Geodesic Mappings of Spaces with Afﬁne Connections onto Generalized Symmetric and Ricci-Symmetric Spaces

: In the paper, we consider geodesic mappings of spaces with an afﬁne connections onto generalized symmetric and Ricci-symmetric spaces. In particular, we studied in detail geodesic mappings of spaces with an afﬁne connections onto 2-, 3-, and m - (Ricci-) symmetric spaces. These spaces play an important role in the General Theory of Relativity. The main results we obtained were generalized to a case of geodesic mappings of spaces with an afﬁne connection onto (Ricci-) symmetric spaces. The main equations of the mappings were obtained as closed mixed systems of PDEs of the Cauchy type in covariant form. For the systems, we have found the maximum number of essential parameters which the solutions depend on. Any m - (Ricci-) symmetric spaces ( m ≥ 1) are geodesically mapped onto many spaces with an afﬁne connection. We can call these spaces projectivelly m- (Ricci-) symmetric spaces and for them there exist above-mentioned nontrivial solutions.


Introduction
The paper is devoted to further study of the theory of geodesic mappings of affinely connected spaces. The theory goes back to the paper [1] by T. Levi-Civita in which the problem on the search for Riemannian spaces with common geodesics was stated and solved in a special coordinate system. We note the remarkable fact that this problem is related to the study of equations of dynamics of mechanical systems.
The spaces with covariantly constant curvature tensor (symmetric spaces) were considered in 1920 by P.A. Shirokov [5], E. Cartan [18], and A. Lichnerowicz [19], and with covariantly parallel curvature tensor (recurrent spaces) [20]. The study of symmetric and recurrent spaces is an extensive part of differential geometry and its applications.
It is well-known that the spaces of constant curvature are symmetric and for them E. Beltrami proved that they admit nontrivial geodesic mappings. In 1954, N.S. Sinyukov [7] began to study geodesic mappings of symmetric, recurrent, and semisymmetric spaces with equiaffine connection onto (pseudo-) Riemannian spaces. Continuation of these studies we can find in the works [21][22][23][24][25], V. Fomin [26], I. Hinterleitner, and J. Mikeš [27]. The above-mentioned results have a negative character in the sense that the space of non-constant curvature does not admit nontrivial geodesic mappings. T. Sakaguchi [28] and V. Domashev, J. Mikeš [29] studied similar tasks for holomorphically projective mappings. In the paper by V.Berezovski et al. [30], it is possible to find the generalized case of geodesic mappings of symmetric spaces.
Later, there were studied more generalized spaces than symmetric and recurrent ones. Generalized symmetric and recurrent spaces were comprehensively studied by V.R. Kaigorodov [31][32][33][34][35][36] from the point of view of the General Theory of Relativity. The paper [35] is a detailed analysis of this issue; it contains 97 citations. In another direction, symmetric spaces are generalized, for example, in works [37,38].
The above-mentioned results with proofs are in the works [12,13,15,17]. In our work, we continue the study of geodesic mappings of generalized symmetric spaces with an affine connection.
We suppose that all spaces under consideration are spaces with an affine connections without torsion. In addition, we assume that all geometric objects under consideration are not only continuous but also sufficiently smooth.

Basic Concepts of the Theory of Geodesic Mappings of Spaces with Affine Connections
A diffeomorphism between two spaces with an affine connections is an one-to-one differentiable mapping, and the inverse mapping is differentiable too. Among diffeomophisms, there are very important ones which are referred to as geodesic mappings.
Let us suppose that a space A n with an affine connection ∇ admits a diffeomorphism f onto another spaceĀ n with an affine connection∇ and locally the spaces are referred to a common coordinate system x, x = (x 1 , x 2 , . . . , x n ).
Assume P =∇ − ∇ and a in local coordinate system where Γ h ij (x) andΓ h ij (x) are components of affine connections ∇ and∇ of the spaces A n andĀ n , respectively, expressed with respect to the common coordinate system x. The tensor P is called a deformation tensor of the connections ∇ and∇ with respect to the mapping f .
A curve : x = x(t) in a space A n with an affine connection ∇ is a geodesic when its tangent vector λ(t) = dx(t)/dt satisfies the equations where ∇ t denotes the covariant derivative along and ρ(t) is some function.
A diffeomorphism f : A n →Ā n is an geodesic mapping if any geodesic of A n is mapped under f onto a geodesic inĀ n .
The most known equations of geodesic mappings are the Levi-Civita equations. He has obtained the equations for Riemannian spaces [1]. For the case of affinely connected spaces, the equation was later obtained by H. Weyl [4]. In the paper [59], the authors present alternative proofs for the Levi-Civita equation.
It is known [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] that the mapping f of a space A n onto a spaceĀ n is geodesic, if and only if in a common coordinate system x the deformation tensor has the form (the Levi-Civita equation) where δ h i is the Kronecker delta and ψ i is a covector. A geodesic mapping is called non-trivial if ψ i ≡ 0. It is obvious that any space A n with an affine connection admits a non-trivial geodesic mapping onto spaceĀ n with an affine connection. However, generally speaking, the similar statement would be wrong for geodesic mappings between Riemannian spaces. In particular, there are classes of Riemannian spaces which do not admit non-trivial geodesic mappings onto other Riemannian spaces.
In the general case, the main equations of geodesic mappings of spaces with an affine connections can not be reduced to closed systems of differential equations of Cauchy-type since the general solutions depend on n arbitrary functions ψ i (x).
N.S. Sinyukov [8,9] proved that the main equations for geodesic mappings of (pseudo-) Riemannian spaces are equivalent to some linear system of differential equations of Cauchy-type in covariant derivatives.
J. Mikeš and V. Berezovski [50] proved that the main equations for geodesic mapping of space with an affine connection onto a (pseudo-) Riemannian space can also be reduced to a closed system of PDE's of Cauchy type. In the case of geodesic mappings of an equiaffine space onto a (pseudo-) Riemannian space, the main equations are equivalent to some linear system of Cauchy-type in covariant derivatives. This property for all spaces with an affine connection follows from the results by J.M. Thomas [3], see [15,16] that any space with an affine connection is projectivelly equivalent to an equiaffine space.
Refs. [46][47][48] were devoted to geodesic mappings of spaces with an affine connections onto Ricci-symmetric and 2-Ricci-symmetric spaces. The main equations for the mappings were also obtained as closed systems of PDE's of Cauchy type. A more detailed description of the theory of partial differential equations (PDE's) of the Cauchy type can be found in the books ( [15] pp. 100-105) and ( [17] pp. 130-134).
It is known [7,[12][13][14][15][16][17] that, in a common coordinate system x, respective to the mapping, the components of the Riemannian tensors R h ijk andR h ijk of spaces with an affine connections A n andĀ n , respectively, are in the relation Throughout the paper, the comma denotes the covariant derivative with respect to the connection ∇ of the space A n . Taking account of (2), from (3), we obtain Contracting the Equation (4) for h and k, we get where R ij andR ij are the Ricci tensors of the spaces with an affine connections A n andĀ n , respectively. From the Equation (5), it follows that In particular, Equation (6) was obtained in the papers [46][47][48].
One of the most general generalizations are generalized m-recurrent (D m n ), m-recurrent (K m n ) and m-symmetric (S m n ) spaces, which are in turn characterized by relations [35] defined these spaces and studied them in detail. The natural generalizations of these spaces are generalized m-Ricci-recurrent (RicD m n ), m-Ricci-recurrent (RicK m n ), and m-Ricci-symmetric (RicS m n ) spaces, which are in turn characterized by relations Our work is devoted to the study of the m-symmetric and m-Ricci symmetric spaces. Therefore, we present an example of four-dimensional pseudo-Riemannian m-symmetric spaces, which is not -symmetric, = 1, 2, . . . , m − 1, see ( [35] p. 192): where β p are function on x 4 and α pq are polynoms (1) a pq ≡ 0. We construct an example of 4-dimensional pseudo-Riemannian Ricci m-symmetric spaces which is not Ricci -symmetric, = 1, 2, . . . , m − 1. These spaces are with the above-mentioned metric with function β p of variable x 4 , α pq (p.q = 2, 3) are m times differentiable function of x 4 and α 22 + α 33 is the polynom It is easy to construct more dimensional m-symmetric and m-Ricci symmetric spaces as product spaces of above-mentioned spaces and also trivial spaces which are e.g., spaces of constant curvature.
Recall the main results of geodesic mappings onto m-symmetric and Ricci m-symmetric spaces: 1. N.S. Sinyukov [7]: If equiaffine symmetric and recurrent spaces admit non-trivial geodesic mappings onto (pseudo-) Riemannian spacesV n thenV n is the space of constant curvature.

Geodesic Mappings of Spaces with Affine Connections onto m-Symmetric Spaces
1. We study geodesic mappings f of a space A n with an affine connection ∇ onto 2-symmetric spaceĀ n with an affine connection∇, which are characterized by the following condition [35]: where the symbol " ; " denotes a covariant derivative with respect to the connection of the spaceĀ n . SinceR From Equations (2) and (9), we get Let us differentiate (10) with respect to x ρ in the space A n . Taking into account (6), we get where Transforming (12) and taking into account (2) and (5), we get where Let us introduce a tensorR h ijkm defined bȳ In this case, we suppose that in (14) covariant derivatives of the tensorR h ijk with respect to the connection of the space A n are expressed according to (15).
From (11) and (13), we get Let us assume that the spaceĀ n is 2-symmetric. Hence, from (16), take into account (8) and (15), we findR Obviously, in the space A n , Equations (6), (15) and (17) form a closed mixed system of PDE's of Cauchy type with respect to functions ψ i (x),R h ijk (x) andR h ijkm (x). The functionsR h ijk (x) andR h ijkm (x) must satisfy the algebraic conditions (Ricci and Bianchi identities): Hence, we have given the proof.
Theorem 1. A space A n with an affine connection admits a geodesic mapping onto a 2-symmetric spaceĀ n if and only if the mixed system of differential equations of Cauchy type in covariant derivatives (6), (15), (17) and (18) has a solution with respect to the functions Obviously, the number of components of ψ i (x),R h ijk (x),R h ijkm (x) is n + n 4 + n 5 . Therefore, a general solution of Cauchy type system (6), (15), (17) and (18) depends on the initial conditions of these components at some point x 0 . This means that the solution depends on a finite number of essential parameters. However, from conditions (19), this number of parameters is reduced, and even more so when we take into account the integrability conditions. Estimation of the parameters is in the following corollary.
2. Now, we study geodesic mapping of space A n onto 3-symmetric spaceĀ n , which are characterized by the following conditions [35]: Let us covariantly differentiate (16) with respect to x l in the space A n and on the left-hand side express the covariant derivative with respect to the connection of A n in terms of the covariant derivative with respect to the connection ofĀ n , using the formula Let us assume that the spaceĀ n is 3-symmetric. Hence, from the obtained equation because of (19), using substitutions and transformations, we find where θ h ijkmρl is some tensor depending on unknown tensors ψ i ,R h ijk ,R h ijkm ,R h ijkmρ , and on some tensors, which are assumed to be known.
Obviously, in the space A n , Equations (6), (15), (20) and (21) form a closed mixed system of PDE's of Cauchy type with respect to functions . In addition, the algebraic conditions (18) have to be satisfied.
Hence, we have proved Theorem 2. A space A n with an affine connection admits a geodesic mapping onto a 3-symmetric spaceĀ n if and only if the mixed system of differential equations of Cauchy type in covariant derivatives (6), (15), (20), (21) and (18) has a solution with respect to the functions The following parameters estimation follows from the Ricci identity of curvature tensor and its derivatives.
3. Finally, we study geodesic mappings of space A n onto m-symmetric spaceĀ n , which are characterized by the following condition [35]: Let us differentiate (21) covariantly (m − 2) times with respect to the connection of the space A n and on the left-hand side express the covariant derivative with respect to the connection of A n in terms of the covariant derivative with respect to the connection ofĀ n , using the formula The formula holds because of (1).
Let us assume that the spaceĀ n is m-symmetric (m > 3). Hence, from the obtained equation because of (22), using substitutions and transformations, taking account of (15), (20), (23), we get where θ h ijkρ 1 ...ρ m−1 ρ m is some tensor depending on unknown tensors , and on some tensors, which are assumed to be known.
Obviously, in the space A n the Equations (6), (15), (20), (23), (24) form a closed mixed system of PDE's of Cauchy type with respect to functions In addition, the algebraic conditions (18) have to be satisfied.
Hence, we have given the proof.

Theorem 3.
A space A n with an affine connection admits a geodesic mapping onto a m-symmetric spaceĀ n if and only if the mixed system of differential equations of Cauchy type in covariant derivatives (6), (15), (18), (20), (23) and (24) has a solution with respect to the functions

Geodesic Mappings of Spaces with Affine Connections onto m-Ricci-Symmetric Spaces
1. Here, we study geodesic mappings of space A n onto 2-Ricci-symmetric spaceĀ n , which are characterized by the following condition:R ij;km = 0, (25) whereR ij is the Ricci tensor ofĀ n . Let us contract the Equation (16) for h and k. Because of expressions for the tensors B h ijkmρ and C h ijkmρ , we find where (ij) denotes an operation called symmetrization without division with respect to the indices i and j.
Let us introduce a tensorR ijmR ij,m =R ijm .
Hence, we have given the proof.

Theorem 4.
A space A n with an affine connection admits a geodesic mapping onto a 2-Ricci-symmetric spacē A n if and only if the closed system of differential equations of Cauchy type in covariant derivatives (6), (27) and (28) has a solution with respect to the functions ψ i (x),R ij (x), andR ijk (x).
Systems (6), (27) and (28) have no more than one solution for initial conditions of components The number of parameters of ψ i (x 0 ),R ij (x 0 ) andR ijk (x 0 ) are n + n 2 + n 3 . Therefore, the following corollary holds.

Corollary 4.
The general solution of the system of Cauchy type (6), (27) and (28) depends on no more than n + n 2 + n 3 essential parameters.
2. Now, we study geodesic mappings of space A n onto 3-Ricci-symmetric spaceV n , which are characterized by the condition:R ij;kml = 0.
Let us covariantly differentiate (26) with respect to x l in the space A n and on the left-hand side express the covariant derivative with respect to the connection of A n in terms of the covariant derivative with respect to the connection ofĀ n , using the formula (R ij;mk ) ,l =R ij;mkl + P α ilRαj;mk + P α jlRiα;mk + P α mlRij;αk + P α klR ij;mα .
Using the formulas for transition from the covariant derivatives with respect to the connection of the spaceĀ n to the the covariant derivatives with respect to the connection of the space A n , we find where Ω ijmkl is some tensor, which depends on unknown tensors ψ i ,R ij ,R ij,k ,R ij,km and, on some tensors, which are assumed to be known. Let us introduce a tensorR ijmk defined bȳ Let us assume that the spaceĀ n is 3-symmetric. Hence, from (30), taking into account (27) and (31), we findR where the tensor Ω ijmkl depends on the unknown tensors ψ i ,R ij ,R ijk , andR ijkm . Obviously, in the space A n , Equations (6), (27), (31) and (32) form a closed system of PDE's of Cauchy type with respect to functions ψ i (x),R ij (x),R ijk (x) andR ijkm (x).
Hence, we have proved Theorem 5. A space A n with an affine connection admits a geodesic mapping onto a 3-Ricci-symmetric spacē A n if and only if the system of differential equations of Cauchy type in covariant derivatives (6), (27), (31) and (32) has a solution with respect to the functions ψ i (x),R ij (x),R ijk (x), andR ijkm (x).
It is essentially that 2-and 3-Ricci-symmetric spaces are particular cases of m-Ricci-symmetric spaces.
Hence, we have given the proof.

Conclusions
The above new results are the generalization of analogical results for geodesic mappings onto symmetric and Ricci-symmetric spaces and for the case of m-symmetric and m-Ricci symmetric spaces (m > 1). The generalization of m-recurrent, m-Ricci-recurrent and also D m n and RicD m n studied by Kaigorodov probably does not exist in the general case.
Author Contributions: Investigation, V.B., Y.C., I.H. and P.P. All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.