# Some General New Einstein Walker Manifolds

Authors: Mehdi Nadjafikhah; Mehdi Jafari
Journal: Advances in Mathematical Physics
 (2013)

---

## Abstract

Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.


---

## Body

## 1. Introduction

As it is well known, the symmetry group method has an important role in the analysis of differential equations. The theory of Lie symmetry groups of differential equations was developed for the first time by Lie at the end of nineteenth century [1]. By this method, we can construct new solutions from known ones, reduce the order of ODEs, and investigate the invariant solutions (for more information about the other applications of Lie symmetries, see [2–4]). Lie’s method led to an algorithmic approach to find special solutions of differential equation by its symmetry group. These solutions that called group invariant solutions are obtained by solving the reduced system of differential equation having less independent variables than the original system. The fact that for some PDEs, the symmetry reductions are unobtainable by the Lie symmetry method caused the creation of some generalizations of this method. These generalizations are called nonclassical symmetry method and was described in these references [5–8].In this paper, we apply the Lie symmetry method to find the invariant solutions of a system of PDEs that determines a spacial kind of Walker manifolds, calledEinstein Walker manifolds. We continue this section by giving the theoretical notions needed to introduce a system of PDEs that determines Einstein Walker manifolds.Let〈·,·〉 be a nondegenerate inner product on a vector space V. We can choose a basis {ei} for V so that
(1)〈ei,ej〉={0i≠j±1i=j.
We set εi:=〈ei,ei〉. Let p be the number of indices i with εi=-1 and q=dimV-p. The inner product is then said to have signature (p,q). We can extend this argument on manifolds. Let ℳ=(M,g) where M is an m-dimensional manifold and g is a symmetric nondegenerate smooth bilinear form on TM of signature (p,q), such a manifold is called a pseudo-Riemannian manifold of signature (p,q). So we have p+q=m. If p=0 then g is positive definite and ℳ is a Riemannian manifold. If (x1,…,xm) is a system of local coordinates on M, we may express g=∑i,j‍gijdxi∘dxj where gij:=g(∂xi,∂xj) and “∘’’ is symmetric product. Let M be a pseudo-Riemannian manifold of signature (p,q). Suppose that a splitting of the tangent bundle in the form TM=V1⊕V2 is given, where V1 and V2 are smooth subbundles which are called distribution. This defines two complementary projection π1 and π2 of TM onto V1 and V2, respectively. V1 is called a parallel distribution if ∇π1=0. Equivalently this means that if X1 is any smooth vector field taking values in V1, then ∇X1 again takes values in V1. We say that V1 is a null parallel distribution if V1 is parallel and if the metric restricted to V1 vanishes identically. Manifolds which admit null parallel distributions are called Walker manifolds. Let D be a connection on manifold M. The curvature operator ℛ is defined by the formula:
(2)ℛ(X,Y)Z:=(DXDY-DYDX-D[X,Y])Z,X,Y,Z∈C∞(TM).
Also the associated Ricci tensor ρ is defined by
(3)ρ(X,Y):=Tr{Z→ℛ(Z,X)Y}.
A Walker manifold is said to be an Einstein Walker manifold if its Ricci tensor is a scaler multiple of the metric at each point; that is, there is a constant k so that ρ=kg. Walker showed that a canonical form for a 2n-dimensional pseudo-Riemannian Walker manifold which admits an n-dimensional distribution is given by the metric tensor as
(4)(gij)=(0IdnIdnB),
where Idn is the n×n identity matrix and B is a symmetric n×n matrix whose entries are functions of the (x1,…,x2n)  (for more details see [9, 10]).Ifn=2 we adopt the following result for 4-dimensional Walker manifolds. Let Ma,b,c:=(𝒪,ga,b,c), where 𝒪 be an open subset of ℝ4 and a,b,c∈C∞(𝒪) are smooth functions on 𝒪. Then we can express the general form of metric tensor for 4-dimensional Walker manifolds as follows:
(5)(ga,b,c)ij=(0010000110ac01cb),
that is given in the coordinates form as
(6)ga,b,c:=2(dx1∘dx3+dx2∘dx4)+a(x1,x2,x3,x4)dx3∘dx3+b(x1,x2,x3,x4)dx4∘dx4+2c(x1,x2,x3,x4)dx3∘dx4.
We can see that Ma,b,c is Einstein if and only if the functions a, b, and c verify the following system of PDEs [9, page 81]:
(7)a11-b22=0,b12+c11=0,a12+c22=0,a1c2+a2b2-a2c1-c22+2ca12+ba22-2a24-ac12+2c23=0,a2b1-c1c2+ca11-a14-b23-ac11-cc12+c13-bc22+c24=0,a1b1-b1c2+b2c1-c12+ab11+2cb12-2b13-bc12+2c14=0.
This system is hard to handle, so we consider a spacial case in this paper, where a, b, and c only depend on x1 and x2. Therefore the following system must be solved:
(8)a11-b22=0,b12+c11=0,a12+c22=0,a1c2+a2b2-a2c1-c22+2ca12+ba22-ac12=0,a2b1-c1c2+ca11-ac11-cc12-bc22=0,a1b1-b1c2+b2c1-c12+ab11+2cb12-bc12=0.
This work is organized as follows. In Section 2, some preliminary results about Lie symmetry method are presented. In Section 3, the infinitesimal generators of symmetry algebra of system (8) are determined and some results will be obtained. In Section 4, the optimal system of subalgebras is constructed. In Section 5, we obtain the invariant solutions corresponding to the infinitesimal symmetries of system (8).


## 2. Lie Symmetries Method

In this section, we recall the procedure for finding symmetries of a system of PDEs (see [2, 11]). To begin, we start with a general case of a system of PDEs of order nth with p independent and q dependent variables such as
(9)Δμ(x,u(n))=0,μ=1,…,r,
involving x=(x1,…,xp) and u=(u1,…,uq) as independent and dependent variables, respectively and all the derivatives of u with respect to x from order 0 to n. We consider a one parameter Lie group of transformations which acts on the all variables of (9)
(10)x~i=xi+εξi(x,u)+o(ε2),i=1,…,p,u~j=uj+εϕj(x,u)+o(ε2),j=1,…,q,
where ξi and ϕj are the infinitesimals of the transformations for the independent and dependent variables, respectively and ε is the parameter of transformation.The most general form of infinitesimal generator associated with the above group of transformations is(11)X=∑i=1p‍ξi(x,u)∂xi+∑j=1q‍ϕj(x,u)∂uj.
Transformations which map solutions of a differential equation to other solutions are called symmetries of this equation. The nth order prolongation of X is defined by
(12)Pr(n)X=X+∑α=1q‍∑J‍ϕαJ(x,u(n))∂uJα,
where J=(j1,…,jk), 1≤jk≤p, 1≤k≤n and the sum is over all J’s of order 0<#J≤n. If #J=k, the coefficient ϕαJ of ∂uJα will only depend on kth and lower order derivatives of u, and
(13)ϕαJ(x,u(n))=DJ(ϕα-∑i=1p‍ξiuiα)+∑i=1p‍ξiuJ,iα,
where uiα:=∂uα/∂xi and uJ,iα:=∂uJα/∂xi. Now, according to Theorem 2.36 of [2], the invariance of the system (9) under the infinitesimal transformations leads to the invariance conditions:
(14)Pr(n)X[Δμ(x,u(n))]=0,μ=1,…,r,wheneverΔμ(x,u(n))=0.
Also, if the system (9) is a nondegenerate system which is locally solvable and of maximal rank, we can conclude that (14) is a necessary and sufficient condition for G to be a (strong) symmetry group of (9) (Theorem 2.71 of [2] and Theorem 1 of [5]).


## 3. Symmetries of System (8)

In this section, we consider one parameter Lie group of infinitesimal transformations: (x1=x, x2=t, u1=a, u2=b, u3=c, we use x and t instead of x1 and x2, resp., in order not to use index)
(15)x~=x+εξ1(x,t,a,b,c)+o(ε2),t~=t+εξ2(x,t,a,b,c)+o(ε2),a~=a+εϕ1(x,t,a,b,c)+o(ε2),b~=b+εϕ2(x,t,a,b,c)+o(ε2),c~=c+εϕ3(x,t,a,b,c)+o(ε2).
The symmetry generator of the above group of transformations is of the form
(16)X=ξ1(x,t,a,b,c)∂x+ξ2(x,t,a,b,c)∂t+ϕ1(x,t,a,b,c)∂a+ϕ2(x,t,a,b,c)∂b+ϕ3(x,t,a,b,c)∂c,
and, its second prolongation is the vector field
(17)Pr(2)X=X+ϕ1x∂ax+ϕ1t∂at+ϕ2x∂bx+ϕ2t∂bt+ϕ3x∂cx+ϕ3t∂ct+ϕ1xx∂axx+ϕ2xx∂bxx+ϕ3xx∂cxx+ϕ1xt∂axt+ϕ2xt∂bxt+ϕ3xt∂cxt+ϕ1tt∂att+ϕ2tt∂btt+ϕ3tt∂ctt.
Let (for convenience) Q1=ϕ1-ξ1ax-ξ2at,Q2=ϕ2-ξ1bx-ξ2bt and Q3=ϕ3-ξ1cx-ξ2ct, then by using (13), we can compute the coefficients of (17) as
(18)ϕ1x=DxQ1+ξ1axx+ξ2axt,ϕ2x=DxQ2+ξ1bxx+ξ2bxt,ϕ3x=DxQ3+ξ1cxx+ξ2cxt,ϕ1t=DtQ1+ξ1axt+ξ2att,ϕ2t=DtQ2+ξ1bxt+ξ2btt,ϕ3t=DtQ3+ξ1cxt+ξ2ctt,ϕ1xx=Dx2Q1+ξ1axxx+ξ2axxt,ϕ2xx=Dx2Q2+ξ1bxxx+ξ2bxxt,ϕ3xx=Dx2Q3+ξ1cxxx+ξ2cxxt,ϕ1tt=Dt2Q1+ξ1axtt+ξ2attt,ϕ2tt=Dt2Q2+ξ1bxtt+ξ2bttt,ϕ3tt=Dt2Q3+ξ1cxtt+ξ2cttt,ϕ1xt=DxDtQ1+ξ1axxt+ξ2axtt,ϕ2xt=DxDtQ2+ξ1bxxt+ξ2bxtt,ϕ3xt=DxDtQ3+ξ1cxxt+ξ2cxtt,
where Dx and Dt are the total derivatives with respect to x and t, respectively. By (14) the invariance condition is equivalent with the following equations:
(19)Pr(2)X[axx-btt]=0,axx-btt=0,Pr(2)X[bxy+cxx]=0,bxy+cxx=0,⋮Pr(2)X[axbx-bxct+btcx-cx2+abxx+2cbxt-bcxt]=0,axbx-bxct+btcx-cx2+abxx+2cbxt-bcxt=0.
After substituting Pr(2)X in the six equations above, we obtain six polynomial equations involving the various derivatives of a, b, and c, whose coefficients are certain derivatives of ξ1, ξ2, ϕ1, ϕ2, and ϕ3. Since ξ1, ξ2, ϕ1, ϕ2 and ϕ3 depend only on x, t, a, b, c we can equate the individual coefficients to zero, leading to the determining the following equations:
(20)a3ξa1=0,b2ξb1=0,aξb2=0,b2ξb2=0,acϕ2a=0,cbξb1=0,…abc(-3ξt1-ϕ3b)+2c2a(-ξt2+ξx1)+ba2(ϕ2b-ϕ3c-ξx1+ξt2)+ca2(-2ξx2+ϕ2c)+ϕ1(ab-2c2)-a2ϕ2+2caϕ3+4c3ξt1=0.
We write some of these equations whereas the number of the whole is 410. By solving this system of PDEs, we have the following.Theorem 1.
The Lie group of point symmetries of system (8) has the Lie algebra generated by (16), which its coefficient functions are
(21)ξ1=k3x+k1t+k2,ϕ2=(2k4-2k3+k7)b+2k6c,ϕ1=2k1c+k7a,ξ2=k6x+k4t+k5,ϕ3=(k7+k4-k3)c+k6a+k1b,
where ki’s, i=1,…,7 are arbitrary constants.Corollary 2.
Infinitesimal generator of any one-parameter Lie group of point symmetries of (8) is an ℝ-linear combination of
(22)X1=∂x,X2=∂t,X3=x∂x-2b∂b-c∂c,X4=x∂t+2c∂b+a∂c,X5=t∂x+2c∂a+b∂c,X6=t∂t+2b∂b+c∂c,X7=a∂a+b∂b+c∂c.It is worthwhile to note that, the Lie group obtained from this method, is a strong symmetry group of system (8). So we can transform solutions of the system to other solutions as well as reduce the system and obtain G-invariant solutions.Let𝔤 be the Lie algebra generated by (22). The commutator table of 𝔤 is given in Table 1, where the entry in the ith row and jth column is [Xi,Xj]=XiXj-XjXi, i,j=1,…,7.Table 1
The commutator table of𝔤.
[
,
]
X
1
X
2
X
3
X
4
X
5
X
6
X
7
X
1
0
0
X
1
X
2
0
0
0
X
2
0
0
0
0
X
1
X
2
0
X
3
-
X
1
0
0
X
4
-
X
5
0
0
X
4
-
X
2
0
-
X
4
0
X
3
-
X
6
+
2
X
7
X
4
0
X
5
0
-
X
1
X
5
-
X
3
+
X
6
-
2
X
7
0
-
X
5
0
X
6
0
-
X
2
0
-
X
4
X
5
0
0
X
7
0
0
0
0
0
0
0The group transformation which is generated byXi=ξi1∂x+ξi2∂t+ϕ1i∂a+ϕ2i∂b+ϕ3i∂c for i=1,…,7 is obtained by solving the seven systems of ODEs:
(23)dx~(s)ds=ξi1(x~(s),t~(s),a~(s),b~(s),c~(s)),x~(0)=x,dt~(s)ds=ξi2(x~(s),t~(s),a~(s),b~(s),c~(s)),t~(0)=t,da~(s)ds=ϕ1i(x~(s),t~(s),a~(s),b~(s),c~(s)),a~(0)=a,db~(s)ds=ϕ2i(x~(s),t~(s),a~(s),b~(s),c~(s)),b~(0)=b,dc~(s)ds=ϕ3i(x~(s),t~(s),a~(s),b~(s),c~(s)),c~(0)=c.dc~(s)ds=ϕ3i(x~(s),t~(s),a~(s),b~(s),c~(s)),i=1,…,7.
By exponentiating the infinitesimal symmetries (22), we obtain the one-parameter groups gk(s) generated by Xk, k=1,…,7 as follows:
(24)g1(s):(x,t,a,b,c)↦(x+s,t,a,b,c),g2(s):(x,t,a,b,c)↦(x,t+s,a,b,c),g3(s):(x,t,a,b,c)↦(xes,t,a,be-2s,ce-s),g4(s):(x,t,a,b,c)↦(x,sx+t,a,as2+2cs+b,as+c),g5(s):(x,t,a,b,c)↦(ts+x,t,bs2+2cs+a,b,bs+c),g6(s):(x,t,a,b,c)↦(x,tes,a,be2s,ces),g7(s):(x,t,a,b,c)↦(x,t,aes,bes,ces).
Consequently, we can state the following theorem.Theorem 3.
Ifa=f=f(x,t), b=h=h(x,t), and c=k=k(x,t), is a solution of (8), so are the functions
(25)g1(s)·f=f(x-s,t),g1(s)·h=h(x-s,t),g1(s)·k=k(x-s,t),g2(s)·f=f(x,t-s),g2(s)·h=h(x,t-s),g2(s)·k=k(x,t-s),g3(s)·f=f(xe-s,t),g3(s)·h=h(xe-s,t)e-2s,g3(s)·k=k(xe-s,t)e-s,g6(s)·f=f(x,te-s),g6(s)·h=h(x,te-s)e2s,g6(s)·k=k(x,te-s)es,g7(s)·f=f(x,t)es,g7(s)·h=h(x,t)es,g7(s)·k=k(x,t)es,g4(s)·f=f(x,t-sx),g4(s)·h=f(x,t-sx)s2+2k(x,t-sx)s+h(x,t-sx),g4(s)·k=f(x,t-sx)s+k(x,t-sx),g5(s)·f=h(x-st,t)s2+2k(x-st,t)s+f(x-st,t),g5(s)·h=h(x-st,t),g5(s)·k=h(x-st,t)s+k(x-st,t).This theorem is applied to obtain new solutions from known ones.Example 4.
Let(26)a=f(x,t)=r1x+r2,b=h(x,t)=r32r1x-r2r32r12ln(r1x+r2),c=k(x,t)=r3x+r4,
be a solution of (8), where r1,r2,r3,r4∈ℝ are arbitrary constants, we conclude that the functions gi(s)·f,gi(s)·h, and gi(s)·k are also solutions of (8) for i=1,…,7. For example,
(27)g5(s)·f=(r32r1(x-ts)-r2r32r12ln(r1(x-ts)+r2))s2g5(s)·f=+2(r3(x-ts)+r4)s+r1(x-ts)+r2,g5(s)·h=r32r1(x-ts)-r2r32r12ln(r1(x-st)+r2),g5(s)·k=(r32r1(x-ts)-r2r32r12ln(r1(x-ts)+r2))sg5(s)·k=+r3(x-ts)+r4
is a set of new solutions of (8) and we can obtain many other solutions by arbitrary combination of gi(s)’s for i=1,…,7. So we obtain infinite number of Einstein Walker manifolds just from this example.


## 4. One-Dimensional Optimal System of (8)

In this section, we obtain the one-parameter optimal system of (8) by using symmetry group. Since every linear combination of infinitesimal symmetries is an infinitesimal symmetry, there is an infinite number of one-dimensional subalgebras for the differential equation. So it is important to determine which subgroups give different types of solutions. Therefore, we must find invariant solutions which are not related by transformation in the full symmetry group. This procedure led to the concept of optimal system for subalgebras. For one-dimensional subalgebras, this classification problem is the same as the problem of classifying the orbits of the adjoint representation [2]. This problem is solved by the simple approach of taking a general element in the Lie algebra and simplify it as much as possible by imposing various adjoint transformation on it ([12, 13]). Optimal set of subalgebras is obtaining from taking only one representative from each class of equivalent subalgebras. Adjoint representation of each Xi, i=1,…,7 is defined as follows:
(28)Ad(exp(s·Xi)·Xj)=Xj-s·[Xi,Xj]+s22·[Xi,[Xi,Xj]]-⋯,
where s is a parameter and [Xi,Xj] is the commutator of 𝔤 defined in Table 1, for i,j=1,…,7 [2, page 199]. We can write the adjoint action for 𝔤  and show the following.Theorem 5.
A one-dimensional optimal system of (8) is given by
(29)(1)X7,(2)X1+aX7,(3)X2+aX7,(4)εX1+X6+aX7,(5)εX2+X5+aX6+bX7,(6)εX1+X4+aX5+bX6+cX7,(7)εX2+X3+ε′X4+aX5+bX6+cX7,
where ε and ε′ are ±1 or zero and a,b,c∈ℝ are arbitrary numbers.Proof.
Attending to Table1, we understand that the center of 𝔤 is the subalgebra 〈X7〉, so it is enough to specify the algebras of 〈X1,X2,X3,X4,X5,X6〉. Each adjoint transformation is a linear map Fis:𝔤→𝔤 that defined by X↦Ad(exp(sXi)·X), for i=1,…,7. The matrix Mis of Fis,  i=1,…,7, with respect to basis {X1,…,X7} is
(30)M1s=[10000000100000-s0100000-s01000000010000000100000001],M2s=[1000000010000000100000001000-s0001000-s000100000001],M3s=[es00000001000000010000000e-s0000000es0000000100000001],M4s=[1s000000100000001s000000100000-s-s21s-2s000-s0100000001],M5s=[1000000s1000000010-s0000s1-s2-s2s00001000000s100000001],M6s=[10000000es000000010000000es0000000e-s0000000100000001],
respectively and M7s is the 7×7 identity matrix. Let X=∑i=17‍aiXi, then
(31)F7s7∘⋯∘F1s1:X↦((1+s5s4)es3a1+es6+s3s4a2)·X1+⋯+(-s5s2a1-es6s2a2+⋯+(1+s5s4)a6-2s5s4a7)X6+a7X7.
Now, we can simplify X as follows.Ifa1=a2=⋯=a6=0, then X is reduced to the case (1), that is center of 𝔤.Ifa3=⋯=a6=0 and a1≠0, then we can make the coefficient of X2 vanish by F4s4; by setting s4=a2/a1. Scaling X if necessary, we can assume that a1=1. So, X is reduced to the case (2).Ifa1=a3=⋯=a6=0, then we can make a2=±1 by F6s6; by setting s6=ln|a2|. So, X is reduced to case (3).Ifa3=⋯=a5=0 and a6≠0, then we can make the coefficient of X2 vanish by F2s2; by setting s2=-a2/a6. Also the coefficient of X1 can be vanished or be ±1 by F3s3; by setting s3=ln|a1|. Scaling X if necessary, we can assume that a6=1. So, X is reduced to the case (4).Ifa3=a4=0 and a5≠0, then we can make the coefficient of X1 vanish by F2s2; by setting s2=-a1/a5. Also the coefficient of X2 can be vanished or be ±1 by F6s6; by setting s6=ln|a2|. Scaling X if necessary, we can assume that a5=1. So, X is reduced to the case (5).Ifa3=0 and a4≠0, then we can make the coefficient of X2 vanish by F1s1; by setting s1=-a2/a4. Also the coefficient of X1 can be vanished or be ±1 by F3s3; by setting s3=ln|a1|. Scaling X if necessary, we can assume that a4=1. So, X is reduced to the case (6).Ifa3≠0, then we can make the coefficient of X1 vanish by F1s1; by setting s1=-a1/a3. Also the coefficients of X2 and X4 can be vanished or be ±1 by F3s3 and F6s6; by setting s3=-ln|a4| and s6=ln|a2|, respectively. Scaling X if necessary, we can assume that a3=1. So, X is reduced to the case (7).


## 5. Similarity Reduction of System (8)

The system (8) is expressed in the coordinates (x,t,a,b,c), so for reducing this system we must search for its form in suitable coordinates. Those coordinates will be obtained by searching for independent invariants w, f, h, k corresponding to the infinitesimal generator. Hence by using the chain rule, the expression of the system in the new coordinate leads to the reduced system. Now, we find some of nontrivial solution of system (8). First, consider the operator X3=x∂x-2b∂b-c∂c. For determining independent invariants I, we ought to solve the homogeneous first-order PDE X3(I)=0, that is,
(32)(x∂x-2b∂b-c∂c)I=x∂I∂x+0∂I∂t+0∂I∂a-2b∂I∂b-c∂I∂c.
So we must solve the associated characteristic ODE
(33)dxx=dt0=da0=db-2b=dc-c.
Hence, four functionally independent invariants w=t, f=a, h=bx2, and k=cx are obtained. If we treat f, h, and k as functions of w, we can compute formulae for the derivatives of a, b, and c with respect to x and t in terms of w, f, h, k, and the derivatives of f, h, and k with respect to w. We have a=f(w)=f(t), b=x-2h(w)=x-2h(t) and c=x-1k(w)=x-1k(t). So by using the chain rule, we have
(34)a1=∂a∂x=∂f∂w∂w∂x=∂f∂t∂t∂x=0,a2=at=⋯=ft,b1=-2x-3h,b2=x-2ht,c1=-x-2k,c2=x-1kt,a11=axx=0,a12=0,a22=ftt,b11=6x-4h,b12=-2x-3ht,b22=x-2htt,c11=2x-3k,c12=-x-2kt,c22=x-1ktt.
Substituting these in the system (8), the reduced system is obtained as follows:
(35)htt=0,3hkt-5htk-k2+6fh=0,ftht+ftk-kt2+hftt+fkt=0,ktt=0,ht-k=0,2fth-2kkt+2fk+hktt=0,
which is a system of ODEs. Two types of solutions are obtained by solving this system
(36)(1){f=f(t)h=0k=0,(2){f=r22r2t+r1h=r2t+r1k=r2,
where r1 and r2 are arbitrary constants and f(t) is an arbitrary function. We can compute all of invariant solutions for other symmetry generators in a similar way. Some of infinitesimal symmetries and their Lie invariants are listed in Table 2.Table 2
Lie invariants and similarity solutions.
i
V
i
w
i
f
i
(
w
)
h
i
(
w
)
k
i
(
w
)
a
i
b
i
c
i
1
X
1
t
a
b
c
f
(
w
)
h
(
w
)
k
(
w
)
2
X
2
x
a
b
c
f
(
w
)
h
(
w
)
k
(
w
)
3
X
6
x
a
b
t
-
2
c
t
-
1
f
(
w
)
t
2
h
(
w
)
t
k
(
w
)
4
X
1
+
X
7
t
a
e
-
x
b
e
-
x
c
e
-
x
e
x
f
(
w
)
e
x
h
(
w
)
e
x
k
(
w
)
5
X
2
+
X
7
x
a
e
-
t
b
e
-
t
c
e
-
t
e
t
f
(
w
)
e
t
h
(
w
)
e
t
k
(
w
)
6
X
6
+
X
7
x
a
t
-
1
b
t
-
3
c
t
-
2
t
f
(
w
)
t
3
h
(
w
)
t
2
k
(
w
)In Table3, we list the reduced form of system (8) corresponding to infinitesimal generators of Table 2 and obtain some of its invariant solutions.Table 3
Reduced systems and their group invariant solutions.
i
Reduced system
Group invariant solutions
(
1
)
h
t
t
=
0 ktt=0 ftht-kt2+hftt=0 hktt=0
f
=
f
(
t
)
f
=
(
r
1
r
3
-
r
4
r
5
2
r
3
2
)
ln
(
r
3
t
+
r
4
)
+
r
5
2
t
r
3
+
r
2
h
=
0
h
=
r
3
t
+
r
4
k
=
r
1
k
=
r
5
t
+
r
6
(
2
)
f
x
x
=
0 kxx=0 fxhx-kx2+fhxx=0 kfxx-fkxx=0
f
=
0
f
=
r
3
x
+
r
4
h
=
h
(
x
)
h
=
(
r
1
r
3
-
r
4
r
5
2
r
3
2
)
ln
(
r
3
x
+
r
4
)
+
r
5
2
x
r
3
+
r
2
k
=
r
1
k
=
r
5
x
+
r
6
(
3
)
f
x
x
-
2
h
=
0 2kxk-kfxx+fkxx=0 2hx+kxx=0 fxk-k2-fkx=0 kx2-fxhx-3hxk-hkx-fhxx=0
k
=
r
2
x
+
r
3
k
=
128
-
r
1
3
(
x
+
r
2
)
2
+
r
3
x
+
r
4
f
=
k
(
x
+
r
1
)
f
=
-
4
k
k
x
x
k
x
x
x
h
=
r
2
h
=
k
x
k
x
x
x
-
2
k
x
x
2
k
x
x
x
(
4
)
h
t
t
-
f
=
0,  ht+k=0,  ft+ktt=0 ftht+ftk-kt2+hftt=0 hktt-fth+2kkt=0 2fh-2hkt+3htk-k2=0
k
=
0
k
=
r
2
e
r
1
t
h
=
r
1
h
=
-
r
2
r
1
e
r
1
t
f
=
0
f
=
-
r
1
r
2
e
r
1
t
(
5
)
f
x
x
-
h
=
0,  hx+kxx=0,  fx+k=0 2hf-2fkx+3fxk-k2=0 fhx-2kxk+kfxx-fkxx-hk=0 hxfx+hxk-kx2+fhxx=0
k
=
0
k
=
r
2
e
r
1
x
f
=
r
1
f
=
-
r
2
r
1
e
r
1
x
h
=
0
h
=
-
r
1
r
2
e
r
1
x
(
6
)
f
x
x
-
6
h
=
0,  fx+2k=0 3fkx-4fxk-3fh+4k2=0 3hx+kxx=0 4kxk-fhx-kfxx+fkxx+2hk=0 fxhx+4hxk+hkx-kx2+fhxx=0
k
=
0
k
=
-
27
(
r
1
x
+
r
2
)
3
h
=
0
h
=
-
1
3
k
x
f
=
r
1
f
=
-
3
k
2
k
x
wheref′,… are derivatives with respect to wand ri’s are arbitrary constants.Now by using the information of   Table2, we can obtain corresponding solutions for system (8). For example, consider the second invariant solution of case 5. Attending to the Table 2, the following solution is obtained:
(37)a=-r2r1er1xet,b=-r1r2er1xet,c=r2er1xet,
for system (8). Consequently by substituting a, b, and c in (6), we can determine the general form for metric of Einstein Walker manifolds for case 5. In addition, we can construct many other solutions from this one, by using Theorem 3, as we did in Example 4.


## 6. Conclusion

In this paper, by applying the method of Lie symmetries, we find the Lie point symmetry group of system (8). This work has been done by applying the criterion of invariance for the system under the prolonged infinitesimal generators. Also, we have obtained the one-parameter optimal system of subalgebras for system (8). Then the Lie invariants and similarity reduced systems and some of the invariant solutions are obtained. In conclusion, we specified a large class of Einstein Walker manifolds.



---

*Source: 591852-2013-02-24.xml*