Geodesic mappings of compact quasi-Einstein spaces, II


The paper treats geodesic mappings of quasi-Einstein spaces with gradient defining vector. 
 
Previously the authors defined three types of these spaces. 
In the present paper it is proved that there are no quasi-Einstein spaces of special type. 
It is demonstrated that quasi-Einstein spaces of main type are closed with respect to geodesic mappings. 
The spaces of particular type are proved to be geodesic $D$-symmetric spaces. 
 
 


INTRODUCTION
We will consider a pseudo-Riemannian space V n (n ą 2) with a metric tensor g ij . In this space we construct an Einstein tensor, which is defined by the following expression: where R ij is the corresponding Ricci tensor, R ij def = R α ijα , R the scalar curvature R αβ g αβ = R, R h ijk the Riemannian tensor. The tensor D ij defined by is called the defect of Einstein tensor [3].
In what follows we will treat pseudo-Riemannian spaces V n , where or in other words the spaces distinct from Einstein spaces. An Einstein tensor is an inner object of pseudo-Riemannian space V n , and thus it is defined by a metric tensor.
By defining special type of tensor D ij we can differentiate several types of special pseudo-Riemannian spaces. For example, when D ij is a linear combination of a metric tensor and a covariant derivative of a vector, depending on the coefficients of this combination, we get either φ(Ric) or Ricci solitons [2].
By imposing some additional conditions on the space, we are able to obtain various special spaces.
When D ij is a simple bivector, which is called "defining", then we face a quasi-Einstein space [6].
A bijection between points of pseudo-Riemannian spaces V n with a metric tensor g ij andV n with a metric tensorḡ ij is called a geodesic mapping, when every geodesic line of V n transforms into a geodesic line ofV n .
The works [4,5,7] are devoted to the research on geodesic mappings of quasi-Einstein spaces with gradient defining vector. The above-mentioned spaces are divided into three types: main, particular and special [1].
In the present paper we obtain local results which can be applied to the study on compact quasi-Einstein spaces of the main type "in general" [11], and the paper [17] solved the same issues in respect to compact quasi-Einstein spaces of a constant scalar curvature [18].
There are numerous works referring to the topics of geodesic and conformal mappings and their application in general relativity theory. It underlines the importance of the topic of this paper [3,8].
However, the plentiful works omitted the spaces of special type due to the absence of examples of these spaces.
This paper aims at the study of geodesic mappings of quasi-Einstein spaces of a special type and at the clarification of the tensor characteristics of spaces of other types.

GEODESIC MAPPINGS OF THE QUASI-EINSTEIN SPACES
Suppose that there exists a solution of the following equation in V n : with respect to a symmetrical tensor a ij ‰ cg ij and a vector λ i , λ i = λ ,i ‰ 0, [10,15].
Here the comma «,» is a sign of a covariant derivative in respect to the connection of V n . Equation (2.1) is called a linear form of the main equations of the theory of geodesic mappings.
An existence of non-trivial solutions of equations (2.1) with respect to a tensor a ij and a vector λ i is a necessary and sufficient condition for the space V n to permit non-trivial geodesic mappings [14, p. 108].
Analyzing the integrability conditions of equations (2.1) we can see that the tensor a ij as well as the Ricci tensor both comply to the following condition: a iα R α j´a αj R α i = 0. Meanwhile, taking into account (1.1) and (1.2) we get that where R h i = R αi g αh , D h i = D αi g αh , and g ij are elements of the matrix inverse to g ij . Differentiate the latter equation and taking into account (2.1) we obtain The equation (2.2) can be rewritten as follows Here (2.5) Pseudo-Riemannian spaces, where the following conditions are true are called geodesic D-symmetric.
In geodesic D-symmetric pseudo-Riemannian spaces equations (2.3) can also be written down as follows: Take into account that the tensor D ij satisfies the following condition: D αβ g αβ = 0. Then, we can wrap (2.7) and arrive at Then (2.7) can be written in the following form: Multiplying (2.9) by λ i = λ α g αi , wrapping it by index i, and taking into account (2.8), we get λ α λ α D jk = 0. Multiplying further (2.9) by D j m we obtain: and permits a non-trivial geodesic mappings, then the vector As far as λ i ı 0, we can select a vector ξ i in such a way, that ξ α λ α = 1. Then, from (2.9) we can obtain the following where τ i = ξ α D αi . Multiplying the latter by ξ i , we arrive at and (2.11) can we rewritten as follows: 3), we see that conditions (2.6) are satisfied. Thus we have proved the following statement.
Theorem 2.1. In order to define a pseudo-Riemannian space as geodesic D-symmetric, it is necessary and sufficient that the conditions (2.12) hold in that space.
On the other hand, pseudo-Riemannian spaces V n (n ą 2) satisfying the following conditions: where u i is a gradient vector by definition, are called quasi-Einstein spaces.
It is proved in [11] that if a quasi-Einstein space V n permits non-trivial geodesic mappings, then the following conditions hold for a vector λ i 14) or and v is some invariant. Thus, quasi-Einstein spaces, that permit non-trivial geodesic mappings, are divided into three types [5]: (1) main type if condition (2.14) hold, while (2.15) fails; (2) particular type if condition (2.15) holds, while (2.14) fails; (3) special type if both (2.14) and (2.15) hold.
Not that particular and special spaces are geodesic D-symmetric spaces. Moreover, in order to define a geodesic D-symmetric space as a quasi-Einstein space, it is necessary, that a vector ? wλ i is gradient.

QUASI-EINSTEIN SPACES OF SPECIAL TYPE
We will consider here geodesic mappings of quasi-Einstein spaces for which both equations (2.14), (2.15) are true.
The following statement holds: There are no quasi-Einstein spaces of special type.
Proof. The vector λ i of quasi-Einstein spaces of the special type is isotropic, namely: λ α λ α = 0.
Differentiating this identity and using (3.2) we get: In other words, the equations (2.14) imply that Suppose that R ‰ 0. Then, it follows from (3.3) and (2.14) that where a = a αβ g αβ . From (2.1) we see that a ,i = 2λ i . Then (3.4) and (3.1) also imply the following identity: The latter contradicts to the statement that R ‰ 0.
Hence the above-mentioned spaces have zero scalar curvature. Together with (3.2) the latter imply that µ = 0, whence (2.14) can be rewritten as follows Substituting (2.15) into (2.13) and taking into account that R = 0 we obtain where ρ def = ln 2 v The spaces satisfying conditions (3.6) are called recurrent. They do not admit non-trivial geodesic mappings as long as they are not spaces of a constant curvature [14, p.131]. This proves Theorem 3.1. □

QUASI-EINSTEIN SPACES OF MAIN TYPE
Consider no quasi-Einstein spaces of main type. These are quasi-Einstein spaces that permit non-trivial geodesic mappings and satisfying conditions (2.14) and (2.16) hold.
Pseudo-Riemannian spaces V n andV n permiting geodesic mappings one onto other are called spaces in a geodesic correspondence or the ones belonging to the same geodesic class.
Due to [14, p.107] pseudo-Riemannian spaces V n andV n permit geodesic mappings one onto other if and only if or, equivalently, taking into account that metric tensor is covariantly constant whenḡ where φ i is a certain vector (which is necessarily gradient), Γ h ij ,Γ h ij are Christoffel symbols of V n andV n relatively, and δ h i is the Kronecker symbol. Solutions of (4.2) and (2.1) are the following relation a ij = e 2φḡαβ g αi g βj (4.3) whereḡ ij are elements of a matrix inverse to the metric tensorV n of a space which is in a geodesic correspondence with V n . Applying covariant differentiation of equations (4.4) and taking into account (4.2) we get Substituting further (2.14) into (4.5), taking into account (4.3), and multiplying by e´2 φ , we obtain: (4.6) Multiplying (4.6) by g iαḡβk we get whereB = φ α φ βḡ αβ´e2φ µ. It is proved in [11] that in this caseB is some uniquely defined constant.
The following equations are necessary conditions for a geodesic mapping: where φ ij = φ i, j´φi φ j , R h ijk and R ij are Riemannian and Ricci tensors respectively.
Substituting (4.7) into (4.8) and taking into account (2.13), we obtain or, in other formĒ WhenB =R n(n´1) , thenV n is a quasi-Einstein space of the same type and with the same defining vector.
The following statement is true. Let us note, that mappings preserving Einstein tensor were studied in [9,13].

QUASI-EINSTEIN SPACES OF PARTICULAR TYPE
The mobility of a space with respect to geodesic mappings is a number of spaces onto which the given space permits non-trivial geodesic mappings.
The degree of mobility r is a number of non-trivial solutions of equations (2.1) and their differential extensions.
Suppose the following condition holds for a vector λ i from the equations (2.1): Here we proved that quasi-Einstein spaces cannot belong to spaces V n (B) with B = const. Thus, their degree of mobility in respect to geodesic mappings cannot exceed 2.
The known spaces having a degree of mobility equal to 2 with respect to geodesic mappings are conformal flat spaces and spaces L n . Both abovementioned types of spaces permit canonical solutions of equations (2.1), see [16].
The solutions for a system (2.1) are called canonical whenever Further, we consider the following problem: do quasi-Einstein spaces of particular type permit canonical solutions? Namely we will prove the following statement. Proof. For quasi-Einstein spaces of the particular type the equations (5.2) can be rewritten as follows: Applying covariant differentiation of (5.3) we get: Wrapping by indices i, j and taking into account that the vector λ i is isotropic we further obtain that Hence we can rewrite (5.4) as follows Here It is well-known, [5], that geodesic symmetrical spaces are geodesic Riccisymmetric spaces. Moreover, a quasi-Einstein space is a space of particular type if and only if it is geodesic Ricci-symmetric also satisfying conditions (2.14). Then, taking into account Theorem 4.1, the condition of falsity of (2.14) can be abandoned. Finally, geodesic symmetrical spaces belong to particular type whenever their scalar curvature is not constant.
For quasi-Einstein spaces with constant curvature the classes of geodesic D-symmetrical and geodesic Ricci-symmetric spaces coincide.

CONCLUSION
In the present paper it is proved that there exist only two types of quasi-Einstein spaces permitting geodesic mappings.
Quasi-Einstein spaces of main type belong to the spaces V n (B). They are closed in respect to geodesic mappings, or in other words, they permit geodesic mappings only onto quasi-Einstein spaces of main type. Moreover, Einstein tensor is invariant under such mappings.
Quasi-Einstein spaces of special type do not exist.
Quasi-Einstein spaces of particular type have a degree of mobility with respect to geodesic mappings not exceeding 2. On the other hand, these spaces do not permit canonical solutions of the main system of equations.