On Canonical Almost Geodesic Mappings of Type π2(e)

In the paper, we consider canonical almost geodesic mappings of type π 2 ( e ) . We have found the conditions that must be satisfied for the mappings to preserve the Riemann tensor. Furthermore, we consider canonical almost geodesic mappings of type π 2 ( e ) of spaces with affine connections onto symmetric spaces. The main equations for the mappings are obtained as a closed mixed system of Cauchy-type Partial Differential Equations. We have found the maximum number of essential parameters which the solution of the system depends on.


Introduction
The paper develops some new ideas in the theory of almost geodesic mappings of spaces with the affine connection. This theory can be dated back to the paper [1] of T. Levi-Civita, where he formulated and solved the problem of finding a Riemannian space with common geodesics, note that the problem was solved in a special coordinate system. It is worth noting that this problem is related to the study of equations of dynamics of mechanical systems.
Issues, arisen by the exploration, were studied by V.F. Kagan, G. Vrançeanu, Ya.L. Shapiro, and D.V. Vedenyapin et al. The authors discovered special classes of (n − 2)-flat spaces.
The first person to introduce the notion of quasi-geodesic mappings was Petrov, see [2]. Principally, the holomorphically projective mappings of Kählerian spaces are special quasi-geodesic mappings; these were examined by T. Otsuki and Y. Tashiro, M. Prvanović, and others.
In 1962, A.Z. Petrov [2] studied quasi-geodesic mappings, where he showed that it is possible to simulate physical processes and electromagnetic fields. Similar results are presented in the paper of C.-L. Bejan and O. Kowalski [37]. The mappings π 2 (e) are similar to those mentioned above. All these spaces are connected with some affinor structure F which can be interpreted as a force field.
In a 2019 paper [38] by A. Kozak and A. Borowiec, the authors studied a new physical interpretation of almost geodesic mappings, that are special transformations which genuinely preserve geodesics on the space-time.
Special almost geodesic mappings π 2 are mappings of type π 2 (e), which are related to e-structure F (F 2 = e · Id, e = ±1, 0), defined on the manifold, see [5]. The paper is devoted to studying the conditions guaranteeing that the Riemann tensor is invariant with respect to the canonical almost geodesic mappings of type π 2 (e). Additionally, we study canonical almost geodesic mappings of type π 2 (e) of spaces with affine connections onto symmetric spaces. The main equations for the mappings are obtained as a closed, mixed system of Cauchy-type Partial Differential Equations in covariant derivatives.
The investigations use local coordinates. We assume that all functions under consideration are sufficiently differentiable.

Basic Definitions of Almost Geodesic Mappings of Spaces with Affine Connections
Let us recall the basic definition, formulas, and theorems of the theory presented in [5,6,[33][34][35]. Consider a space A n with an affine connection ∇ without torsion. The space is referred to with coordinates x = (x 1 , x 2 , . . . , x n ).
A curve : x = x(t) in the space A n is a geodesic when its tangent vector λ(t) = dx(t)/dt satisfies the equations ∇ t λ = ρ(t) · λ, where ρ(t) is a certain function of t and ∇ t is a derivative along . Now, more often used are equations in the form ∇ t λ = 0. From our point of view, the parameter t is canonical, for more detail see [34] (pp. 118-121). A curve in the space A n is an almost geodesic when its tangent vector λ satisfies the equations where a(t) and b(t) are certain functions of t.
A diffeomorphism f : A n → A n is called a geodesic mapping if any geodesic of A n is mapped under f onto a geodesic in A n .
A diffeomorphism f : A n → A n is called an almost geodesic if any geodesic curve of A n is mapped under f onto an almost geodesic curve in A n .
Suppose that a space A n with affine connection ∇ admits a mapping f onto space A n with affine connection ∇, and the spaces are referred to with the common coordinate system x = (x 1 , x 2 , . . . , x n ).
The tensor P =∇ − ∇ is called the deformation tensor of the connections ∇ and∇ with respect to the mapping f ; in common coordinates x, components of P have the following form: where Γ h ij (x) and Γ h ij (x) are components of affine connections of the spaces A n and A n , respectively.
According to [5], a necessary and sufficient condition for the mapping f : A n → A n to be almost geodesic is that the deformation tensor P h ij (x) of the mapping f must satisfy the condition where λ h is an arbitrary vector and a and b are certain functions of variables x 1 , x 2 , . . . , x n and λ 1 , λ 2 , . . . , λ n . The tensor A h ijk is defined as We denote by comma " , " a covariant derivative with respect to the connection of the space A n . Almost geodesic mappings of spaces with affine connections were introduced by N. S. Sinyukov in [5]. He distinguished three kinds of almost geodesic mappings, namely, π 1 , π 2 , and π 3 , characterized by following conditions for the deformation tensor P: where δ h i is the Kronecker symbol, the round parentheses of indices denote an operation called symmetrization without division, and The types of almost geodesic mappings π 1 , π 2 , π 3 can intersect. The problem of completeness of classification had long remained unresolved. Berezovsky and Mikeš [14] proved that, for n > 5, other types of almost geodesic mappings except π 1 , π 2 , and π 3 do not exist.
As it was proved in [25], in case e = ±1, the basic equations of the mappings π 2 (e) can be written as Equation (1), and are unknown functions, and R h ijk is the Riemann tensor of the space A n . We denote by the brackets [i, k], an operation called antisymmetrization (or alternation) without division with respect to the indices i and k.
Obviously, right-hand sides of Equation (3) depend on unknown functions F h i , F h ij , µ i , µ ij and on the components Γ h ij of the space A n . Then, Equations (3) and (4) form a closed, mixed system of PDEs of Cauchy-type with respect to functions F h i , F h ij , µ i , µ ij . The general solution of the system, Equations (3) and (4), depends on no more than 1 2 n(n + 1) 2 essential parameters. In addition, the mapping π 2 (e) depends on unknown functions ψ i , ϕ j (see Equation (1)).

Canonical Almost Geodesic Mappings π 2 (e) (e = ±1) Preserving the Riemann Tensor
An almost geodesic mapping π 2 for which ψ i = 0 is called canonical. It is known that any almost geodesic mapping π 2 can be written as the composition of a canonical almost geodesic mapping and a geodesic mapping. The latter may be referred to as a trivial almost geodesic mapping.
Hence, a canonical almost geodesic mapping π 2 (e) (e = ±1) is determined by the equation and also by Equations (3) and (4). We proved [11] that Riemann tensor is preserved by the diffeomorphism if and only if the tensor A h ijk satisfies the conditions If the deformation tensor P h ij is expressed by Equation (5), then for π 2 (e) (e = ±1) taking account of (2), (3), and (4) we get Now, we require that A h ijk satisfies (6). Hence, where Let us multiply (7) by F j h and contract for indices h and j. Hence, we have Symmetrizing (8) in i and k, we obtain Equations (8) and (9) can be written as Hence, we get the theorem.

Theorem 1.
In order for space A n , with affine connection preserving the Riemann tensor, to admit an almost geodesic mappings of type π 2 (e) (e = ±1) onto space A n with affine connection, it is necessary and sufficient that the mixed system of differential equations of Cauchy-type in covariant derivatives (3) and (10) has a solution with respect to unknown functions F h i , F h ij , µ i , µ ij , ϕ i which must satisfy the algebraic conditions (4).

Canonical Almost Geodesic Mappings π 2 (e) of Spaces with Affine Connection onto Symmetric Spaces
A space A n with affine connection is called (locally) symmetric if its Riemann tensor is absolutely parallel. Symmetric spaces were introduced by É. Cartan in 1932 [44]. These spaces are also described in many monographs, i.e., S. Helgason [45]. Let us note that in the 1920's, P.A. Shirokov studied spaces where the Riemannian tensor is absolutely parallel, see reference paper [46]. Thus, the symmetric spaces A n are characterized by where R h ijk is the Riemann tensor of the space A n . By the symbol " | " we denote covariant derivative with respect to the connection of the space A n .
Let us consider the canonical almost geodesic mappings of type π 2 (e) (e = ±1) of spaces A n with affine connection onto symmetric spaces A n , which are determined by Equations (5), (3), and (4). Suppose that the spaces are referred to the common coordinate system x 1 , x 2 , . . . , x n . Since then taking account of (2) we can obtain In what follows, we understand that the space A n is symmetric. Taking account of (5) and (11), we have from (12) that It is known [5] that the Riemann tensors of the spaces A n and A n are related to each other by the equations Since the deformation tensor of the mapping P h ij (x) is represented by Equation (5), it follows from (14) that where Let us multiply (15) by F k h and contract for h and k. Hence, we have Symmetrizing (16) in i and j, we obtain Equations (16) and (17) can be written as Obviously, Equations (3), (13), and (18) Hence, we have proved the theorem.

Theorem 2.
In order for space A n with affine connection to admit an almost geodesic mappings of type π 2 (e) (e = ±1) onto symmetric space A n , it is necessary and sufficient that the mixed system of differential equations of Cauchy-type in covariant derivatives (3), (13), and (18) has a solution with respect to unknown functions F h i , F h ij , µ i , µ ij , R h ijk , ϕ i which must satisfy the algebraic conditions (4) and (19).
It is obvious that the general solution of the mixed system of Cauchy-type depends on no more than 1 3 n 2 (n 2 − 1) + 1 2 n(n + 1) 2 + n essential parameters.
Author Contributions: All authors contributed equally and significantly in writing this article. All authors have read and agreed to the published version of the manuscript.