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Killing a with Keywords manifold a such in studied Cer are structure. this possess to shown popula are the spacetimes Kottler others, Among connection. metric symmetric semi Abstract scle h soitd1fr and 1-form associated the called is colo ahmtc n ttsis iuUiest fTe of University Qilu Statistics, and Mathematics of School M n , 1 ntepeetpprw nrdc eiqaiEnti man semi-quasi-Einstein a introduce we paper present the In nsm-us-isenManifold semi-Quasi-Einstein On )( 32,53D11. 53C20, us-isenMnfl;Sm-ymti ercconnecti metric Semi-symmetric Manifold; Quasi-Einstein n > S )i endt eaqaiEnti aiodi h ic tens Ricci the if manifold quasi-Einstein a be to defined is 2) ( X , Y a ) , b = , a -al [email protected] E-mail: 1 -al [email protected] E-mail: ,where 0, -al [email protected] E-mail: ( X ia 533 P.R.China, 250353, Jinan aag400 Malaysia 43000, Kajang , eba Zhao Peibiao aln Han Yanling Y ) vkDe Avik + b 1 η η ξ ( sannzr -omsc that such 1-form non-zero a is X scle h eeao ftemanifold. the of generator the called is ) iiTnuAdlRha,Bna ugiLong, sungai Bandar Rahman, Abdul Tunku siti η n ehooy ajn 104 .R China R. P. 210094, Nanjing Technology, and e hooySadn cdm fSciences), of Academy chnology(Shandong ( Y ) , ∀ S / X 1 / , 2019 Y S ancraueconditions curvature tain ie us-isenspace quasi-Einstein mixed ( ∈ X edeutosadalso and equations field s / , o-a Riemannian non-flat A . STG06 cwrshl and Schwarzschild r TM d fteespace-time these of udy no hn (No.11871275; China of on Y ag,sc squasi- as such range, r ) ensor. ensfil equations field tein’s = iaino h Ein- the of ximation / UM a fl rma from ifold 1 1 nteter of theory the in ( n Ricci- on; ltvt sa4- a is elativity / X 02 , / Y 6. o scalar for ) 1 ( X ξ , ) or = models can help us better understand the evolution of the universe. Quasi-Einstein manifold was defined by M. C. Chaki and R. K. Maity[2], later it was widely studied in [1, 4, 6, 7, 8, 9, 10, 11]. Z.L.Li [15] discussed the basic properties of quasi-Einstein manifold, obtained some geometric characteristics and proved the non-existence of some quasi-Eistein manifolds. U.C.De and G. C. Ghosh [9] proved a theorem for ex- istence of a QE-manifold and give some examples about QE-manifolds. V.A.Kiosak [14] considered the conformal mappings of quasi-Einstein spaces and proved that they are closed with respect to concircular mappings. M.M. Tripathi and J.S. Kim [13] studied a quasi-Einstein manifold whose generator belongs to the k-nullity distri- bution N(k) and have proved that conformally quasi-Einstein manifolds are certain N(k)-quasi Einstein manifolds. The purpose of this paper is to investigate quasi-Einstein manifolds from a rather different perspective. By replacing the Levi-Civita connection with a semi-symmetric metric connection [18], we define a semi-quasi-Einstein manifold which generalizes the concept of quasi-Einstein manifolds. Semi-symmetric metric connections are widely used to modify Einstein’s gravity theory in form of Einstein-Cartan gravity and also in unifying the gravitational and electromagnetic forces. Recently in [12] the field equations are derived from an action principle which is formed by the scalar curvature of a semi-symmetric metric connection. The derived equations contain the Einstein and Maxwell equations in vacuum. In fact, Schouten criticized Einstein’s argument for using a symmetric connection and used the notion of semi-symmetric connections in his approach [17]. We further prove the existence of such structure by analyzing the Schwarzschild and Kottler spacetimes and obtain a necessary and sufficient conditions for such a manifold to be an Einstein space. In [5] Murathan and Ozgur considered the semi-symmetric metric connection with unit parallel vector field P, and proved that R.R¯ = 0 if and only if M is semi- symmetric; if R.R¯ = 0 or R.R¯ − R¯ .R = 0 or M is semi-symmetric and R¯ .R¯ = 0, then M is conformally flat and quasi-Einstein. Motivated by this we further investigate the manifolds for Ricci symmetric and Ricci semi-symmetric criterion under similar ambience. The paper is organized as follows: Section 2 recalls the form and some curvature properties of semi-symmetric metric connection. Section 3 gives the main defini- tion of semi-quasi-Einstein manifold (S(QEn)) and discusses the relations between S(QEn), quasi-Einstein and generalized quasi-Einstein manifolds. Section 4 concerns with some physical and geometric characteristics of S(QEn) manifold under certain curvature conditions. Some interesting examples of S(QEn) manifold is constructed in the last section.

2 Preliminaries

Let (Mn, ∇) be a Riemannian manifold of dimension n and ∇ be the Levi-Civita connection compatible to the metric 1. A semi-symmetric metric connection is defined by ∇¯ XY = ∇XY + π(Y)X − 1(X, Y)P, ∀X, Y ∈ TM

2 where π is any given 1-form and P is the associated vector field, 1(X, P) = π(X). By direct computations, one can obtain R¯ (X, Y)Z = R(X, Y)Z + 1(AX, Z)Y − 1(AY, Z)X +1(X, Z)AY − 1(Y, Z)AX,

= ∇ − + 1 1(AY, Z) ( Yπ)(Z) π(Y)π(Z) 2 π(P)1(Y, Z) = ∇ − + 1 1( YP, Z) π(Y)π(Z) 2 π(P)1(Y, Z),

1(∇¯ YP, Z) − 1(∇¯ ZP, Y) = 1(∇YP, Z) − 1(∇ZP, Y)

S¯(Y, Z) = S(Y, Z) − (n − 2)(∇Yπ)(Z) + (n − 2)π(Y)π(Z)

−{(n − 2)π(P) + i 1(∇iP, ei)}1(Y, Z), (2.1) further one has P R¯ (X, Y)Z = −R¯ (Y, X)Z, R¯ (X, Y)Z + R¯ (Y, Z)X + R¯ (Z, X)Y = (1(AZ, Y) − 1(AY, Z))X +1(AX, Z) − 1(AZ, X))Y +(1(AY, X) − 1(AX, Y))Z. It is obvious that the Ricci tensor of the semi-symmetric metric connection is not symmetric unless under certain conditions. For example, if π is closed, then A is symmetric, thus S¯ is also symmetric. In particular, if a unit P satisfies Killing’s equation 1(∇XP, Y) + 1(∇YP, X) = 0, then 1 1(∇ P, Y) = 0, 1(∇ P, X) = 0, AX = ∇ P − π(X)P + X, P X X 2 R¯ (X, Y)P = R(X, Y)P + π(Y)∇XP − π(X)∇YP

R¯ (X, P)Y = R(X, P)Y + 1(∇XP, Y) − π(Y)∇XP,

R¯ (X, P)P = R(X, P)P + ∇XP,

S¯(Y, Z) = S(Y, Z) − (n − 2)1(∇YP, Z) + (n − 2)π(Y)π(Z) − (n − 2)1(Y, Z). S¯(Y, P) = S(Y, P) = S(P, Y) = S¯(P, Y). And if P is a unit parallel vector field with respect to ∇, 1 1(AX, Y) = 1(AY, X), AX = −π(X)P + X, 2 R¯ (X, Y, Z, W) = −R¯ (X, Y, W, Z), R¯ (X, Y)Z + R¯ (Y, Z)X + R¯ (Z, X)Y = 0, R(P, X, Y, Z) = R(X, Y, P, Z) = 0, R¯ (X, Y)P = R(X, Y)P = 0, R¯ (P, X, Y, Z) = R¯ (X, Y, P, Z) = 0, S¯(Y, Z) = S(Y, Z) + (n − 2)π(Y)π(Z) − (n − 2)π(P)1(Y, Z) = S¯(Z, Y), S¯(Y, P) = S(Y, P) = 0.

3 3 Semi-quasi-Einstein Manifolds

In this section, we define a new structure using the semi-symmetric metric con- nection. Definition 3.1. A Riemannian manifold (Mn, ∇¯ ), n = dimM ≥ 3, is said to be a semi-quasi- Einstein manifold S(QEn) if the Ricci curvature tensor components are non-zero and of the forms Sˆ(Y, Z) = symS¯(Y, Z) = a1(Y, Z) + bη(Y)η(Z), where a and b are scalars and η is a non-zero 1-form. If Sˆ(Y, Z) is identically zero, we say the manifold (Mn, ∇¯ ) is semi-Ricci flat. Theorem 3.1. If the generator P of manifold (Mn, ∇¯ ) is a Killing vector field with respect to ∇, then an Einstein manifold (Mn, ∇) isaS(QEn)manifold. Proof. In view of the relation (2.1), 1 Sˆ(Y, Z) = {S¯(Y, Z) + S¯(Z, Y)} 2 = S(Y, Z) − (n − 2)[(∇Yπ)(Z) + (∇Zπ)(Y)] + (n − 2)π(Y)π(Z)

−{(n − 2)π(P) + 1(∇iP, ei)}1(Y, Z). Xi

Notice that (∇Yπ)(Z) + (∇Zπ)(Y) = 1(∇YP, Z) + 1(∇ZP, Y), if P is a Killing vector field with respect to ∇, then one has

Sˆ(Y, Z) = S(Y, Z) + (n − 2)π(Y)π(Z) − (n − 2)π(P)1(Y, Z). (3.1)

Therefore, if (Mn, ∇) is Einstein, then (Mn, ∇¯ )isa S(QEn)manifold.  Remark 3.1. The above result isalso true if we replacethe Einstein manifold with a Ricci-flat manifold. Remark 3.2. If the generator P is a unit parallel vector, Sˆ(X, Y) = S¯(X, Y). Theorem 3.2. For a unit vector field P, (Mn, ∇¯ ) isaS(QEn) manifold, if for some scalar ρ

Sˆ(Y, Z)Sˆ(X, W) − Sˆ(X, Z)Sˆ(Y, W) = ρ(1(Y, Z)1(X, W) − 1(X, Z)1(Y, W)). (3.2)

Proof. Taking X = W = P in (3.2) we have

Sˆ(Y, Z)Sˆ(P, P) − Sˆ(P, Z)Sˆ(Y, P) = ρ{1(Y, Z) − 1(P, Y)1(P, Z)}

We denote by α = Sˆ(P, P), Sˆ(P, Z) = 1(QZˆ , P) = π(QZˆ ) = η(Z), then ρ ρ 1 Sˆ(Y, Z) = 1(Y, Z) − π(Y)π(Z) + η(Y)η(Z). (3.3) α α α On the other hand, putting X = P in (3.2) we have

Sˆ(Y, Z)η(W) − Sˆ(Y, W)η(Z) = ρ{1(Y, Z)π(W) − 1(Y, W)π(Z)} (3.4)

4 Substituting (3.3) into (3.4) we obtain ρ ρ {1(Y, Z)η(W) − 1(Y, W)η(Z)}− {η(W)η(Z)π(Y) − π(W)π(Y)η(Z)} α α = ρ{1(Y, Z)π(W) − 1(Y, W)π(Z)} (3.5)

Now in (3.5) , put Y = Z = ei and take sum about i, we get η(W) = απ(W) (3.6) we further obtain by substituting (3.6) into (3.3) ρ ρ Sˆ(Y, Z) = 1(Y, Z) + (α − )π(Y)π(Z). α α The proof is finished. 

4 (Mn, ∇¯ )-manifold with unit generator

In this section we consider a (Mn, ∇¯ )-manifold with unit generator kPk = 1. Definition 4.1. (Mn, ∇¯ ) is said to be ∇¯ -symmetric if the curvature tensor satisfies ∇¯ R¯ = 0; ∇¯ -Ricci symmetric if ∇¯ Sˆ = 0; semi-symmetric if R¯ · R¯ = 0; Ricci semi-symmetric if R¯ · Sˆ = 0. If H is a(0, k)-type tensor field, we define the operation R.H by

(R(X, Y) · H)(W1, ··· , Wk) = −H(R(X, Y)W1, ··· , Wk)

−···− H(W1, ··· , R(X, Y)Wk). If θ is a symmetric (0, 2)-type tensor, the following formula can define a (0, k+2)-type tensor

Q(θ, H)(W1, ··· , Wk; X, Y) = −H((X ∧θ Y)W1, ··· , Wk)

−···− H(W1, ··· , (X ∧θ Y)Wk), (4.1)

where X ∧θ Y is given by (X ∧θ Y)Z = θ(Y, Z)X − θ(X, Z)Y. When P is a unit Killing field, based on the relation (3.1) we get

Sˆ(Y, Z) = S(Y, Z) + (n − 2)π(Y)π(Z) − (n − 2)1(Y, Z). By direct calculations we obtain

(∇¯ XSˆ)(Y, Z) = (∇XS)(Y, Z) + (n − 2){(∇Xπ)(Y)π(Z) + (∇Xπ)(Z)π(Y)} −π(Y)[S(X, Z) + (n − 2)π(X)π(Z) − (n − 2)1(X, Z)] −π(Z)[S(Y, X) + (n − 2)π(X)π(Y) − (n − 2)1(X, Y)] +π(QY)1(X, Z) + π(QZ)1(X, Y) (4.2)

= (∇XS)(Y, Z) +1(X, Z)S¯(Y, P) − 1(Y, P)S¯(X, Z) + 1(X, Y)S¯(Z, P) − 1(Z, P)S¯(X, Y) which derivates the following:

5 Theorem 4.1. If the generator P is unit Killing, (Mn, ∇) is Ricci symmetric if and only if

(∇¯ XSˆ)(Y, Z) = 1(X, Z)S¯(Y, P) − 1(Y, P)S¯(X, Z) + 1(X, Y)S¯(Z, P) − 1(Z, P)S¯(X, Y).

Corollary 4.2. Let P be a unit parallel vector field, then (Mn, ∇) is Ricci symmetric if and only if

(∇¯ XSˆ)(Y, Z) = −π(Y)S¯(X, Z) − π(Z)S¯(X, Y).

Alternately, for a specific form of the Ricci curvature S(X, Y) we have the follow- ing:

Theorem 4.3. Let S(Y, Z) = (n − 2)1(Y, Z) − (n − 2)π(Y)π(Z), kPk = 1. If the manifold (Mn, ∇) is Ricci symmetric then (Mn, ∇¯ ) is ∇¯ -Ricci symmetric.

Proof. Let S(Y, Z) = (n−2)1(Y, Z)−(n−2)π(Y)π(Z) and(∇xS)(Y, Z) = 0. These together imply (∇Xπ)(Y)π(Z) + (∇Xπ)(Z)π(Y) = 0 which means ∇P = 0 for a unit vector field P. Hence, S¯(Y, P) = 0. Therefore, using (4.2) we conclude

(∇¯ XSˆ)(Y, Z) = (∇XS)(Y, Z) − π(Y)[S(X, Z) + (n − 2)π(X)π(Z) − (n − 2)1(X, Z)] − π(Z)[S(Y, X) + (n − 2)π(X)π(Y) − (n − 2)1(X, Y)]

Hence the result. 

Theorem 4.4. In general, if (Mn, ∇¯ ) is a ∇¯ -Ricci symmetric S(QEn) manifold, Sˆ(X, Y) = a1(X, Y)+bη(X)η(Y) with a unit vector ξ associated to the one-form η, then a+b = constant.

Proof. We have

(∇¯ XSˆ)(Y, Z) = X(a)1(Y, Z) + X(b)η(Y)η(Z)

+b[(∇¯ Xη)(Y)η(Z) + (∇¯ Xη)(Z)η(Y)]. (4.3)

Putting Y = Z = ξ in (4.3), we finish the proof. 

Theorem 4.5. Let the generator P in (Mn, ∇¯ ) be a unit parallel vector field and R¯ · S¯ = −Q(1, S¯), if (Mn, ∇) is Ricci semi-symmetric then M is a quasi-Einstein manifold of the form

S(X, Y) = (n − 2)1(X, Y) − (n − 2)π(X)π(Y).

Conversely, if the Ricci curvature satisfies S(X, Y) = (n − 2)1(X, Y) − (n − 2)π(X)π(Y), then M is Ricci semi-symmetric.

6 Proof. we have −(R¯ (X, Y) · S¯)(Z, W) = S¯(R¯ (X, Y)Z, W) + S¯(Z, R¯ (X, Y)W) = S(R(X, Y)Z, W) + S(Z, R(X, Y)W) +1(AX, Z)S(Y, W) + 1(AX, W)S(Y, Z) −1(AY, Z)S(X, W) − 1(AY, W)S(X, Z) +1(X, Z)S(AY, W) + 1(X, W)S(AY, Z) −1(Y, Z)S(AX, W) − 1(Y, W)S(AX, Z) +(n − 2)[π(W)R¯ (X, Y, Z, P) + π(Z)R¯ (X, Y, W, P)] = −(R(X, Y) · S)(Z, W) + Q(1, S)(Z, W; X, Y) +π(Y)π(W)S(X, Z) − π(X)π(W)S(Y, Z) +π(Y)π(Z)S(X, W) − π(X)π(Z)S(Y, W) = −(R(X, Y) · S)(Z, W) + Q(1, S¯)(Z, W; X, Y) +[S(X, Z) − (n − 2)1(X, Z)]π(Y)π(W) −[S(Y, Z) − (n − 2)1(Y, Z)]π(X)π(W) +[S(X, W) − (n − 2)1(X, W)]π(Y)π(Z) −[S(Y, W) − (n − 2)1(Y, W)]π(X)π(Z), where the last equality follows from Q(1, S¯)(Z, W; X, Y) = −S¯((X ∧ Y)Z, W) − S¯(Z, (X ∧ Y)W) = Q(1, S)(Z, W; X, Y) + (n − 2)[1(X, Z)π(Y)π(W) −1(Y, Z)π(X)π(W) + 1(X, W)π(Y)π(Z) −1(Y, W)π(X)π(Z)]. Since R¯ · S¯ = −Q(1, S¯), we have (R(X, Y) · S)(Z, W) = [S(X, Z) − (n − 2)1(X, Z)]π(Y)π(W) − [S(Y, Z) − (n − 2)1(Y, Z)]π(X)π(W) + [S(X, W) − (n − 2)1(X, W)]π(Y)π(Z) − [S(Y, W) − (n − 2)1(Y, W)]π(X)π(Z). (4.4) Since R · S = 0, using Y = W = P in the above equation we get the result. The converse part is quite obvious at this point. 

5 Someexamples

Example 5.1. The first example we consider is the popular four dimensional Schwarzschild spacetime M with the Ricci-flat metric 2 m 1 1 = − 1 dt ⊗ dt + − dr ⊗ dr + r2dθ ⊗ dθ + r2 sin (θ)2 dφ ⊗ dφ, r 2 m −   r 1! ∂ and a Killing vector ∂t in this spacetime.

7 The corresponding nontrivial Levi-Civita connection components are:

Γ t = − m Γ r = − 2 m2−mr Γ r = m Γ r = − Γ φ = cos(θ) tr 2 mr−r2 , t t r3 , rr 2 mr−r2 , θθ 2 m r, θ φ sin(θ) , Γ r = − 2 Γ θ = 1 Γ θ = − Γ φ = 1  φ φ (2 m r) sin (θ) , r θ r , φ φ cos (θ) sin (θ) , r φ r .   We define a semi-symmetric metric connection corresponding to the one-form = 2m−r ∂ π ( r )dt associated to the Killing vector ∂t by ¯ k k k k k ik Γij = Γij + πjδi − 1ijπ , π = 1 πi.

We calculate the nontrivial components as:

Γ¯ t = − m Γ¯ t = − m Γ¯ t = r Γ¯ t = − 2 Γ¯ t = − 2 2 tr 2 mr−r2 , r t 2 mr−r2 , rr 2 m−r , θθ r , φ φ r sin (θ) , r 2 m2−mr r 2 m−r r m r r 2 Γ¯ = − 3 , Γ¯ = , Γ¯ = 2 , Γ¯ = 2 m − r, Γ¯ = (2 m − r) sin (θ) ,  t t r r t r rr 2 mr−r θθ φ φ Γ¯ θ = 1 Γ¯ θ = 2 m−r Γ¯ θ = 1 Γ¯ θ = − Γ¯ φ = 1 Γ¯ φ = cos(θ)  r θ r , θ t r , θ r r , φ φ cos (θ) sin (θ) , r φ r , θ φ sin(θ) ,  Γ¯ φ = 2 m−r Γ¯ φ = 1 Γ¯ φ = cos(θ)  t r , r r , sin(θ) .  φ φ φθ   The corresponding Ricci curvature components S¯ij are: − m 0 r2 0 0 m  r2 10 0   2 −   0 0 r 2mr 0   2 2   00 0 − 2 mr − r sin (θ)    
 = r−2m =  For a r and b 1, we check that:

0 = Sˆtt = a1tt + bπtπt

1 = Sˆrr = a1rr + bπrπr 2 r − 2mr = Sˆθθ = a1θθ + bπθπθ 2 2 −(2mr − r )sin θ = Sˆφφ = a1φφ + bπφπφ

Hence, (M, ∇¯ )isa S(QEn) manifold.

Example 5.2. We can further consider more complicated 4-dimenisonal Kottler spacetime with metric [19]

1 2 m 3 1 = Λr2 + − 1 dt⊗dt+ − dr⊗dr+r2dθ⊗dθ+r2 sin (θ)2 dφ⊗dφ 3 r Λ 2 + 6 m −   r r 3! as anexampleof anEinsteinspacecarryingaS(QEn) structure with a Killing generator. This spacetime satisfies Einstein’s field equations of general relativity with positive cosmological constant for a vacuum space around a center of spherical symmetry. The manifold is also called the Schwarzschild-de Sitter spacetime and provides us with a two-parameter family of

8 static spacetime with compactspacelikeslices, whichare locally (but not globally) conformally flat. The Levi-Civita components of this metric are:

Γ t = Λr3−3 m Γ r = Λ2r6+3 Λmr3−3 Λr4−18 m2+9 mr Γ r = − Λr3−3 m tr Λr4+6 mr−3 r2 , t t 9 r3 , rr Λr4+6 mr−3 r2 Γ r = 1 Λr3 + 2 m − r, Γ r = 1 Λr3 + 6 m − 3 r sin (θ)2 , Γ θ = 1  θθ 3 φ φ 3 r θ r Γ θ = − Γ φ = 1 Γ φ = cos(θ)  φ φ cos (θ) sin (θ) , r φ r , θ φ sin(θ)
  The Ricci tensor Sij is calculated as − Λ2r3+6 Λm−3 Λr 3 r 00 0 3 Λr  0 Λr3+6 m−3 r 0 0   −Λ 2   0 0 r 0   2 2   0 00 −Λr sin (θ)      Clearly Sij = −Λ1ijin this spacetime. Hence it is an Einstein space.  = Λr3+6 m−3 r The semi-symmetric metric connection corresponding to the one-form π ( 3 r )dt ∂ associated to the Killing vector ∂t is given by ¯ k k k k k ik Γij = Γij + πjδi − 1ijπ , π = 1 πi. We calculate its nontrivial components as:

Γ¯ t = Λr3−3 m Γ¯ t = Λr3−3 m Γ¯ t = 3 r Γ¯ t = − 2 Γ¯ t = − 2 2 tr Λr4+6 mr−3 r2 , r t Λr4+6 mr−3 r2 , rr Λr3+6 m−3 r , θθ r , φ φ r sin (θ) 2 6 3 4 2 3 3 Γ¯ r = Λ r +3 Λmr −3 Λr −18 m +9 mr , Γ¯ r = Λr +6 m−3 r , Γ¯ r = − Λr −3 m , Γ¯ φ = cos(θ)  t t 9 r3 r t 3 r rr Λr4+6 mr−3 r2 φθ sin(θ)  3 Γ¯ r = 1 Λ 3 + − Γ¯ r = 1 Λ 3 + − 2 Γ¯ θ = 1 Γ¯ θ = Λr +6 m−3 r  θθ 3 r 2 m r, φ φ 3 r 6 m 3 r sin (θ) , r θ r , θ t 3 r  3 Γ¯ θ = − Γ¯ φ = 1 Γ¯ φ = cos(θ) Γ¯ φ = Λr +6 m−3 r Γ¯ φ = 1 Γ¯ θ = 1  cos (θ) sin (θ) , r , ( ) ,
r , r , r  φ φ r φ θ φ sin θ φ t 3 φ r θ r   The corresponding Ricci curvature components S¯ij are: − Λ2r3+6 Λm−3 Λr Λr3−3 m 3 r 3 r2 0 0 3  − Λr3−3m Λr +3(Λ−1)r+6m  3r2 Λr3+6m−3r 0 0  4 2   − Λr +3(Λ−1)r +6mr   0 0 3 0   4 2   − (Λr +3(Λ−1)r +6mr) 2   0 0 0 3 sin(θ)     2   = −Λ − Λr +6m−3r = ¯  For a 3r and b 1, we check that the nontrivial Sij components satisfy: Λ2r3 + 6Λm − 3Λr = Sˆ = a1 + bπ π 3r tt tt t t Λr3 + 3(Λ − 1)r + 6m = Sˆ = a1 + bπ π Λr3 + 6m − 3r rr rr r r Λr4 + 3(Λ − 1)r2 + 6mr = Sˆ = a1 + bπ π 3 θθ θθ θ θ Λr4 + 3(Λ − 1)r2 + 6mr sin2 θ = Sˆ = a1 + bπ π 3 φφ φφ φ φ Hence, (M, ∇¯ ) isaS(QEn) manifold.

9 Example 5.3. We consider a three-dimensional manifold M3 endowed with a metric

i j x1 1 2 2 2 3 2 1ijdx dx = e [(dx ) − (dx ) ] + (dx )

1 x1− 1 x2 1 1 x1− 1 x2 2 3 and a non-zero 1-form π = e 2 2 dx − e 2 2 dx + dx . Γ1 = The non-zero components of the corresponding Levi-Civita connnection are 11 Γ2 = Γ1 = 1 12 22 2 and the manifold is Ricci flat. Furthermore, π is a parallel one-form with respect to this connection. The semi-symmetric metric connection corresponding to the 1-form π is given by

¯ k k k k Γij = Γij + πjδi − 1ijπ , and the non-zero coefficients are

1 1 2 1 x1− 1 x2 3 x1 1 1 x1− 1 x2 2 1 1 Γ¯ = Γ¯ = − 2 2 Γ¯ = − Γ¯ = − 2 2 Γ¯ = Γ¯ = 11 2 , 11 e , 11 e , 12 e , 12 2 , 13 1, 2 2 1 1 x1− 1 x2 1 1 1 x1− 1 x2 3 x1 3 1 x1− 1 x2 Γ¯ = 1, Γ¯ = + e 2 2 , Γ¯ = + e 2 2 , Γ¯ = e , Γ¯ = e 2 2 ,  23 21 2 22 2 22 31  3 1 x1− 1 x2 1 − 1 x1− 1 x2 2 Γ¯ = −e 2 2 , Γ¯ = −e 2 2 = Γ¯ .  32 33 33   The Ricci curvature components S¯ij satisfies

x1 x1−x2 Sˆ11 = −e + e = −111 + π1π1, Sˆ = −ex1−x2 = −1 +  12 12 π1π2,  1 1− 1 2  ˆ 2 x 2 x S13 = e = −113 + π1π3,   ˆ x1 x1−x2 S22 = e + e = −122 + π2π2,  1 1− 1 2 ˆ 2 x 2 x S23 = −e = −123 + π2π3,  Sˆ = 0 = −1 + π π ,  33 33 3 3   now we take a = −1 and b = 1, then

S¯ij = a1ij + bπiπj, i, j = 1, 2, 3 therefore (M3, ∇¯ )isa S(QEn) manifold.

6 Acknowledgments

The third author would like to thank Professor H. Z. Li for his encouragement and help!

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