometric to one of the spacesin Examples 1-5, as follows.Furthermore, h satisfies the following equation (2.23) h ′′ + Rn ( n − h = c h − ( n − for a constant c and (2.24) ( h ′ ) + 2 c n − h − ( n − + Rn ( n − h = k. The non-constant lapse fucntion f satisfies (2.25) h ′ f ′ − f h ′′ = 0 . FENGJIANG LI
To compare with Kobayashi’s work [20] on locally conformally flat vacuum staticspaces, we note that the only difference is that ˜ g is an Einstein metric in the D -flatcase, while it is of constant sectional curvature in the locally conformally flat case.Therefore, as noted in [23], one only needs to replace the constant curvature spacefactor of the warped products in [20] with corresponding Einstein manifold factorto obtain the following five D -flat examples. Example 1 : Let N ( k ) be an ( n − n − k . On a Riemannian product R × N n − ( k ), k = 0,(2.26) f ( s ) = (cid:26) c sin p ( n − ks + c cos p ( n − ks + x, if k > ,c sinh p − ( n − ks + + c cosh p − ( n − ks + x, if k < , where c and c are constants. Example 2 : Compact quotients of Example 1 with k >
0, with an explicitisometry group Γ l,φ generated by ( s, x ) → ( s + 2 lπ/ p ( n − k, φ ( x )), where l isany natural number and φ ∈ Isom ( N ( k )). Example 3 . A warped product R × h N n − (1), where h is a periodic solutionto (2.23) with a > , R > , k > k = R ( n − n − ( n ( n − aR ) /n (as in Proposition 2.6of Kobayashi [20]), and non-trivial f satisfying f ( s ) = ch ′ for some constant c . Example 4 . Since f and h in Example 3 have a common period, we obtaincompact spaces with non-trivial solutions to (1.1) in a similar way as Example2. These spaces were first found by Ejiri [13] as counterexamples for a differentproblem. Example 5 . A warped product R × h N n − ( k ), where h, k are as in Proposition2.5 of Kobayashi [20] (i.e, h, k satisfies one of the conditions (IV.1)-(IV.4)), andnon-trivial f ( s ) = ch ′ for some constant c . Remark . Suppose that ( M n , g ) is a warped product (cid:0) I × h N n − , ds + h ( s )˜ g (cid:1) ,where h is not constant and (cid:0) N n − , h ( s )˜ g (cid:1) is Einstein with the Einstein constant( n − k . Then there exists a smooth function f depending only on s satisfying(1.1) if and only if ( M n , g ) is of constant scalar curvature, as well as functions f and h satisfy (2.23) and (2.25).One can now pick up compact spaces in Theorem 2.2. Theorem 2.4. (Qing-Yuan [23] ) Let ( M n , g, f ) be a compact D -flat vacuumstatic space satisfying (1.1) . Then M is isometric to the Euclidean sphere S n , orthe quotient of ( S , ds ) × ( N n − , g ) , or the quotient of a product torus ( S × N n − , ds + h ( s ) g ) , where ( N n − , g ) is an Einstein manifold. The basic local structure for n -dimensional vacuum static spaceswith harmonic curvature In the following sections, in a similar way to that of Ricci soliton [17], we willgive the (local) classification of vacuum static space with harmonic curvature. Inthis section, the goal is to derive the integrability conditions (ODEs) for a vacuumstatic space with harmonic curvature, and then show that, locally, the metric is amultiply warped product.
SS WITH HARMONIC CURVATURE 7
Let ( M n , g, f ), n ≥
4, be an n -dimensional vacuum static space with harmoniccurvature satisfying (1.1). First of all, we recall the next lemma (see also Lemma2.5 in [19]). Lemma 3.1.
In some neighborhood U of each point in {∇ f = 0 } , we choose anorthonormal frame field { E = ∇ f |∇ f | , E , · · · , E n } with the dual frame field { ω = df |∇ f | , ω , · · · , ω n } . Then we have following properties (i) E = ∇ f |∇ f | is an eigenvector field of the Ricci tensor. (ii) The 1-form ω = df |∇ f | is closed. So the distribution V = Span { E , · · · , E n } is integrable from the Frobenius theorem. We denote by L and N the integrablecurve of the vector field E and the integrable submanifold of V respectively. Thenit follows locally M = L × N and there exists a local coordinates ( s, x , · · · , x n ) of M such that ds = df |∇ f | , E = ∇ s , V = Span { ∂∂x , · · · , ∂∂x n } and g = ds + P a ω a . (iii) f and R can be considered as functions of the single variable s .Proof. Firstly, it follows from (2.12) that the Ricci tensor of a Riemannian metricwith harmonic curvature is a Codazzi tensor, i.e. R ij,k = R ik,j . Moreover, forany n -dimensional manifold ( M n , g, f ) with harmonic curvature satisfying (1.1), itholds that(3.1) f l R lijk = f k (cid:18) R ij − Rn − δ ij (cid:19) − f j (cid:18) R ik − Rn − δ ik (cid:19) where the Ricci identity (2.7) was used. Noting that f = |∇ f | 6 = 0, f a = 0 for2 ≤ a ≤ n , setting i = j = 1, and k = a in (3.1), we have f R a = 0. Hence R a = 0, i.e., E = ∇ f |∇ f | is an eigenvector field of the Ricci curvature.Next, R a = 0 leads to f ,a = f ( R a − Rn − δ a ) = 0, and then (cid:16) |∇ f | (cid:17) a = 2 f f ,a = 0 . Therefore dω = d (cid:18) df |∇ f | (cid:19) = − |∇ f | d (cid:0) |∇ f | (cid:1) ∧ df = 0 , that is ω = df |∇ f | is closed.Note that f can be considered as a function of the single variable s . From (2.5),it follows that f , = f ′′ . Since f = 0 on an open subset, we have R = f − (cid:18) f , + Rn − (cid:19) depending only on s . We have completed the proof of this lemma. (cid:3) For any n -dimensional ( n ≥
3) Riemannian manifold with harmonic curvature,the Ricci tensor satisfies the Codazzi equation. As described on Codazzi tensorsby Derdzi´nski [11, 12], for any point x in M , let E Ric ( x ) be the number of distincteigenvalues of Ric x , and set M Ric = { x ∈ M | E Ric is constant in a neighborhood of x } . Then M Ric is an open dense subset of M and in each connected component of M Ric , the eigenvalues are well-defined and differentiable functions. A Riemannianmanifold with harmonic curvature is real analytic in harmonic coordinates [11], i.e.,
FENGJIANG LI f is real analytic (in harmonic coordinates). Then if f is not constant, {∇ f = 0 } is open and dense in M .Therefore, in some neighborhood U of each point in M Ric ∩ {∇ f = 0 } , thenumber of distinct Ricci-eigenvalues is constant, and we assume there are m dis-tinct Ricci-eigenvalues of multiplicities r , r , · · · , r m , respectively, excluding theone with corresponding to the eigenvector ∇ f |∇ f | . Here 1 + r + r + · · · + r m = n .Therefore, we can choose an orthonormal frame field { E = ∇ f |∇ f | , E , · · · , E n } , withthe dual { ω = df |∇ f | , ω , · · · , ω n } such that(3.2) R ij = λ i δ ij ,λ = · · · = λ r +1 , λ r +2 = · · · = λ r + r +1 , · · · λ r + r + ··· + r m − +2 = · · · = λ n , and λ , λ r +2 , · · · , λ r + r + ··· + r m − +2 are distinct. Next, we need the followinglemma to prove that all Ricci-eigenfunctions λ i ( i = 1 , · · · , n ) depend only on thelocal variable s . Lemma 3.2.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . For the above local frame field { E i } in M Ric ∩ {∇ f = 0 } , we have (3.3) f ′′ = f λ − Rn − , (3.4) ω a = ξ a ω a and (3.5) R aa, = ( λ − λ a ) ξ a , where (3.6) ξ a := 1 |∇ f | (cid:18) f λ a − Rn − (cid:19) . Proof.
It follows from (2.5) that for 2 ≤ a ≤ n , f , = f ′′ and f ′ ω a = f a,j ω j . Substituting f i,j = (cid:16) f λ i − Rn − (cid:17) δ ij into the above equations, we immediately have(3.3) and (3.4). Directly calculation by using of (2.9) gives us R , = λ ′ and R a,a = ( λ − λ a ) ξ a . Hence harmonic condition R aa, = R a,a yields (3.7), and we have completed theproof of this lemma. (cid:3) Making use of the above lemmas, it is easy to prove the following lemma, whichwas proved by Kim-Shin [19] for the four dimensional case.
Lemma 3.3.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . For the above local frame field { E i } in M Ric ∩ {∇ f = 0 } , the Ricci eigenfunctions λ i ( i = 1 , · · · , n ) depend only on thelocal variable s , so do the functions ξ a for ≤ a ≤ n . SS WITH HARMONIC CURVATURE 9
Subsequently, we will obtain the local structure of the metric for an n -dimensionalvacuum static space with harmonic curvature. First, we denote [ a ] = { b | λ b = λ a and b = 1 } for 2 ≤ a ≤ n and make the following conventions on the range ofindices: 2 ≤ a, b, c · · · ≤ n and 2 ≤ α, β, · · · ≤ n where [ a ] = [ b ], [ α ] = [ β ] and [ a ] = [ α ]. Lemma 3.4.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . For the above local frame field { E i } in M Ric ∩ {∇ f = 0 } , we see that (3.7) ω aα = 0 , (3.8) R a b = − (cid:0) ξ ′ a + ξ a (cid:1) δ ab and (3.9) R aαbβ = − ξ a ξ α δ ab δ αβ . Proof.
Using the same method to that of Lemma 3.4 in [17], we can easily provethis lemma. (cid:3)
At last, we are ready to derive the integrability conditions and prove the localstructure of the metric as a multiply warped product.
Theorem 3.5.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . For each point p ∈ M Ric ∩ {∇ f = 0 } ,there exists a neighborhood U of p such that U = L × h L × · · · × h l L l × h l +1 N l +1 × · · · × h m N m is a multiply warped product furnished with the metric (3.10) g = ds + h ( s ) dt + · · · + h l ( s ) dt l + h l +1 ( s )˜ g l +1 + · · · + h m ( s )˜ g m , where h j ( s ) are smooth positive functions for ≤ j ≤ m , dim L ν =1 for ≤ ν ≤ l , and ( N µ , ˜ g µ ) is an r µ -dimensional Einstein manifold of the Einstein constant ( r µ − k µ for l + 1 ≤ µ ≤ m .Moreover, let λ i ( i = 1 , · · · , n ) be the Ricci-eigenvalues, λ being the Ricci-eigenvalue with respect to the gradient vector ∇ f , and functions ξ a be given by (3.6) . Then the following integrability conditions hold (3.11) ξ ′ a + ξ a = λ a − Rn − , (3.12) λ ′ a − ( λ − λ a ) ξ a = 0 , (3.13) λ = f − (cid:18) f ′′ + f Rn − (cid:19) = − n X i =2 (cid:0) ξ ′ i + ξ i (cid:1) , and (3.14) λ a = f − (cid:18) f ′ ξ a + f Rn − (cid:19) = − ξ ′ a − ξ a n X i =2 ξ i + ( r − kh , Here ≤ a ≤ n and ξ a = h ′ /h ; in (3.14) , when a = 2 , . . . , l + 1 , then r = 1 and h = h a − ; when a ∈ [ l + r l +1 + · · · + r µ − + 2] , then r = r µ , h = h µ and k = k µ .Proof. In a similar way to that of Theorem 3.4 in [17], we can easily prove thislemma. Here, we only prove the partially integrability conditions, different fromthat of Ricci solitons.Making use of (3.1) again, we see R a a = − R aa + Rn − R aa, = λ ′ into (3.5) implies harmonic condition(3.12). The rest proof can be referred to the paper [17]. (cid:3) The local structure of the case other than two distinctRicci-eigenfunctions
In this section and the next section, we will prove the local classification of n -dimensional vacuum static spaces with harmonic curvature according to howmany distinct Ricci-eigenvalues and their multiplicities. To avoid repetition, unlessstated otherwise, the Ricci-eigenvalues mentioned in the following discussions donot include λ , which is the eigenvalue with respect to the gradient vector ∇ f ofthe lapse function. Also, we denote the other Ricci-eigenvalues by λ a , a = 2 , · · · , n .Firstly, we treat the case that all Ricci-eigenfunctions { λ a } , 2 ≤ a ≤ n , areequal. Type (i) of Theorem 1.1 will come from this case. Theorem 4.1.
Suppose that ( M n , g, f ) is an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . If all Ricci-eigenfunctions { λ a } , ≤ a ≤ n , are equal. Then locally the metric g is a warped product of the form g = ds + h ( s )˜ g with a certain positive warping function h , where the Riemannian metric ˜ g is Ein-stein. In particular, the D -tensor of ( M n , g, f ) vanishes (and so does the Bachtensor).Proof. The first part of this theorem has been obtained by Theorem 3.5. For thesecond part, since the harmonicity implies that the Cotton tensor C ijk = 0, withthe relationships (2.21) and (2.22), it is easy to check that the vacuum static spaceis D -flat and Bach-flat of constant scalar curvature. The detail also can be referredto [17]. (cid:3) Remark . In combination with Theorem 2.2, it follows that for a vacuum staticspace ( M n , g, f ) satisfying (1.1), ( M n , g ) is a locally warped product, described bythe hypothesis of Theorem 4.1, if and only if the D -tensor vanishes. In addition,from the proof of Theorem 4.1, we have that D = 0 implies B = 0, but the reverseis not true.Next, we shall study the case when λ , λ , · · · , λ n are at least three mutuallydifferent. As we shall see below, it turns out that this case cannot occur.First of all, we analyze the integrability conditions in Theorem 3.5. Assume that λ a and λ α are mutually different of multiplicities r and r , i.e., λ a := λ = · · · = λ r +1 and λ α := λ r +2 = · · · = λ r + r +1 ; SS WITH HARMONIC CURVATURE 11
Here we make the following conventions on the range of indices:2 ≤ a, b, · · · ≤ ( r + 1); ( r + 2) ≤ α, β, · · · ≤ ( r + r + 1);Denote ξ a := X and ξ α := Y , from Section 3, and then they satisfy the followingintegrability conditions:(4.1) X ′ + X − λ a = Y ′ + Y − λ α = ξ ′ i + ξ i − λ i = − Rn − , (4.2) λ = f − (cid:18) f ′′ + Rn − f (cid:19) = − n X i =2 (cid:0) ξ ′ i + ξ i (cid:1) , (4.3) λ a = f − (cid:18) f ′ X + Rn − f (cid:19) = − ( X ′ + X ) − X n X i =2 ξ i + ( r − k h + X , (4.4) λ α = f − (cid:18) f ′ Y + Rn − f (cid:19) = − ( Y ′ + Y ) − Y n X i =2 ξ i + ( r − k h + Y and(4.5) λ ′ a − ( λ − λ a ) X = λ ′ α − ( λ − λ α ) Y = 0 . By using the above basic facts, it is easy to get the following equations:
Lemma 4.3.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature. In some neighborhood U of p ∈ M Ric ∩ {∇ f = 0 } , let λ a and λ α are mutually different Ricci-eigenvalues with multiplicities r and r . Thenthe following identities hold: (4.6) ( f ′ + f " n X i =2 ξ i − ( X + Y ) ( X − Y ) = f (cid:20) ( r − k h − ( r − k h (cid:21) and (4.7) − f − f ′ n X i =2 ξ i + n X i =2 ξ ′ i + 2 n X i =2 ξ i = ( X + Y ) n X i =2 ξ i . Proof.
Subtracting (4.3) from (4.4) gives us(4.8) λ a − λ α = f − f ′ ( X − Y )and λ a − λ α = − ( X − Y ) ′ − ( X − Y ) n X i =2 ξ i + (cid:20) ( r − k h − ( r − k h (cid:21) . From the first equality of (4.1), it follows that λ a − λ α = ( X − Y ) ′ + ( X − Y ).Then respectively comparing with (4.8) and the above equation, we have(4.9) X ′ − Y ′ = ( X − Y )[ f − f ′ − ( X + Y )]and 2( λ a − λ α ) = − ( X − Y ) " n X i =2 ξ i − ( X + Y ) + (cid:20) ( r − k h − ( r − k h (cid:21) . Putting (4.8) into the above equation, we can immediately see (4.6). Then differ-entiating the above equation yields2( λ a − λ α ) ′ = − ( X − Y ) ′ n X i =2 ξ i − ( X − Y ) n X i =2 ξ ′ i − (cid:20)(cid:18) ( r − k h − X ′ (cid:19) X − (cid:18) ( r − k h − Y ′ (cid:19) Y (cid:21) , where X = h ′ /h and Y = h ′ /h were used. On the other hand, it follows from(4.5) that λ ′ a − λ ′ α = λ ( X − Y ) − λ a X + λ α Y = − ( X − Y ) n X i =2 (cid:0) ξ ′ i + ξ i (cid:1) + ( X − Y ) n X i =2 ξ i − (cid:20)(cid:18) ( r − k h − X ′ (cid:19) X − (cid:18) ( r − k h − Y ′ (cid:19) Y (cid:21) , where (4.1) and (4.2) were used. Comparing with the above two equations leads to − [( X − Y ) ′ + 2( X − Y )] n X i =2 ξ i + ( X − Y ) n X i =2 ξ ′ i + 2( X − Y ) n X i =2 ξ i = 0 . Putting ( X − Y ) ′ + 2( X − Y ) = ( X − Y )( f − f ′ + X + Y ), rewritten by (4.9), intothe above equation, we see (4.7) since X and Y are different. We have completedthe proof of this lemma. (cid:3) Next, we will apply equations (4.1), (4.2) and (4.7) to deal with the case when λ , λ , · · · , λ n are at least three mutually different values. Theorem 4.4.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature. Then in some neighborhood U of p ∈ M Ric ∩ {∇ f = 0 } ,the Ricci-eigenvalues λ , λ , · · · , λ n cannot be more than two distinct values.Proof. If not, we assume that λ , λ , · · · , λ n are at least three mutually differ-ent values, and denote by λ a , λ α and λ p with multiplicities r , r and r . Forconvenience, we also denote ξ a := X , ξ α := Y and ξ p := Z . By (4.7), with theassumption of Lemma 4.3, we see that( X + Y ) n X i =2 ξ i = ( X + Z ) n X i =2 ξ i = ( Y + Z ) n X i =2 ξ i . If P ni =2 ξ i = 0, the above equation implies that X = Y = Z and this contradictsthe hypothesis. If P ni =2 ξ i = 0, (4.7) implies P ni =2 ξ i = 0. Hence ξ i = 0 for each i = 2 , . . . , n , which is a contradiction. We have completed the proof of Theorem4.4. (cid:3) The local structure of the case with exactly two distinctRicci-eigenfunctions
In this section we begin to study the case when there are exactly two distinctRicci values in the eigenvalues λ , · · · , λ n . Types (ii), (iii) and (iv) of Theorem 1.1come from this section. SS WITH HARMONIC CURVATURE 13
First of all, we need the following lemma to prepare for the local structure of thecase.
Lemma 5.1.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static space withharmonic curvature. Assume in some neighborhood U of p ∈ M Ric ∩ {∇ f = 0 } ,there are exactly two distinct values in the Ricci-eigenvalues λ , · · · , λ n , denoted by λ a and λ α of multiplicities r and r := n − r − . Then one of functions X and Y vanishes.Proof. In this case, equation (4.6) can be rewritten as − ( X ′ − Y ′ )( r X + r Y ) + ( X − Y )( r X ′ + r Y ′ )+ 2( r X + r Y )( X − Y ) − X − Y )( r X + r Y ) = 0 . Directly simplifying the above equation yields that X ′ Y − XY ′ + 2 XY ( X − Y ) = 0 . Meanwhile, the integrability conditions (3.11) and (3.14) imply that f ξ ′ i = f ′ ξ i − f ξ i for each i = 2 , . . . , , n . Then putting f X ′ = f ′ X − f X and f Y ′ = f ′ Y − f Y into the above equation, we see XY = 0, which means that one of functions X and Y vanishes. We have completed the proof of this lemma. (cid:3) From now on, without loss of generality, we assume that X = 0 and Y = 0.Meanwhile, Y = 0 implies that h is a constant.Next, we will discuss two cases according to whether one of the two Ricci-eigenfunctions is of single multiplicity.5.1. One of the multiplicities of two Ricci-eigenfunctions λ a and λ α issingle. In this subsection, we will study the case that one of the multiplicitiesof two Ricci-eigenfunctions λ a and λ α is single in some neighborhood U of p in M Ric ∩ {∇ f = 0 } . Types (ii), (iii) and (iv) for r = n − Subcase I. r = 1 and r = n − ≥ h = h and then locally the metric is given by g = ds + h ( s ) dt + g .Firstly, we claim R = 0. If not, then R = 0. (4.1) means λ a = X ′ + X , while (4.3)means λ a = − ( X ′ + X ). Hence we see λ a = X ′ + X = 0 and λ α = 0 by (4.4).This is a contradiction.Next, making use of (4.1) and (4.3) again, we see X ′ + X = − R n − = 0 and h ′′ h = − R n − = 0. On the other hand, (4.4) gives us ( r − k h = Rn − , whichimplies that g = h ˜ g is an Einstein metric with the Einstein constant Rn − .If R >
0, setting r = q R n − , then, h = C sin( r s ) for some constant C = 0and X = r cot( r s ). From f ′ X = f ( X ′ + X ), it follows that f = c cos( r s ) forsome nonzero constant c . Thus g = ds + sin ( r s ) dt + g by absorbing a constantinto dt and using λ α = k h = Rn − . Here s is the distance on S (cid:16) R n − (cid:17) from apoint.If R <
0, we set r = q − R n − . One can argue similarly as above, and get g = ds + sinh ( r s ) dt + g and f = c cosh( r s ) for some nonzero constant c . Consequently, we have a conclusion as follows.
Theorem 5.2.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature. Aussme locally the metric is given by g = ds + h ( s ) dt + g , and g is an Einstein metric. Then g is of nonzero scalar curvature R ; (i) when R > , ( M n , g ) is locally isometric to the Riemannian product S (cid:16) R n − (cid:17) × N n − and f = c cos (cid:16)q R n − s (cid:17) for some nonzero constant c , where s is thedistance on S (cid:16) R n − (cid:17) from a point and the Einstein manifold (cid:0) N n − , g (cid:1) is ofpositive Einstein constant Rn − . (ii) when R < , ( M n , g ) is locally isometric to the Riemannian product H (cid:16) R n − (cid:17) × N n − , where the Einstein manifold (cid:0) N n − , g (cid:1) is of negative Einstein constant Rn − ,and f = c cosh (cid:16)q − R n − s (cid:17) for some nonzero constant c . Subcase II. r = 1 and r = n − ≥ r = n − h = h and then locally the metric is g = ds + h ( s ) g + dt . Firstly,from (4.4) we see R = 0. Meanwhile, f ′ h ′ h = f ′ X = f ( X ′ + X ) = f h ′′ h implies ch ′ = f for a constant c = 0. By (4.2), we have ch ′′′ = f ′′ = − ( n − f ( X ′ + X ) = − ( n − ch ′ h ′′ h and h ( n − h ′′ = c for some constant c = 0. In fact, if c = 0, h ′′ = 0 and X ′ + X = h ′′ /h = 0. It follows from (4.3) that λ a = X ′ + X = 0 and λ α = 0.This is a contradiction.By (4.1) and (4.3), we see0 = 2( X ′ + X ) + ( n − X − kh ) = 2 h ′′ h + ( n − h ′ − kh , which implies 2 hh ′′ + ( n − h ′ − k ) = 0. Combining with h ( n − h ′′ = c , weobtain that h ′ + 2 c n − h − ( n − = k. Hence, in comparing with (2.23), it is easy to see that ( W n − = R × N , ¯ g = ds + h ( s ) g ) is an ( n − D tensor, and the scalar curvature ¯ R = R − λ n = 0.Consequently, we have the following conclusion in this subcase. Theorem 5.3.
Let ( M n , g, f ) , n ≥ , be a vacuum static space with harmoniccurvature satisfying (1.1) . Assume locally the metric is given by g = ds + h ( s ) g + dt , and g is an Einstein metric. Then ( M, g ) is locally isometric to a domain in ( W n − × R , g W + dt ) , where ( W n − , g W ) is an ( n − -dimensional D -flat vacuumstatic space of zero scalar curvature and f = ch ′ . SS WITH HARMONIC CURVATURE 15
The multiplicities of two Ricci-eigenfunctions λ a and λ α are morethan one. In this subsection, we will study the multiplicities of two Ricci eigen-functions λ a and λ α are more than one in some neighborhood U of a point p in M Ric ∩ {∇ f = 0 } . We have the local classification results, which form type (iv) ofTheorem 1.1 for 3 ≤ r ≤ n − r , r ≥ Y = 0 and X = 0. From f ′ ξ i = f ( ξ ′ i + ξ i ), we have f ′ h ′ h = f ′ X = f ( X ′ + X ) = f h ′′ h , and ch ′ = f for some constant c = 0. By using of equations (4.1) and (4.3), we see ch ′′′ = f ′′ = − f (cid:18) r ( X ′ + X ) + Rn − (cid:19) = − ch ′ r h ′′ h + Rn − ! . for some constant c , and(5.1) h ′′ + R ( n − r + 1) h = c h − r . From (4.1) and (4.3), we see0 = 2( X ′ + X ) + ( r − X − kh ) + Rn − h ′′ h + ( r − h ′ − kh + Rn − , which implies 2 h ′′ + ( r − h ′ − k ) + Rn − . Putting (5.1) into the above equation, we get(5.2) ( h ′ ) + 2 c r − h − ( r − + R ( n − r + 1) h = k. Now, we consider the manifold W r +1 = R × N r with ¯ g = ds + h g . Thescalar curvature is given by ¯ R = R − r λ α = r n − R . From (5.1) and (5.2), we haverespectively, h ′′ + ¯ Rr ( r + 1) h = c h − r and ( h ′ ) + 2 c r − h − ( r − + ¯ Rr ( r + 1) h = k. Moreover, f = ch ′ . It is now easy to see that ( W r +1 , ¯ g ) is D -flat vacuum staticspace explained in Section 2.Consequently, we have the following conclusion in this subcase. Theorem 5.4.
Let ( M n , g, f ) , ( n ≥ , be an n -dimensional vacuum static spacewith harmonic curvature satisfying (1.1) . Assume locally the metric is given by g = ds + h ( s ) g + g , where g and g are both Einstein metrics. Then ( M n , g ) is locally isometric to a domain in ( W r +1 × N n − − r , g = ¯ g + g ) , where ≤ r ≤ n − , ( W r +1 , ¯ g ) is an ( r + 1) -dimensional D -flat vacuum static space and (cid:0) N n − − r , g (cid:1) is an Einstein manifold of the Einstein constant Rn − . Finally, from Theorems 4.1, 2.2, 5.2, 5.3 and 5.4, it follows that among types(i)-(iv) Theorem 1.1, each type is different from the other three types. Therefore,Theorem 1.1 holds from continuity argument of complete Riemannian metrics.
Acknowledgments.
This work was completed while the author was visitingLehigh University from August, 2019 to August, 2020. She would like to thankher advisor Professor Huai-Dong Cao for his invaluable guidance, constant encour-agement and support. She is grateful to her advisors Professor Yu Zheng andProfessor Zhen Guo for their constant encouragement and support. She also wouldlike to thank Junming Xie, Jiangtao Yu, and other members of the geometry groupat Lehigh for their interest, helpful discussions, and suggestions during the prepa-ration of this paper. She also would like to thank the China Scholarship Council(No: 201906140158) for the financial support, and the Department of Mathemat-ics at Lehigh University for hospitality and for providing a great environment forresearch.
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