# Multiply Warped Products with a Semisymmetric Metric Connection

Authors: Yong Wang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We study the Einstein multiply warped products with a semisymmetric metric connection and the multiply warped products with a semisymmetric metric connection with constant scalar curvature, and we apply our results to generalized Robertson-Walker space-times with a semisymmetric metric connection and generalized Kasner space-times with a semisymmetric metric connection and find some new examples of Einstein manifolds with a semisymmetric metric connection and manifolds with constant scalar curvature with a semisymmetric metric connection.


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## Body

## 1. Introduction

The (singly) warped productB×bF of two pseudo-Riemannian manifolds (B,gB) and (F,gF) with a smooth function b:B→(0,∞) is the product manifold B×F with the metric tensor g=gB⊕b2gF. Here, (B,gB) is called the base manifold and (F,gF) is called the fiber manifold and b is called the warping function. Generalized Robertson-Walker space-times and standard static space-times are two well-known warped product spaces. The concept of warped products was first introduced by Bishop and O’Neill (see [1]) to construct examples of Riemannian manifolds with negative curvature. In Riemannian geometry, warped product manifolds and their generic forms have been used to construct new examples with interesting curvature properties since then. In [2], Dobarro and Dozo had studied from the viewpoint of partial differential equations and variational methods the problem of showing when a Riemannian metric of constant scalar curvature can be produced on product manifolds by a warped product construction. In [3], Ehrlich et al. got explicit solutions to warping function to have a constant scalar curvature for generalized Robertson-Walker space-times. In [4], explicit solutions were also obtained for the warping function to make the space-time as Einstein when the fiber is also Einstein.One can generalize singly warped products to multiply warped products. Briefly, a multiply warped product(M,g) is a product manifold of form M=B×b1F1×b2F2⋯×bmFm with the metric g=gB⊕b12gF1⊕b22gF2⋯⊕bm2gFm, where for each i∈{1,…,m},bi:B→(0,∞) is smooth and (Fi,gFi) is a pseudo-Riemannian manifold. In particular, when B=(c,d) with the negative definite metric gB=-dt2 and (Fi,gFi) is a Riemannian manifold, we call M the multiply generalized Robertson-Walker space-time. In [5], Dobarro and Ünal studied Ricci-flat and Einstein-Lorentzian multiply warped products and considered the case of having constant scalar curvature for multiply warped products and applied their results to generalized Kasner space-times.Singly warped products have a natural generalization. A twisted product(M,g) is a product manifold of form M=B×bF, with a smooth function b:B×F→(0,∞), and the metric tensor g=gB⊕b2gF. In [6], they showed that mixed Ricci-flat twisted products could be expressed as warped products. As a consequence, any Einstein twisted products are warped products. In this paper, we define the multiply twisted products as generalizations of multiply warped products and twisted products. A multiply twisted product (M,g) is a product manifold of form M=B×b1F1×b2F2⋯×bmFm with the metric g=gB⊕b12gF1⊕b22gF2⋯⊕bm2gFm, where for each i∈{1,…,m},bi:B×Fi→(0,∞) is smooth.The definition of a semisymmetric metric connection was given by Hayden in [7]. In 1970, Yano [8] considered a semisymmetric metric connection and studied some of its properties. He proved that a Riemannian manifold admitting the semisymmetric metric connection has vanishing curvature tensor if and only if it is conformally flat. Motivated by the Yano result, in [9], Sular and Özgür studied warped product manifolds with a semisymmetric metric connection; they computed curvature of semisymmetric metric connection and considered Einstein warped product manifolds with a semisymmetric metric connection. In this paper, we considered multiply twisted products with a semisymmetric metric connection and computed the curvature of a semisymmetric metric connection. We showed that mixed Ricci-flat multiply twisted products with a semisymmetric metric connection can be expressed as multiply warped products which generalizes the result in [6]. We also studied the Einstein multiply warped products with a semisymmetric metric connection and multiply warped products with a semisymmetric metric connection with constant scalar curvature; we applied our results to generalized Robertson-Walker space-times with a semisymmetric metric connection and generalized Kasner space-times with a semisymmetric metric connection and we found some new examples of Einstein affine manifolds and affine manifolds with constant scalar curvature. We also classified generalized Einstein Robertson-Walker space-times with a semisymmetric metric connection and generalized Einstein Kasner space-times with a semisymmetric metric connection.Semisymmetric metric connections have some physical applications. In [10, 11], they considered orthogonal connections with arbitrary torsion on compact Riemannian manifolds. For the induced Dirac operators, twisted Dirac operators, and Dirac operators of Chamseddine-Connes type, they computed the spectral action. In addition to the Einstein-Hilbert action and the bosonic part of the standard model Lagrangian, they found the Holst term from loop quantum gravity, a coupling of the Holst term to the scalar curvature, and a prediction for the value of the Barbero-Immirzi parameter. For connections whose torsion is not zero, they showed that the Holst action can be recovered from the heat asymptotics for the natural Dirac operator acting on spinor fields. For the physical consequences of the use of torsion connections in Lorentzian geometry, we refer to the classical review [12] and the more recent overview [13] and references therein.This paper is arranged as follows. In Section2, we compute curvature of multiply twisted products with a semisymmetric metric connection. In Section 3, we study the special multiply warped products with a semisymmetric metric connection. In Section 4, we study the generalized Robertson-Walker space-times with a semisymmetric metric connection. In Section 5, we consider the generalized Kasner space-times with a semisymmetric metric connection.


## 2. Preliminaries

LetM be a Riemannian manifold with Riemannian metric g. A linear connection ∇¯ on a Riemannian manifold M is called asemisymmetric connection if the torsion tensor T of the connection ∇¯(1)T(X,Y)=∇¯XY-∇¯YX-[X,Y]
satisfies
(2)T(X,Y)=π(Y)X-π(X)Y,
where π is a one form associated with the vector field P on M defined by π(X)=g(X·P).  ∇¯ is called asemisymmetric metric connection if it satisfies ∇¯g=0. If ∇ is the Levi-Civita connection of M, the semisymmetric metric connection ∇¯ is given by
(3)∇¯XY=∇XY+π(Y)X-g(X,Y)P,
(see [8]). Let R and R¯ be the curvature tensors of ∇ and ∇¯, respectively. Then R and R¯ are related by
(4)R¯(X,Y)Z=R(X,Y)Z+g(Z,∇XP)Y-g(Z,∇YP)X+g(X,Z)∇YP-g(Y,Z)∇XP+π(P)[g(X,Z)Y-g(Y,Z)X]+[g(Y,Z)π(X)-g(X,Z)π(Y)]P+π(Z)[π(Y)X-π(X)Y],
for any vector fields X,Y,Z on M [8]. By (3) and Proposition 2, we have amultiply twisted product(M,g) which is a product manifold of form M=B×b1F1×b2F2⋯×bmFm with the metric g=gB⊕b12gF1⊕b22gF2⋯⊕bm2gFm, where for each i∈{1,…,m},bi:B×Fi→(0,∞) is smooth.Here,(B,gB) is called the base manifold and (Fi,gFi) is called the fiber manifold and bi is called the twisted function. Obviously, twisted products and multiply warped products are the special cases of multiply twisted products.Proposition 1 (compare with [5, Proposition 2.2]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y∈Γ(TB) and U∈Γ(TFi), W∈Γ(TFj) and P∈Γ(TB). Then(1)
∇¯XY=∇¯XBY;(2)
∇¯XU=(X(bi)/bi)U;(3)
∇¯UX=[X(bi)/bi+π(X)]U;(4)
∇¯UW=0ifi≠j;(5)
∇¯UW=U(ln⁡bi)W+W(ln⁡bi)U-(gFi(U,W)/bi)gradFibi-bigFi(U,W)gradBbi+∇UFiW-g(U,W)Pifi=j,where ∇¯B denotes the semisymmetric metric connection on B and gradBbi,gradFibi denote the gradient vector fields on B and Fi, respectively.Proof.
Similar to Proposition 2.2 in [5] and Proposition 1 in [6], we have(1)
∇XY=∇XBY;(2)
∇XU=∇UX=(X(bi)/bi)U;(3)
∇UW=0 if i≠j;(4)
∇UW=U(ln⁡bi)W+W(ln⁡bi)U-(gFi(U,W)/bi)gradFibi-bigFi(U,W)gradBbi+∇UFiW if i=j.So by (3), we get Proposition 1.Similar to the proof of Proposition1, we get the following.Proposition 2 (compare with [5, Proposition 2.2]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y∈Γ(TB) and U∈Γ(TFi), W∈Γ(TFj) and P∈Γ(TFk) for a fixed k. Then(1)
∇¯XY=∇XBY-g(X,Y)P;(2)
∇¯XU=(X(bi)/bi)U+g(P,U)X;(3)
∇¯UX=(X(bi)/bi)U;(4)
∇¯UW=g(W,P)Uifi≠j;(5)
∇¯UW=U(ln⁡bi)W+W(ln⁡bi)U-(gFi(U,W)/bi)gradFibi-bigFi(U,W)gradBbi+∇UFiW+π(W)U-g(U,W)Pifi=j.Remark 3.
When for eachi∈{1,…,m},bi:B→(0,∞) is smooth and P=0, we get Proposition 2.2 in [5] by Propositions 1 and 2.Define the curvature, Ricci curvature, and scalar curvature as follows:(5)R(X,Y)Z=∇X∇Y-∇Y∇X-∇[X,Y],Ric(X,Y)=∑k‍εk〈R(X,Ek)Y,Ek〉,S=∑k‍εkRic(Ek,Ek),
where Ek is an orthonormal base of M with 〈Ek,Ek〉=εk. The Hessian of f is defined by Hf(X,Y)=XYf-(∇XY)f.Proposition 4 (compare with [5, Proposition 2.4]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y,Z∈Γ(TB) and V∈Γ(TFi), W∈Γ(TFj), U∈Γ(TFk), and P∈Γ(TB). Then(1)
R¯(X,Y)Z=R¯B(X,Y)Z;(2)
R¯(V,X)Y=-[(HBbi(X,Y)/bi)+(P(bi)/bi)g(X,Y)+π(P)g(X,Y)+g(Y,∇XP)-π(X)π(Y)]V;(3)
R¯(X,V)W=R¯(V,W)X=R¯(V,X)W=0ifi≠j;(4)
R¯(X,Y)V=0;(5)
R¯(V,W)X=VX(ln⁡bi)W-WX(ln⁡bi)Vifi=j;(6)
R¯(V,W)U=0ifi=j≠kori≠j≠k;(7)
R¯(U,V)W=-g(V,W)(gB(gradBbi,gradBbk)/bibk)U-g(V,W)((P(bi)/bi)+(P(bk)/bk))U-π(P)g(V,W)U,ifi=j≠k;(8)
R¯(X,V)W=[WX(ln⁡bi)]V-g(W,V)·[(∇XB(gradBbi)/bi)+(gradFi(Xln⁡bi)/bi2)+(P(bi)/bi)X+∇XP+π(P)X-π(X)P]ifi=j;(9)
R¯(U,V)W=g(U,W)gradB(V(ln⁡bi))-g(V,W)gradB(U(ln⁡bi))+RFi(U,V)W-((|gradBbi|B2/bi2)+2(P(bi)/bi)+π(P))(g(V,W)U-g(U,W)V)ifi=j=k.Proof.
Similar to Proposition 2.4 in [5], we have(1)
R(X,Y)Z=RB(X,Y)Z;(2)
R(V,X)Y=-(HBbi(X,Y)/bi)V;(3)
R(X,V)W=R(V,W)X=R(V,X)W=0 if i≠j;(4)
R(X,Y)V=0;(5)
R(V,W)X=VX(ln⁡bi)W-WX(ln⁡bi)V if i=j;(6)
R(V,W)U=0 if i=j≠k or i≠j≠k;(7)
R(U,V)W=-g(V,W)(gB(gradBbi,gradBbk)/bibk)U, if i=j≠k;(8)
R(X,V)W=-(g(V,W)/bi)∇XB(gradBbi)+[WX(ln⁡bi)]V-gFi(W,V)gradFi(Xln⁡bi) if i=j;(9)
R(V,W)U=g(V,U)gradB(W(ln⁡bi))-g(W,U)gradB(V(ln⁡bi))+RFi(V,W)U-(|gradBbi|B2/bi2)(g(W,U)V-g(V,U)W) if i=j=k.Then by (4), we get Proposition 4.Similarly we may get the following.Proposition 5 (compare with [5, Proposition 2.4]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y,Z∈Γ(TB) and V∈Γ(TFi), W∈Γ(TFj), U∈Γ(TFk), and P∈Γ(TFl) for a fixed l. Then(1)
R¯(X,Y)Z=RB(X,Y)Z+[g(X,Z)(Ybl/bl)-g(Y,Z)(Xbl/bl)]P+π(P)[g(X,Z)Y-g(Y,Z)X];(2)
R¯(V,X)Y=-(HBbi(X,Y)/bi)V-π(P)g(X,Y)Vifi≠l;(3)
R¯(V,X)Y=-(HBbi(X,Y)/bi)V-π(V)(Y(bi)/bi)X-g(X,Y)∇VP-g(X,Y)[π(P)V-π(V)P]ifi=l;(4)
R¯(X,V)W=(X(bl)/bl)π(W)Vifi≠j;(5)
R¯(V,W)X=-δil(π(V)/bi)X(bi)W+δjl(π(W)/bj)X(bj)Vifi≠j;(6)
R¯(X,Y)V=π(V)[(X(bl)/bl)Y-(Y(bl)/bl)X];(7)
R¯(V,W)X=VX(ln⁡bi)W-WX(ln⁡bi)V-δil(X(bi)/bi)[π(V)W-π(W)V]ifi=j;(8)
R¯(V,W)U=0ifi=j≠k or i≠j≠k;(9)
R¯(U,V)W=-g(V,W)(gB(gradBbi,gradBbk)/bibk)U-g(W,∇VP)U-g(V,W)∇UP-π(P)g(V,W)U+g(V,W)π(U)P+π(W)[π(V)U-π(U)V],ifi=j≠k;(10)
R¯(X,V)W=[WX(ln⁡bi)]V-g(W,V)(∇XB(gradBbi)/bi)-gradFi(Xln⁡bi)gFi(W,V)+(X(bl)/bl)π(W)V-g(W,∇VP)X-g(V,W)(X(bl)/bl)P-g(V,W)π(P)X+π(V)π(W)Xifi=j;(11)
R¯(U,V)W=g(U,W)gradB(V(ln⁡bi))-g(V,W)gradB(U(ln⁡bi))+RFi(U,V)W-(|gradBbi|B2/bi2)(g(V,W)U-g(U,W)V)+π(P)[g(U,W)V-g(V,W)U]ifi=j=k≠l;(12)
R¯(U,V)W=g(U,W)gradB(V(ln⁡bi))-g(V,W)gradB(U(ln⁡bi))+RFi(U,V)W-(|gradBbi|B2/bi2)(g(V,W)U-g(U,W)V)+g(W,∇UP)V-g(W,∇VP)U+g(U,W)∇VP-g(V,W)∇UP+π(P)[g(U,W)V-g(V,W)U]+[g(V,W)π(U)-g(U,W)π(V)]P+π(W)[π(V)U-π(U)V]ifi=j=k=l, where δil denotes the Kronecker symbol.Remark 6.
When for eachi∈{1,…,m},bi:B→(0,∞) is smooth and P=0, we get Proposition 2.4 in [5] by Propositions 4 and 5.By Propositions4 and 5 and the definition of the Ricci curvature tensor, we have the following.Proposition 7 (compare with [5, Proposition 2.5]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y,Z∈Γ(TB) and V∈Γ(TFi), W∈Γ(TFj), and P∈Γ(TB). Then(1)
Ric¯(X,Y)=Ric¯B(X,Y) + ∑i=1m‍li[(HBbi(X,Y)/bi) + (P(bi)/bi)g(X,Y)+π(P)g(X,Y) + g(Y,∇XP) - π(X)π(Y)];(2)
Ric¯(X,V)=Ric¯(V,X)=(li-1)[VX(ln⁡bi)];(3)
Ric¯(V,W)=0ifi≠j;(4)
Ric¯(V,W)=RicFi(V,W) + [(ΔBbi/bi) + (li-1)(|gradBbi|B2/bi2)  +   ∑j≠i‍lj(gB(gradBbi,gradBbj)/bibj) + (n¯-2)π(P) + ∑k=1n‍εk〈∇EkP,Ek〉 + ∑j≠i‍lj(Pbj/bj) + (n¯+li-2)(Pbi/bi)]g(V,W)ifi=j,where Ek,1≤k≤n is an orthonormal base of B with εk=g(Ek,Ek) and dim⁡B=n,dim⁡M=n¯.By(2) and (3) in Proposition 7, similar to the proof of Theorem 1 in [6], we have the following.Corollary 8.
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and dim⁡Fi>1 and P∈Γ(TB); then (M,∇¯) is mixed Ricci-flat if and only if M can be expressed as a multiply warped product. In particular, if (M,∇¯) is Einstein, then M can be expressed as a multiply warped product.Similar to Proposition7, we have the following.Proposition 9 (compare with [5, Proposition 2.5]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and let X,Y,Z∈Γ(TB) and V∈Γ(TFi), W∈Γ(TFj), and P∈Γ(TFr) for a fixed r. Then(1)
Ric¯(X,Y)=RicB(X,Y)+∑i=1m‍li(HBbi(X,Y)/bi)+g(X,Y)π(P)(n¯-2)   +  g(X,Y)∑q=1lr‍εqg(∇EqP,Eq),whereEqfor1≤q≤lris an orthonormal basis ofTFr;(2)
Ric¯(X,V)=(li-1)[VX(ln⁡bi)]+(n¯-2)(X(br)/br)π(V);(3)
Ric¯(V,X)=(li-1)[VX(ln⁡bi)]+(2-n¯)(X(br)/br)π(V);(4)
Ric¯(V,W)=0ifi≠j;(5)
Ric¯(V,W)=RicFi(V,W)+g(V,W)[(ΔBbi/bi) + (li-1)(|gradBbi|B2/bi2) + ∑k,k≠i‍lk(gB(gradBbi,gradBbk)/bibk) + (n¯-2)π(P)]+ (n¯-2)g(W,∇VP)+(2-n¯)π(V)π(W)+g(V,W)div⁡FrPifi=j.Remark 10.
When for eachi∈{1,…,m},bi:B→(0,∞) is smooth and P=0, we get Proposition 2.5 in [5] by Propositions 7 and 9.Corollary 11.
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and dim⁡Fi>1 and P∈Γ(TFr); then (M,∇¯) is mixed Ricci-flat if and only if M can be expressed as a multiply warped product and br is only dependent on Fr. In particular, if (M,∇¯) is Einstein, then M can be expressed as a multiply warped product.Proof.
By(2) and (3) in Proposition 9, we have that (M,∇¯) is mixed Ricci-flat if and only if [VX(ln⁡bi)]=0,(X(br)/br)π(V)=0. By [VX(ln⁡bi)]=0, similar to the proof of Corollary 8, we get that M can be expressed as a multiply warped product. When i≠r, π(V)=0. When i=r, by (X(br)/br)π(V)=0, then br depends only on Mr.By Proposition7 and the definition of the scalar curvature, we have the following.Proposition 12 (compare with [5, Proposition 2.6]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and P∈Γ(TB); then the scalar curvature S¯ has the following expression:(6)S¯=S¯B+2∑i=1m‍libiΔBbi+∑i=1m‍SFibi2+∑i=1m‍li(li-1)|gradBbi|B2bi2+∑i=1m‍∑j≠i‍liljgB(gradBbi,gradBbj)bibj+∑i=1m‍li(n+n¯+li-2)P(bi)bi+∑i=1m‍∑j≠i‍liljP(bj)bj+∑i=1m‍li(n+n¯-3)π(P)+2∑i=1m‍lidiv⁡BP.By Proposition9 and the definition of the scalar curvature, we have the following.Proposition 13 (compare with [5, Proposition 2.6]).
LetM=B×b1F1×b2F2⋯×bmFm be a multiply twisted product and P∈Γ(TFr); then the scalar curvature S¯ has the following expression:
(7)S¯=SB+2∑i=1m‍libiΔBbi+∑i=1m‍SFibi2+∑i=1m‍li(li-1)|gradBbi|B2bi2+∑i=1m‍∑j≠i‍liljgB(gradBbi,gradBbj)bibj+π(P)(n¯-1)(n¯-2)+2(n¯-1)div⁡FrP.Remark 14.
When for eachi∈{1,…,m},bi:B→(0,∞) is smooth and P=0, we get Proposition 2.6 in [5] by Propositions 12 and 13.


## 3. Special Multiply Warped Product with a Semisymmetric Connection

LetM=I×b1F1×b2F2⋯×bmFm be a multiply warped product with the metric tensor -dt2⊕b12gF1⊕⋯⊕bm2gFm and I is an open interval in ℝ and bi∈C∞(I).Theorem 15.
LetM=I×b1F1×b2F2⋯×bmFm be a multiply warped product with the metric tensor -dt2⊕b12gF1⊕⋯⊕bm2gFm and P=(∂/∂t). Then (M,∇¯) is Einstein with the Einstein constant λ if and only if the following conditions are satisfied for any i∈{1,…,m}:(1)
(Fi,∇Fi) is Einstein with the Einstein constant λi, i∈{1,…,m};(2)
∑i=1m‍li(bi′/bi-bi′′/bi)=λ;(3)
λi-bibi′′-(li-1)bi′2+(bi2-bibi′)∑j≠i‍lj(bj′/bj)+(2-n¯)bi2+(n¯+li-2)bibi′=λbi2.Proof.
By Proposition7, we have
(8)Ric¯(∂∂t,∂∂t)=-∑i=1m‍li(bi′bi-bi′′bi);Ric¯(∂∂t,V)=Ric¯(V,∂∂t)=0;Ric¯(V,W)=RicFi(V,W)+gFi(V,W)×[-bibi′′-(li-1)bi′2+(bi2-bibi′)∑j≠i‍ljbj′bj+(2-n¯)bi2+(n¯+li-2)bibi′∑j≠i‍ljbj′bj].
By (8) and the Einstein condition, we get Theorem 15.Theorem 16.
LetM=I×b1F1×b2F2⋯×bmFm be a multiply warped product with the metric tensor -dt2⊕b12gF1⊕⋯⊕bm2gFm and P∈Γ(TFr) with gFr(P,P)=1 and n¯>2. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if the following conditions are satisfied for any i∈{1,…,m}:(1)
(Fi,∇Fi)(i≠r) is Einstein with the Einstein constant λi, i∈{1,…,m};(2)
br is a constant and ∑i=1m‍li(bi′′/bi)=μ0;div⁡FrP=μ1,μ0-μ1+λ=(2-n¯)br2, where μ0,μ1 are constants;(3)
RicFr(V,W)+λ¯gFr(V,W)=(n¯-2)[π(V)π(W)-g(W,∇VP)], for V,W∈Γ(TFr);(4)
λi-bibi′′+(n¯-2)bi2br2-bibi′∑j≠i‍lj(bj′/bj)-(li-1)bi′2=(λ-μ1)bi2.Proof.
By Proposition9(2) and gFr(P,P)=1, we have that br is a constant. By Proposition 9, then
(9)Ric¯(∂∂t,∂∂t)=∑i=1m‍libi′′bi+(2-n¯)br2-div⁡FrP=-λ.
By variables separation, we have
(10)∑i=1m‍libi′′bi=μ0,div⁡FrP=μ1,μ0-μ1+λ=(2-n¯)br2,Ric¯(V,W)=RicFi(V,W)+bi2gFi(V,W)×[-bi′′bi+(li-1)-bi′2bi2+∑j≠i‍lj-bi′bj′bibj+(n¯-2)π(P)∑j≠i‍lj-bi′bj′bibj]+(n¯-2)g(W,∇VP)+(2-n¯)π(V)π(W)+g(V,W)div⁡FrP.
When i≠r, then ∇VP=π(V)=0, so
(11)Ric¯(V,W)=RicFi(V,W)+bi2gFi(V,W)×[-bi′′bi+(li-1)-bi′2bi2+∑j≠i‍lj-bi′bj′bibj+(n¯-2)br2∑j≠i‍lj-bi′bj′bibj]+μ1bi2gFi(V,W)=λbi2gFi(V,W).
By variables separation, we have that (Fi,∇Fi)(i≠r) is Einstein with the Einstein constant λi and
(12)λi-bibi′′+(n¯-2)bi2br2-bibi′∑j≠i‍ljbj′bj-(li-1)bi′2=(λ-μ1)bi2.
When i=r and br is a constant, then
(13)RicFi(V,W)+br2[(n¯-2)br2+μ1-λ]gFi(V,W)=(n¯-2)[π(V)π(W)-g(W,∇VP)].
So we prove the above theorem.Remark 17.
Comparing with Theorem 3.3 in [5], in Theorems 15 and 16, the unit vector field P emerges. For Theorem 3.3 in [5], P equals zero. So Theorem 3.3 in [5] is not the special case of Theorems 15 and 16. Then equations in our theorems are different from the equations in theorems in [5].WhenM=I×b1F1×b2F2⋯×bmFm is a multiply warped product and P=∂/∂t, by Proposition 12, we have
(14)S¯=-2∑i=1m‍libi′′bi+∑i=1m‍SFibi2+∑i=1m‍li(li-1)-bi′2bi2+∑i=1m‍∑j≠i‍lilj-bibjbibj+∑i=1m‍li(n¯+li-1)bi′bi+∑i=1m‍∑j≠i‍liljbj′bj-∑i=1m‍li(n¯-2).
The following result just follows from the method of separation of variables and the fact that each SFi is function defined on Fi.Proposition 18.
LetM=I×b1F1×b2F2⋯×bmFm be a multiply warped product and P=∂/∂t. If (M,∇¯) has constant scalar curvature S¯, then each (Fi,∇Fi) has constant scalar curvature SFi.WhenP∈Γ(TFr), by Proposition 13, we have
(15)S¯=-2∑i=1m‍libi′′bi+∑i=1m‍SFibi2+∑i=1m‍li(li-1)-bi′2bi2+∑i=1m‍∑j≠i‍lilj-bi′bj′bibj+π(P)(n¯-1)(n¯-2)+2(n¯-1)div⁡FrP.Similarly we have the following.Proposition 19 (compare with [5, Proposition 3.5]).
LetM=I×b1F1×b2F2⋯×bmFm be a multiply warped product and P∈Γ(TFr). If (M,∇¯) has constant scalar curvature S¯, then (1)
each(Fi,∇Fi)(i≠r)has constant scalar curvatureSFi;(2)
moreover,ifgFr(P,P), div⁡FrPare also constants,thenSFris a constant.Remark 20.
WhenP=0 in Proposition 19, we get Proposition 3.5 in [5].


## 4. Generalized Robertson-Walker Space-Times with a Semisymmetric Metric Connection

In this section, we studyM=I×F with the metric tensor -dt2+f(t)2gF. As a corollary of Theorem 15, we obtain the following.Corollary 21.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if the following conditions are satisfied: (1)
(F,∇F) is Einstein with the Einstein constant λF;(2)
l((f′/f)-(f′′/f))=λ;(3)
λF-ff′′+(1-l)f′2+(1-l-λ)f2+(2l-1)f′f=0.Remark 22.
In Theorem 5.1 in [9], they got the Einstein condition of M=I×F with a semisymmetric metric connection, but they did not consider conditions (2) and (3).Corollary 23.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F=1. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if f′′=f′-λf.By Corollary21 (2) and (3), we get the following.Corollary 24.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F>1. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if the following conditions are satisfied: (1)
(F,∇F) is Einstein with the Einstein constant λF;(2)
f′′=f′-(λ/l)f;(3)
(λF/(1-l))+f′2+(1+(λ/l))f2-2ff′=0.By Corollary23 and elementary methods for ordinary differential equations, we get the following.Theorem 25.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F=1. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if (1)
λ<1/4,f(t)=c1e((1+1-4λ)/2)t+c2e((1-1-4λ)/2)t,(2)
λ=1/4,f(t)=c1e(1/2)t+c2te(1/2)t,(3)
λ>1/4,f(t)=c1e(1/2)tcos⁡((4λ-1/2)t)+c2e(1/2)tsin((4λ-1/2)t).Letλ/l=d0,  λF/(1-l)=d0¯,  (1+1-4d0)/2=a0,(1-1-4d0)/2=b0; then, a0+b0=1,d0=a0b0. When dim⁡F>1, by Corollary 24(2), we have the following three cases.Case
(i)(d0<1/4). We have f=c1ea0t+c2eb0t. By Corollary 24(3), then
(16)d0¯+c12(a02+1+a0b0-2a0)e2a0t+c22(b02+1+a0b0-2b0)e2b0t+2c1c2(2a0b0+1-a0-b0)e(a0+b0)t=0.
When b0=0, we get d0=0,a0=1,λ=0. By (16), d0¯+c22=0, so λF=(l-1)c22. In this case f=c1et+c2. When b≠0, then e2a0t, e2b0t, and e(a0+b0)t are linearly independent, so c22(b02+1+a0b0-2b0)=c22(1-b0)=0 and c2=0. Then c1≠0, by c12(a02+1+a0b0-2a0)=c12(1-a0)=0, so a0=1; then d0=λ=0. Thus f=c1et.Case
(ii)  (d0=1/4). One has f=c1e(1/2)t+c2te(1/2)t. By Corollary 24(3), then
(17)d0¯+et[(12c1+c2+12c2t)2+54(c1+c2t)2-2(12c1+c2+12c2t)(c1+c2t)(12c1+c2+12c2t)2]=0.
The coefficient of t2et is (1/2)c22, so c2=0. The coefficient of et is ((1/2)c1+c2)2+(5/4)(c1)2-2c1((1/2)c1+c2), so c1=0; in this case we have no solutions.Case
(iii)  (d0>1/4). One has f(t)=c1e(1/2)tcos⁡(h0t)+c2e(1/2)tsin(h0t), where h0=4d0-1/2. By Corollary 24(3), then
(18)d0¯+et{[(c12+c2h0)cos⁡(h0t)+(c22-c1h0)sin(h0t)]2+(1+d0)(c1cos⁡(h0t)+c2sin(h0t))2-2(c1cos⁡(h0t)+c2sin(h0t))×[(c12+c2h0)cos⁡(h0t)+(c22-c1h0)sin(h0t)][(c12+c2h0)cos⁡(h0t)+(c22-c1h0)sin(h0t)]2}=0.
Considering the coefficients of cos⁡2(h0t)et and sin2(h0t)et, we get
(19)(14+d0)c12+c22h02-c1c2h0=0;(14+d0)c22+c12h02+c1c2h0=0.
Adding the above two equalities, then 1/4+d0+h02=0 and d0=0. There is a contradiction with d0>1/4 and in this case we have no solutions. So we obtain the following theorem.Theorem 26.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F>1. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if λ=0 and f=c1et+c2 and (F,∇F) is Einstein with the Einstein constant (l-1)c22.By (14) and (15), we have the following.Corollary 27.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t. Then (M,∇¯) has constant scalar curvature S¯ if and only if (F,∇F) has constant scalar curvature SF and
(20)S¯=SFf2-2lf′′f-l(l-1)f′2f2+2l2f′f+(1-l)l.Corollary 28.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P∈Γ(TF) and gF(P,P)=c0,div⁡FrP=c0′, where c0 and c0′ are constants. Then (M,∇¯) has constant scalar curvature S¯ if and only if (F,∇F) has constant scalar curvature SF and
(21)S¯=SFf2-2lf′′f-l(l-1)f′2f2+c0(l-1)lf2+2c0′l.In (20), we make the change of variable f(t)=v(t) and have the following equation:
(22)v′′(t)+l-34v′(t)2v(t)-lv′(t)+(l-1+S¯l)v(t)-SFl=0.Theorem 29.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F=l=3. Then (M,∇¯) has constant scalar curvature S¯ if and only if (F,∇F) has constant scalar curvature SF(1)
S¯<3/4 and S¯≠-6,v(t)=c1e((3+1-(4/3)S¯)/2)t+c2e((3-1-(4/3)S¯)/2)t+(SF/(6+S¯));(2)
S¯=3/4,v(t)=c1e(3/2)t+c2te(3/2)t+(SF/(6+S¯));(3)
S¯>3/4,v(t)=c1e(3/2)tcos⁡⁡(((4/3)S¯-1/2)t)+c2e(3/2)tsin⁡(((4/3)S¯-1/2)t)+(SF/(6+S¯));(4)
S¯=-6,v(t)=c1-(SF/9)t+c2e3t.Proof.
Ifl=3, then we have a simple differential equation as follows:
(23)v′′(t)-3v′(t)+(2+S¯3)v(t)-SF3=0.
If S¯≠-6, putting h(t)=(2+S¯/3)v(t)-SF/3, it follows that h′′(t)-3h′(t)+(2+S¯/3)h(t)=0. The above solutions (1)–(3) follow directly from elementary methods for ordinary differential equations. When S¯=-6, then v′′(t)-3v′(t)-SF/3=0, and we get solution (4).Theorem 30.
LetM=I×F with the metric tensor -dt2+f(t)2gF and P=∂/∂t and dim⁡F=l≠3 and SF=0. If (M,∇¯) has constant scalar curvature S¯ if and only if(1)S¯<l/(l+1),  v(t)=(c1e((l+1-((l+1)/l)S¯)/2)t+c2e((l-1-((l+1)/l)S¯)/2)t)4/(l+1);(2)S¯=l/(l+1),v(t)=(c1e(l/2)t+c2te(l/2)t)4/(l+1);(3)S¯>l/(l+1),  v(t)=(c1e(l/2)tcos⁡((((l+1)/l)S¯-1/2)t)+c2e(l/2)tsin((((l+1)/l)S¯-1/2)t))4/(l+1).Proof.
In this case, (22) is changed into the simpler form
(24)v′′(t)v(t)+l-34v′(t)2v(t)2-lv′(t)v(t)+(l-1+S¯l)=0.
Putting v(t)=w(t)4/(l+1), then w(t) satisfies the equation w′′-lw′+((l+1)/4)(l-1+S¯/l)w=0; by the elementary methods for ordinary differential equations we prove the above theorem.Whendim⁡F=l≠3 and SF≠0, putting v(t)=w(t)4/(l+1), then w(t) satisfies the following equation:
(25)w′′-lw′+(l+1)4(l-1+S¯l)w-(l+1)4SFlw1-4/(l+1)=0.


## 5. Generalized Kasner Space-Times with a Semisymmetric Metric Connection

In this section, we consider the scalar and Ricci curvature of generalized Kasner space-times with a semisymmetric metric connection. We recall the definition of generalized Kasner space-times [5].Definition 31.
A generalized Kasner space-time(M,g) is a Lorentzian multiply warped product of the form M=I×ϕp1F1×⋯×ϕpmFm with the metric g=-dt2⊕ϕ2p1gF1⊕⋯⊕ϕ2pmgFm, where ϕ:I→(0,∞) is smooth and pi∈ℝ, for any i∈{1,…,m} and also I=(t1,t2).We introduce the following parametersζ=∑i=1m‍lipi and η=∑i=1m‍lipi2 for generalized Kasner space-times. By Theorem 15 and direct computations, we get the following.Proposition 32.
LetM=I×ϕp1F1×⋯×ϕpmFm be a generalized Kasner space-time and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if the following conditions are satisfied for any i∈{1,…,m}:(1)
(Fi,∇Fi)is Einstein with the Einstein constantλi, i∈{1,…,m};(2)
ζ((ϕ′-ϕ′′)/ϕ)-(η-ζ)(ϕ′2/ϕ2)=λ;(3)
λi/ϕ2pi-pi(ϕ′′/ϕ)-(ζ-1)pi(ϕ′2/ϕ2)+[ζ+(n¯-2)pi](ϕ′/ϕ)=n¯+λ-2.By (14), we obtain the following.Proposition 33.
LetM=I×ϕp1F1×⋯×ϕpmFm be a generalized Kasner space-time and P=∂/∂t. Then (M,∇¯) has constant scalar curvature S¯ if and only if each (Fi,∇Fi) has constant scalar curvature SFi and
(26)S¯=∑i=1m‍SFiϕ2pi-2ζϕ′′ϕ-(η+ζ2-2ζ)ϕ′2ϕ2+2(n¯-1)ζϕ′ϕ+(2-n¯)(n¯-1).Next, we first give a classification of four-dimensional generalized Kasner space-times with a semisymmetric metric connection and then consider Ricci tensors and scalar curvatures of them.Definition 34.
LetM=I×b1F1×⋯×bmFm with the metric g=-dt2⊕b12gF1⊕⋯⊕bm2gFm. Consider the following:(i)
(M,g) is said to be of type (I) if m=1 and dim⁡(F)=3;(ii)
(M,g) is said to be of type (II) if m=2 and dim⁡(F1)=1 and dim⁡(F2)=2;(iii)
(M,g) is said to be of type (III) if m=3 and dim⁡(F1)=1, dim⁡(F2)=1 and dim⁡(F3)=1.By Theorems26 and 29, we have given a classification of type (I) Einstein spaces and type (I) spaces with the constant scalar curvature.


### 5.1. Classification of Einstein Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

LetM=I×ϕp1F1×ϕp2F2 be an Einstein type (II) generalized Kasner space-time and P=∂/∂t. Then ζ=p1+2p2, η=p12+2p22. By Theorem 15, we have(27a)ζ(ϕ′-ϕ′′ϕ)-(η-ζ)ϕ′2ϕ2=λ,(27b)-p1ϕ′′ϕ-(ζ-1)p1ϕ′2ϕ2+[ζ+2p1]ϕ′ϕ=λ+2,(27c)λ2ϕ2p2-p2ϕ′′ϕ-(ζ-1)p2ϕ′2ϕ2+[ζ+2p2]ϕ′ϕ=λ+2,whereλ2 is a constant. Consider the following two cases.Case
(i) (ζ=0). In this case, p2=-(1/2)p1,  η=(3/2)p12. Then by (27a)–(5.2iii), we have(28a)-ηϕ′2ϕ2=λ,(28b)p1(-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ)=λ+2,(28c)λ2ϕ-p1-12p1(-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ)=λ+2.Case
(
i
)
(a) (η=0). One has pi=0; by (28a), λ=0. By (28b), λ+2=0; this is a contradiction.Case
(
i
)
(b) (η≠0). One has pi≠0.Case
(
i
)
(b)
(1) (λ2=0). By (28b) and (28c), λ=-2 and
(29)-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ=0,ϕ′2ϕ2=2η;
then ϕ=c0e±(2/η)t which does not satisfy the first equation in (29); this a contradiction.Case
(
i
)
(b)
(2) (λ2≠0). By (28b) and (28c), we have λ2/ϕ-p1=(3/2)(λ+2), so ϕ is a constant. By (28b), λ+2=0, so λ2=0; this is a contradiction. In a word, we have no solutions when ζ=0.Case
(ii) (ζ≠0). One has η≠0. Putting ϕ=ψζ/η, then ψ′′-ψ′+(λη/ζ2)ψ=0. Hence,(1)
λ<ζ2/4η,
ψ=c1e((1+1-4λη/ζ2)/2)t+c2e((1-1-4λη/ζ2)/2)t,(2)
λ=ζ2/4η,
ψ=c1e(1/2)t+c2te(1/2)t,(3)
λ>ζ2/4η,
ψ=c1e(1/2)tcos⁡((4λη/ζ2-1/2)t)+c2e(1/2)tsin((4λη/ζ2-1/2)t).We make (27a), (5.2ii), (5.2iii) into(30a)ζ2ηψ′-ψ′′ψ=λ,ψ=ϕη/ζ,(30b)-p1ζ(ϕζ)′′ϕζ+ζ+2p1ζ(ϕζ)′ϕζ=λ+2,(30c)λ2ϕ2p2-p2ζ(ϕζ)′′ϕζ+ζ+2p2ζ(ϕζ)′ϕζ=λ+2.Whenp1=p2, type (II) spaces turn into type (I) spaces, so we assume p1≠p2. By (30b) and (30c), then
(31)ψ′=p1λ2η(p2-p1)ζ2ψ1-2p2ζ/η+(λ+2)ηζ2ψ.Case
(ii)
(1)
.Considerλ<ζ2/4η,  ψ=c1eat+c2ebt, where a=(1+1-4λη/ζ2)/2,b=((1-1-4λη/ζ2)/2)t.By (31),
(32)ac1eat+bc2ebt=p1λ2η(p2-p1)ζ2(c1eat+c2ebt)1-2p2ζ/η+(λ+2)ηζ2(c1eat+c2ebt).Case
(ii)
(1)
(a) (c1=0). One has
(33)[b-(λ+2)ηζ2]c2ebt=p1λ2η(p2-p1)ζ2(c2ebt)1-2p2ζ/η.Case
(ii)
(1)
(a)
(1) (b≠0,p1λ2≠0). One has p2=0 and ζ=p1,η=p12 and b=(1-1-4λ)/2 and ψ=c2ebt. By (30b) and b2-b+λ=0, we get -b2+3b=λ+2 and b=1. But b<1/2, and this is a contradiction.Case
(ii)
(1)
(a)
(2) (b≠0,p1λ2=0). If p1=0, then ζ=2p2, η=2p22 and b=(λ+2)η/ζ2, so λ=-4 and b=-1. By (30c), we get λ2=0 and -2b2+4b=λ+2 which is a contradiction.Ifλ2=0, by (30b) and b=(λ+2)η/ζ2, we get λ=0 or -2. When λ=0, then b=2η/ζ2=0; this is a contradiction. There is a similar contradiction for λ=-2.Case
(ii)
(1)
(a)
(3) (b=0). One has ψ=c2; by (30a), λ=0. By (30b), λ=-2; this is a contradiction.Case
(ii)
(1)
(b) (c2=0). One has
(34)[a-(λ+2)ηζ2]c1eat=p1λ2η(p2-p1)ζ2(c1eat)1-2p2ζ/η.Case
(ii)
(1)
(b)
(1) (p1λ2≠0). One has p2=0 and ζ=2p2, η=2p22 and a=(1+1-4λ)/2 and λ2=λ+2-(1+1-4λ)/2. By (30b), then a=1 and λ=0, so λ2=1 and ψ=c1et and ϕ satisfies (30c). In this case, we get p2=0,p1≠0,ϕ=c0et/p1,λ=0,λ2=1.Case
(ii)
(1)
(b)
(2) (p1λ2=0). If p1=0, then ζ=2p2, η=2p22 and ψ=c1eat and a=(λ+2)η/ζ2, so λ=0 and a=1. By (30c), we get λ2=0 and ϕ satisfies (30b) and (30c). In this case, p1=0,p2≠0λ=0,λ2=0,ϕ=c0et/p2.Ifλ2=0, by (30b) and a=(λ+2)η/ζ2, then λ=0 and a=2η/ζ2=1. By (30c), then λ2=0 and ϕ satisfies (30b) and (30c). In this case, p1≠0,p2≠0,λ=λ2=0,  p1=4p2,ϕ=c0et/3p2.Case
(ii)
(1)
(c) (c1≠0,c2≠0,b≠0). If p2≠0, then eat,ebt,(c1eat+c2ebt)1-2p2ζ/η are linearly independent; by (32), then
(35)[a-(λ+2)ηζ2]c1=0,[b-(λ+2)ηζ2]c2=0,p1λ2η(p2-p1)ζ2(c1eat)1-2p2ζ/η.
So a=b=(λ+2)η/ζ2, and this is a contradiction.Ifp2=0, then by (32),
(36)a-(λ+2)ηζ2-p1λ2η(p2-p1)ζ2=0,b-(λ+2)ηζ2-p1λ2η(p2-p1)ζ2=0,
so a=b and we get a contradiction.Case
(ii)
(1)
(d) (c1≠0,c2≠0,b=0). When  1-2p2ζ/η≠0, we have similar discussions. When 1-2p2ζ/η=0, we have (λ+2)η/ζ2=1. By b=0, then λ=0 and 2η=ζ2=4p2ζ, so 4p2=ζ and p1=2p2. But η=2p2ζ; then p1=p2=0. This is a contradiction.Case
(ii)
(2) (λ=ζ2/4η,  ψ=c1e(1/2)t+c2te(1/2)t). By (31), we have
(37)[12c1+c2-a0c1+(c22-a0c2)t]e(1/2)t=p1λ2η(p2-p1)ζ2(c1+c2t)1-2p2ζ/η(e(1/2)t)1-2p2ζ/η,
where a0=1/4+2η/ζ2.Case
(ii)
(2)
(a) (c2≠0). One has (1/2)c1+c2-a0c1+(c2/2-a0c2)t≠0 and p2=0. By (30c), λ2+(ϕζ)′/ϕζ=λ+2, then ϕζ=c0e(-λ2+λ+2)t and (c1e(1/2)t+c2te(1/2)t)ζ2/η=c0e(-λ2+λ+2)t; this is a contradiction with c2≠0.Case
(ii)
(2)
(b) (c2=0). By (37), we have
(38)(12c1-a0c1)e(1/2)t=p1λ2η(p2-p1)ζ2(c1)1-2p2ζ/η(e(1/2)t)1-2p2ζ/η.
If a0=1/2, then p1λ2=0 and λ=2. If p1=0, then ζ=2p2 and by (30b), (ϕζ)′/ϕζ=4 and ϕζ=c0e4t. By (30c),
(39)λ2(c0′e(1/2)t)2p2ζ/η-16p2ζ+4ζ+2p2ζ=4,
so λ2=0 and we have a contradiction by (39).Ifp1≠0, then λ2=0, so ψ=c1e(1/2)t and ϕζ=c0e4t. Then by (30b), we have p1=0 which contradicts with p1≠0.Ifa0≠1/2, then p2=0 and ζ=p1,η=p12 and λ=1/4 and a0=9/4. By ϕξ=c1e(1/2)t and (30b), we have a contradiction. In a word, we have no solutions in case (ii)(2).Case
(ii)
(3)
.One has  λ>ζ2/4η,  ψ=c1e(1/2)tcos⁡(at)+c2e(1/2)tsin(at), where a=4λη/ζ2-1/2. By (31), we have
(40)(c12+ac2)cos⁡(at)+(-ac1+c22)sin(at)=p1λ2η(p2-p1)ζ2(c1cos⁡(at)+c2sin(at))1-2p2ζ/ηe-2p2ζ/η+(λ+2)ηζ2(c1cos⁡(at)+c2sin(at)).
If p2≠0, then p1λ2=0 and
(41)c12+ac2=(λ+2)ηζ2c1,-ac1+c22=(λ+2)ηζ2c2,
so c12+c22=0. This is a contradiction.Ifp2=0, then
(42)c12+ac2=p1λ2η(p2-p1)ζ2c1+(λ+2)ηζ2c1,-ac1+c22=p1λ2η(p2-p1)ζ2c2+(λ+2)ηζ2c2.
Then c12+c22=0. This is a contradiction. By the above discussions, we get the following theorem.Theorem 35.
LetM=I×ϕp1F1×ϕp2F2 be a generalized Kasner space-time and dim⁡F1=1,dim⁡F2=2 and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if (F2,∇F2) is Einstein with the Einstein constant λ2, and one of the following conditions is satisfied:(1)
p2=0,p1≠0,ϕ=c0et/p1,λ=0,λ2=1;(2)
p1=0,p2≠0,λ=0,λ2=0,ϕ=c0et/p2;(3)
p1≠0,p2≠0,λ=λ2=0,p1=4p2,ϕ=c0et/3p2.


### 5.2. Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition33, then (F2,∇F2) has constant scalar curvature SF2 and
(43)S¯=SF2ϕ2p2-2ζϕ′′ϕ-(η+ζ2-2ζ)ϕ′2ϕ2+6ζϕ′ϕ-6.
If ζ=0, when η=0, then p1=p2=0 and S¯=SF2-6. If η≠0, then
(44)ηϕ′2ϕ2=SF2ϕ2p2-(S¯+6).
If ζ≠0, putting ϕ=ψ2ζ/(η+ζ2), we get
(45)-4ζ2η+ζ2ψ′′+12ζ2η+ζ2ψ′-(S¯+6)ψ+SF2ψ1-4p2ζ/(η+ζ2)=0.


### 5.3. Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition33, then
(46)S¯=-2ζϕ′′ϕ-(η+ζ2-2ζ)ϕ′2ϕ2+6ζϕ′ϕ-6.
If ζ=η=0, then p1=p2=p3=0, and we get S¯=-6.Ifζ=0,η≠0, then [(ln⁡ϕ)′]2=-(S¯+6)/η, so when S¯+6>0, there are no solutions, when S¯+6=0, ϕ is a constant, and when S¯+6<0, ϕ=c0e±(-(S¯+6)/η)t.Ifζ≠0, then η≠0; putting ϕ=ψ2ζ/(η+ζ2), then
(47)ψ′′-3ψ′+(S¯+6)(η+ζ2)4ζ2ψ=0.
So, we get (1)
S¯+6<9ζ2/(η+ζ2),
ψ=c1e((3+9-(S¯+6)(η+ζ2)/ζ2)/2)t+c2e((3-9-(S¯+6)(η+ζ2)/ζ2)/2)t,(2)
S¯+6=9ζ2/(η+ζ2),  ψ=c1e(3/2)t+c2te(3/2)t,(3)
S¯+6>9ζ2/(η+ζ2),  ψ=c1e(3/2)tcos⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t)+c2e(3/2)tsin((-9+(S¯+6)(η+ζ2)/ζ2/2)t). So we get the following theorem.Theorem 36.
LetM=I×ϕp1F1×ϕp2F2×ϕp3F3 be a generalized Kasner space-time and dim⁡F1=dim⁡F2=dim⁡F3=1, and P=∂/∂t. Then S¯ is a constant if and only if one of the following cases holds. (1)
One hasζ=η=0,  S¯=-6.(2)
One hasζ=0,η≠0; when S¯+6>0, there are no solutions, when S¯+6=0, ϕ is a constant, and when S¯+6<0, ϕ=c0e±-((S¯+6)/η)t.(3)
Ifζ≠0
(a)
S¯+6<9ζ2/(η+ζ2),
ϕ=(c1e(3+9-(S¯+6)(η+ζ2)/ζ2/2)t+c2e(3-9-(S¯+6)(η+ζ2)/ζ2/2)t)2ζ/(η+ζ2),(b)
S¯+6=9ζ2/(η+ζ2),
ϕ=(c1e(3/2)t+c2te(3/2)t)2ζ/(η+ζ2),(c)
S¯+6>9ζ2/(η+ζ2),
ϕ=(c1e(3/2)tcos⁡⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t)+c2e(3/2)tsin⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t))2ζ/(η+ζ2).


### 5.4.  Einstein Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

By Proposition32, we have(48a)ζ(ϕ′-ϕ′′ϕ)-(η-ζ)ϕ′2ϕ2=λ,(48b)-p1[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2,(48c)-p2[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2,(48d)-p3[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2.Ifζ=η=0, by (48a), λ=0 and by (48b), λ=-2; this is a contradiction.Ifζ=0,η≠0, adding (48b), (48c), and (48d), we get λ=-2. By (48a), ϕ′2/ϕ2=2/η and ϕ=c0e±2/ηt. But by (48b), then ϕ′′/ϕ+(ζ-1)(ϕ′2/ϕ2)-2ϕ′/ϕ=0, and this is a contradiction.Considerζ≠0. If p1=p2=p3, we get type (I), so we may let p1≠p2. By (48b) and (48c), we have (ϕζ)′/ϕζ=λ+2 and (ϕζ)′′/ϕζ-2(ϕζ)′/ϕζ=0, so ϕζ=c0e(λ+2)t and λ=-2 or 0. When λ=-2, ψ is a constant; by (48a), λ=0, this is a contradiction. When λ=0, ψ=c0e2ηt/ζ2, and ψ′′-ψ′=0, so 2η/ζ2=1. In this case, we get when pi≠pj for some i,j∈{1,2,3},  λ=0,2η/ζ2=1,ϕ=c0e2t/ζ. We get the following theorem.Theorem 37.
LetM=I×ϕp1F1×ϕp2F2×ϕp3F3 be a generalized Kasner space-time for pi≠pj for some i,j∈{1,2,3} and dim⁡F1=dim⁡F2=dim⁡F3=1, and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if λ=0,2η/ζ2=1,ϕ=c0e2t/ζ.Remark 38.
Comparing with Proposition 4.3, Proposition 4.11, Section 5 in [5], in Proposition 32, Proposition 33, and Theorems 35–37, the unit vector field P=∂/∂t emerges. For Proposition 4.3, Proposition 4.11, Section 5 in [5], P equals zero. So equations in our theorems are different from the equations in theorems in [5].


## 5.1. Classification of Einstein Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

LetM=I×ϕp1F1×ϕp2F2 be an Einstein type (II) generalized Kasner space-time and P=∂/∂t. Then ζ=p1+2p2, η=p12+2p22. By Theorem 15, we have(27a)ζ(ϕ′-ϕ′′ϕ)-(η-ζ)ϕ′2ϕ2=λ,(27b)-p1ϕ′′ϕ-(ζ-1)p1ϕ′2ϕ2+[ζ+2p1]ϕ′ϕ=λ+2,(27c)λ2ϕ2p2-p2ϕ′′ϕ-(ζ-1)p2ϕ′2ϕ2+[ζ+2p2]ϕ′ϕ=λ+2,whereλ2 is a constant. Consider the following two cases.Case
(i) (ζ=0). In this case, p2=-(1/2)p1,  η=(3/2)p12. Then by (27a)–(5.2iii), we have(28a)-ηϕ′2ϕ2=λ,(28b)p1(-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ)=λ+2,(28c)λ2ϕ-p1-12p1(-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ)=λ+2.Case
(
i
)
(a) (η=0). One has pi=0; by (28a), λ=0. By (28b), λ+2=0; this is a contradiction.Case
(
i
)
(b) (η≠0). One has pi≠0.Case
(
i
)
(b)
(1) (λ2=0). By (28b) and (28c), λ=-2 and
(29)-ϕ′′ϕ+ϕ′2ϕ2+2ϕ′ϕ=0,ϕ′2ϕ2=2η;
then ϕ=c0e±(2/η)t which does not satisfy the first equation in (29); this a contradiction.Case
(
i
)
(b)
(2) (λ2≠0). By (28b) and (28c), we have λ2/ϕ-p1=(3/2)(λ+2), so ϕ is a constant. By (28b), λ+2=0, so λ2=0; this is a contradiction. In a word, we have no solutions when ζ=0.Case
(ii) (ζ≠0). One has η≠0. Putting ϕ=ψζ/η, then ψ′′-ψ′+(λη/ζ2)ψ=0. Hence,(1)
λ<ζ2/4η,
ψ=c1e((1+1-4λη/ζ2)/2)t+c2e((1-1-4λη/ζ2)/2)t,(2)
λ=ζ2/4η,
ψ=c1e(1/2)t+c2te(1/2)t,(3)
λ>ζ2/4η,
ψ=c1e(1/2)tcos⁡((4λη/ζ2-1/2)t)+c2e(1/2)tsin((4λη/ζ2-1/2)t).We make (27a), (5.2ii), (5.2iii) into(30a)ζ2ηψ′-ψ′′ψ=λ,ψ=ϕη/ζ,(30b)-p1ζ(ϕζ)′′ϕζ+ζ+2p1ζ(ϕζ)′ϕζ=λ+2,(30c)λ2ϕ2p2-p2ζ(ϕζ)′′ϕζ+ζ+2p2ζ(ϕζ)′ϕζ=λ+2.Whenp1=p2, type (II) spaces turn into type (I) spaces, so we assume p1≠p2. By (30b) and (30c), then
(31)ψ′=p1λ2η(p2-p1)ζ2ψ1-2p2ζ/η+(λ+2)ηζ2ψ.Case
(ii)
(1)
.Considerλ<ζ2/4η,  ψ=c1eat+c2ebt, where a=(1+1-4λη/ζ2)/2,b=((1-1-4λη/ζ2)/2)t.By (31),
(32)ac1eat+bc2ebt=p1λ2η(p2-p1)ζ2(c1eat+c2ebt)1-2p2ζ/η+(λ+2)ηζ2(c1eat+c2ebt).Case
(ii)
(1)
(a) (c1=0). One has
(33)[b-(λ+2)ηζ2]c2ebt=p1λ2η(p2-p1)ζ2(c2ebt)1-2p2ζ/η.Case
(ii)
(1)
(a)
(1) (b≠0,p1λ2≠0). One has p2=0 and ζ=p1,η=p12 and b=(1-1-4λ)/2 and ψ=c2ebt. By (30b) and b2-b+λ=0, we get -b2+3b=λ+2 and b=1. But b<1/2, and this is a contradiction.Case
(ii)
(1)
(a)
(2) (b≠0,p1λ2=0). If p1=0, then ζ=2p2, η=2p22 and b=(λ+2)η/ζ2, so λ=-4 and b=-1. By (30c), we get λ2=0 and -2b2+4b=λ+2 which is a contradiction.Ifλ2=0, by (30b) and b=(λ+2)η/ζ2, we get λ=0 or -2. When λ=0, then b=2η/ζ2=0; this is a contradiction. There is a similar contradiction for λ=-2.Case
(ii)
(1)
(a)
(3) (b=0). One has ψ=c2; by (30a), λ=0. By (30b), λ=-2; this is a contradiction.Case
(ii)
(1)
(b) (c2=0). One has
(34)[a-(λ+2)ηζ2]c1eat=p1λ2η(p2-p1)ζ2(c1eat)1-2p2ζ/η.Case
(ii)
(1)
(b)
(1) (p1λ2≠0). One has p2=0 and ζ=2p2, η=2p22 and a=(1+1-4λ)/2 and λ2=λ+2-(1+1-4λ)/2. By (30b), then a=1 and λ=0, so λ2=1 and ψ=c1et and ϕ satisfies (30c). In this case, we get p2=0,p1≠0,ϕ=c0et/p1,λ=0,λ2=1.Case
(ii)
(1)
(b)
(2) (p1λ2=0). If p1=0, then ζ=2p2, η=2p22 and ψ=c1eat and a=(λ+2)η/ζ2, so λ=0 and a=1. By (30c), we get λ2=0 and ϕ satisfies (30b) and (30c). In this case, p1=0,p2≠0λ=0,λ2=0,ϕ=c0et/p2.Ifλ2=0, by (30b) and a=(λ+2)η/ζ2, then λ=0 and a=2η/ζ2=1. By (30c), then λ2=0 and ϕ satisfies (30b) and (30c). In this case, p1≠0,p2≠0,λ=λ2=0,  p1=4p2,ϕ=c0et/3p2.Case
(ii)
(1)
(c) (c1≠0,c2≠0,b≠0). If p2≠0, then eat,ebt,(c1eat+c2ebt)1-2p2ζ/η are linearly independent; by (32), then
(35)[a-(λ+2)ηζ2]c1=0,[b-(λ+2)ηζ2]c2=0,p1λ2η(p2-p1)ζ2(c1eat)1-2p2ζ/η.
So a=b=(λ+2)η/ζ2, and this is a contradiction.Ifp2=0, then by (32),
(36)a-(λ+2)ηζ2-p1λ2η(p2-p1)ζ2=0,b-(λ+2)ηζ2-p1λ2η(p2-p1)ζ2=0,
so a=b and we get a contradiction.Case
(ii)
(1)
(d) (c1≠0,c2≠0,b=0). When  1-2p2ζ/η≠0, we have similar discussions. When 1-2p2ζ/η=0, we have (λ+2)η/ζ2=1. By b=0, then λ=0 and 2η=ζ2=4p2ζ, so 4p2=ζ and p1=2p2. But η=2p2ζ; then p1=p2=0. This is a contradiction.Case
(ii)
(2) (λ=ζ2/4η,  ψ=c1e(1/2)t+c2te(1/2)t). By (31), we have
(37)[12c1+c2-a0c1+(c22-a0c2)t]e(1/2)t=p1λ2η(p2-p1)ζ2(c1+c2t)1-2p2ζ/η(e(1/2)t)1-2p2ζ/η,
where a0=1/4+2η/ζ2.Case
(ii)
(2)
(a) (c2≠0). One has (1/2)c1+c2-a0c1+(c2/2-a0c2)t≠0 and p2=0. By (30c), λ2+(ϕζ)′/ϕζ=λ+2, then ϕζ=c0e(-λ2+λ+2)t and (c1e(1/2)t+c2te(1/2)t)ζ2/η=c0e(-λ2+λ+2)t; this is a contradiction with c2≠0.Case
(ii)
(2)
(b) (c2=0). By (37), we have
(38)(12c1-a0c1)e(1/2)t=p1λ2η(p2-p1)ζ2(c1)1-2p2ζ/η(e(1/2)t)1-2p2ζ/η.
If a0=1/2, then p1λ2=0 and λ=2. If p1=0, then ζ=2p2 and by (30b), (ϕζ)′/ϕζ=4 and ϕζ=c0e4t. By (30c),
(39)λ2(c0′e(1/2)t)2p2ζ/η-16p2ζ+4ζ+2p2ζ=4,
so λ2=0 and we have a contradiction by (39).Ifp1≠0, then λ2=0, so ψ=c1e(1/2)t and ϕζ=c0e4t. Then by (30b), we have p1=0 which contradicts with p1≠0.Ifa0≠1/2, then p2=0 and ζ=p1,η=p12 and λ=1/4 and a0=9/4. By ϕξ=c1e(1/2)t and (30b), we have a contradiction. In a word, we have no solutions in case (ii)(2).Case
(ii)
(3)
.One has  λ>ζ2/4η,  ψ=c1e(1/2)tcos⁡(at)+c2e(1/2)tsin(at), where a=4λη/ζ2-1/2. By (31), we have
(40)(c12+ac2)cos⁡(at)+(-ac1+c22)sin(at)=p1λ2η(p2-p1)ζ2(c1cos⁡(at)+c2sin(at))1-2p2ζ/ηe-2p2ζ/η+(λ+2)ηζ2(c1cos⁡(at)+c2sin(at)).
If p2≠0, then p1λ2=0 and
(41)c12+ac2=(λ+2)ηζ2c1,-ac1+c22=(λ+2)ηζ2c2,
so c12+c22=0. This is a contradiction.Ifp2=0, then
(42)c12+ac2=p1λ2η(p2-p1)ζ2c1+(λ+2)ηζ2c1,-ac1+c22=p1λ2η(p2-p1)ζ2c2+(λ+2)ηζ2c2.
Then c12+c22=0. This is a contradiction. By the above discussions, we get the following theorem.Theorem 35.
LetM=I×ϕp1F1×ϕp2F2 be a generalized Kasner space-time and dim⁡F1=1,dim⁡F2=2 and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if (F2,∇F2) is Einstein with the Einstein constant λ2, and one of the following conditions is satisfied:(1)
p2=0,p1≠0,ϕ=c0et/p1,λ=0,λ2=1;(2)
p1=0,p2≠0,λ=0,λ2=0,ϕ=c0et/p2;(3)
p1≠0,p2≠0,λ=λ2=0,p1=4p2,ϕ=c0et/3p2.


## 5.2. Type (II) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition33, then (F2,∇F2) has constant scalar curvature SF2 and
(43)S¯=SF2ϕ2p2-2ζϕ′′ϕ-(η+ζ2-2ζ)ϕ′2ϕ2+6ζϕ′ϕ-6.
If ζ=0, when η=0, then p1=p2=0 and S¯=SF2-6. If η≠0, then
(44)ηϕ′2ϕ2=SF2ϕ2p2-(S¯+6).
If ζ≠0, putting ϕ=ψ2ζ/(η+ζ2), we get
(45)-4ζ2η+ζ2ψ′′+12ζ2η+ζ2ψ′-(S¯+6)ψ+SF2ψ1-4p2ζ/(η+ζ2)=0.


## 5.3. Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection with Constant Scalar Curvature

By Proposition33, then
(46)S¯=-2ζϕ′′ϕ-(η+ζ2-2ζ)ϕ′2ϕ2+6ζϕ′ϕ-6.
If ζ=η=0, then p1=p2=p3=0, and we get S¯=-6.Ifζ=0,η≠0, then [(ln⁡ϕ)′]2=-(S¯+6)/η, so when S¯+6>0, there are no solutions, when S¯+6=0, ϕ is a constant, and when S¯+6<0, ϕ=c0e±(-(S¯+6)/η)t.Ifζ≠0, then η≠0; putting ϕ=ψ2ζ/(η+ζ2), then
(47)ψ′′-3ψ′+(S¯+6)(η+ζ2)4ζ2ψ=0.
So, we get (1)
S¯+6<9ζ2/(η+ζ2),
ψ=c1e((3+9-(S¯+6)(η+ζ2)/ζ2)/2)t+c2e((3-9-(S¯+6)(η+ζ2)/ζ2)/2)t,(2)
S¯+6=9ζ2/(η+ζ2),  ψ=c1e(3/2)t+c2te(3/2)t,(3)
S¯+6>9ζ2/(η+ζ2),  ψ=c1e(3/2)tcos⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t)+c2e(3/2)tsin((-9+(S¯+6)(η+ζ2)/ζ2/2)t). So we get the following theorem.Theorem 36.
LetM=I×ϕp1F1×ϕp2F2×ϕp3F3 be a generalized Kasner space-time and dim⁡F1=dim⁡F2=dim⁡F3=1, and P=∂/∂t. Then S¯ is a constant if and only if one of the following cases holds. (1)
One hasζ=η=0,  S¯=-6.(2)
One hasζ=0,η≠0; when S¯+6>0, there are no solutions, when S¯+6=0, ϕ is a constant, and when S¯+6<0, ϕ=c0e±-((S¯+6)/η)t.(3)
Ifζ≠0
(a)
S¯+6<9ζ2/(η+ζ2),
ϕ=(c1e(3+9-(S¯+6)(η+ζ2)/ζ2/2)t+c2e(3-9-(S¯+6)(η+ζ2)/ζ2/2)t)2ζ/(η+ζ2),(b)
S¯+6=9ζ2/(η+ζ2),
ϕ=(c1e(3/2)t+c2te(3/2)t)2ζ/(η+ζ2),(c)
S¯+6>9ζ2/(η+ζ2),
ϕ=(c1e(3/2)tcos⁡⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t)+c2e(3/2)tsin⁡((-9+(S¯+6)(η+ζ2)/ζ2/2)t))2ζ/(η+ζ2).


## 5.4.  Einstein Type (III) Generalized Kasner Space-Times with a Semisymmetric Metric Connection

By Proposition32, we have(48a)ζ(ϕ′-ϕ′′ϕ)-(η-ζ)ϕ′2ϕ2=λ,(48b)-p1[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2,(48c)-p2[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2,(48d)-p3[ϕ′′ϕ+(ζ-1)ϕ′2ϕ2-2ϕ′ϕ]+ζϕ′ϕ=λ+2.Ifζ=η=0, by (48a), λ=0 and by (48b), λ=-2; this is a contradiction.Ifζ=0,η≠0, adding (48b), (48c), and (48d), we get λ=-2. By (48a), ϕ′2/ϕ2=2/η and ϕ=c0e±2/ηt. But by (48b), then ϕ′′/ϕ+(ζ-1)(ϕ′2/ϕ2)-2ϕ′/ϕ=0, and this is a contradiction.Considerζ≠0. If p1=p2=p3, we get type (I), so we may let p1≠p2. By (48b) and (48c), we have (ϕζ)′/ϕζ=λ+2 and (ϕζ)′′/ϕζ-2(ϕζ)′/ϕζ=0, so ϕζ=c0e(λ+2)t and λ=-2 or 0. When λ=-2, ψ is a constant; by (48a), λ=0, this is a contradiction. When λ=0, ψ=c0e2ηt/ζ2, and ψ′′-ψ′=0, so 2η/ζ2=1. In this case, we get when pi≠pj for some i,j∈{1,2,3},  λ=0,2η/ζ2=1,ϕ=c0e2t/ζ. We get the following theorem.Theorem 37.
LetM=I×ϕp1F1×ϕp2F2×ϕp3F3 be a generalized Kasner space-time for pi≠pj for some i,j∈{1,2,3} and dim⁡F1=dim⁡F2=dim⁡F3=1, and P=∂/∂t. Then (M,∇¯) is Einstein with the Einstein constant λ if and only if λ=0,2η/ζ2=1,ϕ=c0e2t/ζ.Remark 38.
Comparing with Proposition 4.3, Proposition 4.11, Section 5 in [5], in Proposition 32, Proposition 33, and Theorems 35–37, the unit vector field P=∂/∂t emerges. For Proposition 4.3, Proposition 4.11, Section 5 in [5], P equals zero. So equations in our theorems are different from the equations in theorems in [5].



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