Research Article | Open Access
Some General New Einstein Walker Manifolds
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.
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 . 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, called Einstein 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 . We can choose a basis for so that We set . Let be the number of indices with and . The inner product is then said to have signature . We can extend this argument on manifolds. Let where is an -dimensional manifold and is a symmetric nondegenerate smooth bilinear form on of signature , such a manifold is called a pseudo-Riemannian manifold of signature . So we have . If then is positive definite and is a Riemannian manifold. If is a system of local coordinates on , we may express where and “’’ is symmetric product. Let be a pseudo-Riemannian manifold of signature . Suppose that a splitting of the tangent bundle in the form is given, where and are smooth subbundles which are called distribution. This defines two complementary projection and of onto and , respectively. is called a parallel distribution if . Equivalently this means that if is any smooth vector field taking values in , then again takes values in . We say that is a null parallel distribution if is parallel and if the metric restricted to vanishes identically. Manifolds which admit null parallel distributions are called Walker manifolds. Let be a connection on manifold . The curvature operator is defined by the formula: Also the associated Ricci tensor is defined by 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 so that . Walker showed that a canonical form for a -dimensional pseudo-Riemannian Walker manifold which admits an -dimensional distribution is given by the metric tensor as where is the identity matrix and is a symmetric matrix whose entries are functions of the (for more details see [9, 10]).
If we adopt the following result for 4-dimensional Walker manifolds. Let , where be an open subset of and are smooth functions on . Then we can express the general form of metric tensor for 4-dimensional Walker manifolds as follows: that is given in the coordinates form as We can see that is Einstein if and only if the functions , , and verify the following system of PDEs [9, page 81]: This system is hard to handle, so we consider a spacial case in this paper, where , , and only depend on and . Therefore the following system must be solved: 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 th with independent and dependent variables such as involving and as independent and dependent variables, respectively and all the derivatives of with respect to from order to . We consider a one parameter Lie group of transformations which acts on the all variables of (9) where and are the infinitesimals of the transformations for the independent and dependent variables, respectively and is the parameter of transformation.
The most general form of infinitesimal generator associated with the above group of transformations is Transformations which map solutions of a differential equation to other solutions are called symmetries of this equation. The th order prolongation of is defined by where , , and the sum is over all ’s of order . If , the coefficient of will only depend on th and lower order derivatives of , and where and . Now, according to Theorem 2.36 of , the invariance of the system (9) under the infinitesimal transformations leads to the invariance conditions: 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 to be a (strong) symmetry group of (9) (Theorem 2.71 of  and Theorem 1 of ).
3. Symmetries of System (8)
In this section, we consider one parameter Lie group of infinitesimal transformations: (, , , , , we use and instead of and , resp., in order not to use index) The symmetry generator of the above group of transformations is of the form and, its second prolongation is the vector field Let (for convenience) and , then by using (13), we can compute the coefficients of (17) as where and are the total derivatives with respect to and , respectively. By (14) the invariance condition is equivalent with the following equations: After substituting in the six equations above, we obtain six polynomial equations involving the various derivatives of , , and , whose coefficients are certain derivatives of , , , , and . Since , , , and depend only on , , , , we can equate the individual coefficients to zero, leading to the determining the following equations: We write some of these equations whereas the number of the whole is 410. By solving this system of PDEs, we have the following.
Corollary 2. Infinitesimal generator of any one-parameter Lie group of point symmetries of (8) is an -linear combination of
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 -invariant solutions.
The group transformation which is generated by for is obtained by solving the seven systems of ODEs: By exponentiating the infinitesimal symmetries (22), we obtain the one-parameter groups generated by , as follows: Consequently, we can state the following theorem.
Theorem 3. If , , and , is a solution of (8), so are the functions
This theorem is applied to obtain new solutions from known ones.
Example 4. Let be a solution of (8), where are arbitrary constants, we conclude that the functions , and are also solutions of (8) for . For example, is a set of new solutions of (8) and we can obtain many other solutions by arbitrary combination of ’s for . 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 . 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 , is defined as follows: where is a parameter and is the commutator of defined in Table 1, for [2, page 199]. We can write the adjoint action for and show the following.
Theorem 5. A one-dimensional optimal system of (8) is given by where and are or zero and are arbitrary numbers.
Proof. Attending to Table 1, we understand that the center of is the subalgebra , so it is enough to specify the algebras of . Each adjoint transformation is a linear map that defined by , for . The matrix of , , with respect to basis is respectively and is the identity matrix. Let , then Now, we can simplify as follows. If , then is reduced to the case , that is center of . If and , then we can make the coefficient of vanish by ; by setting . Scaling if necessary, we can assume that . So, is reduced to the case . If , then we can make by ; by setting . So, is reduced to case . If and , then we can make the coefficient of vanish by ; by setting . Also the coefficient of can be vanished or be by ; by setting . Scaling if necessary, we can assume that . So, is reduced to the case . If and , then we can make the coefficient of vanish by ; by setting . Also the coefficient of can be vanished or be by ; by setting . Scaling if necessary, we can assume that . So, is reduced to the case . If and , then we can make the coefficient of vanish by ; by setting . Also the coefficient of can be vanished or be by ; by setting . Scaling if necessary, we can assume that . So, is reduced to the case . If , then we can make the coefficient of vanish by ; by setting . Also the coefficients of and can be vanished or be by and ; by setting and , respectively. Scaling if necessary, we can assume that . So, is reduced to the case .
5. Similarity Reduction of System (8)
The system (8) is expressed in the coordinates , so for reducing this system we must search for its form in suitable coordinates. Those coordinates will be obtained by searching for independent invariants , , , 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 . For determining independent invariants , we ought to solve the homogeneous first-order PDE , that is, So we must solve the associated characteristic ODE Hence, four functionally independent invariants , , , and are obtained. If we treat , , and as functions of , we can compute formulae for the derivatives of , , and with respect to and in terms of , , , , and the derivatives of , , and with respect to . We have , and . So by using the chain rule, we have Substituting these in the system (8), the reduced system is obtained as follows: which is a system of ODEs. Two types of solutions are obtained by solving this system where and are arbitrary constants and 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.