Canonical F -Planar Mappings of Spaces with Afﬁne Connection onto m -Symmetric Spaces

: In this paper, we consider canonical F -planar mappings of spaces with afﬁne connection onto m -symmetric spaces. We obtained the fundamental equations of these mappings in the form of a closed system of Chauchy-type equations in covariant derivatives. Furthermore, we established the number of essential parameters on which its general solution depends.


Introduction
In this paper, we further investigate F-planar mappings of spaces with affine connection.Ideologically, the theory concerning these mappings goes back to T. Levi-Civita's work [1], where he posed a problem of finding Riemannian spaces with common geodesic.He solved this problem in the special coordinate system.This problem is closely related to another topic, which is the study of the equations of mechanical system dynamics.
The study continued with a natural generalization of these classes of mappings called almost geodesic mappings.N.S.Sinyukov introduced almost geodesic mappings [3].He also determined three types of almost geodesic mappings, namely, π 1 , π 2 , and π 3 .
As the broadest generalization of geodesic, quasi-geodesic, and holomorphic-procjective mappings, the F-planar mappings were introduced into consideration by J. Mikeš and N.S.Sinyukov [4].At the same time, almost geodesic mappings of the second type π 2 are special F-planar mappings.Substantial refinements of the fundamental concepts of F-planar mappings are in the articles by I. Hinterleitner, J. Mikeš, and P. Peška [5][6][7].
In conclusion, we emphasize that the mappings mentioned above were found as diffeomorphisms preserving special curves: geodesic, holomorphically projective, and F-planar.The work of [7] shows the possibility of formulating the definitions as diffeomorphisms that map all geodesic curves onto the indicated types of curves.Therefore, we can use them to model the physical processes associated with these curves, which are implicitly described in the already mentioned works by Levi-Civita [1], Petrov [2], and Bejan, Kowalski [18].These curves are highly important in physics, especially theoretical mechanics and physics.The meaning of geodesics is widely known.
The study of the physical properties of special F-planar curves is described in the work of Petrov [2] (quasi-geodesics) and also currently in the works of Bejan and Druţȃ-Romaniuc [34] (magnetic curves).These curves are trajectories of the particles on which forces perpendicular to the direction of motion act.As a consequence, an operator F can be used to model magnetic forces.

Basic Concepts of the Theory of F-Planar Mappings of Spaces with Affine Connection
The following definitions and theorems for F-planar mappings are described in detail in the monograph [15,16] and the review article [6].The research is conducted locally, in a class of sufficiently smooth functions.
Consider the n-dimensional space A n with torsion-free affine connection ∇, assigned to the local coordinate system x 1 , x 2 , . . ., x n , in which the affinor structure F (i.e., a tensor field of type (1, 1)) is defined, for which in coordinates F h i = a • δ h i , where δ h i is the Kronecker symbol, a is some function.

Definition 1.
A curve defined by the equation = (t) is called F-planar if its tangent vector λ(t) = d (t)/dt( = 0) remains, under parallel translation along the curve , in the distribution generated by the vector functions λ and Fλ along .
According to this definition, a curve is F-planar if and only if the following condition holds: where ρ 1 (t) and ρ 2 (t) are some functions of the parameter t.
The class of F-planar curves is wide enough.It includes geodesic (if F = ρ Id, where ρ is a function and Id is the identity operator, or a function ρ 2 ≡ 0), quasi-geodesic, planar, and analytically planar curves.
Let A n and A n be two spaces with torsion-free affine connections ∇ and ∇, respectively.Let F and F be affine structures defined on A n and A n , respectively.

Definition 2. The mapping π:
Let us recall what a deformation tensor is, see [9,15,35].Consider the affine connection spaces A n and A n in a common F-planar coordinate system x 1 , x 2 , . . ., x n .The tensor is called a tensor of the deformation of connections.Here, Γ h ij (x) and Γ h ij (x) are components of affine connections ∇ and ∇, respectively.
From Theorems 1 and 2 of [4], and more precisely [6,7], see [16] (Chapter 14), it actually follows that the mapping π: A n → A n (n > 2) will be F-planar if and only if, for the deformation tensor P in the coordinate system x 1 , x 2 , . . ., x n , the following equality holds: where ψ i (x) and ϕ i (x) are some covectors, and the brackets mean symmetrization by the specified indices without division.F-planar mapping is called canonical if ψ i vanishes.Each F-planar mapping can be represented as a composition of a canonical F-planar mapping and a geodesic mapping.The latter can be considered a trivial F-planar.
Thus, canonical F-planar mappings in the common coordinate system x 1 , x 2 , . . ., x n are characterized by the equations Suppose that the affinor F defines in the space A n an e-structure [9] (p.177), which satisfies the condition F 2 = e Id, e = ±1, in coordinates: In this case, F-planar mapping will be denoted π(e).

Properties of Vector ϕ i
It is known [9] that there is a dependence between the Riemann tensors of spaces A n and Given that the deformation tensor of connections (1) has the structure (2), from the Formula (4) after transformations we obtain where Note, that the right hand side of the Equation ( 5) does not depend on the derivatives of ϕ i .Contracting (5) with the affinor F m ρ with respect to the indices ρ and h, we obtain Next, we contract (7) with respect to the indices m and i.As a result, we find After contraction of (7) with respect to the indices m and k, we obtain The Equations ( 9) after taking into account (8) can be written as Note that the Formula ( 10) is obtained for the general case of canonical F-planar mappings π(e) (e = ±1).
Therefore, we proved the following theorem.
Theorem 1.The vector ϕ i , participating in the Equations (2) of canonical F-planar mappings π(e), e = ±1 satisfies the conditions (10), where the tensor B h ijk is defined by the Formulas (6).
The right part of Equation ( 10) depends on the unknown tensor R h ijk , the unknown vector ϕ i , and the known affinor F h i and its covariant derivative F h i,k in A n .

Canonical F-Planar Mappings π(e) (e = ±1) of Spaces with Affine Connection onto 2-Symmetric Spaces
Space A n with affine connection is called (locally) symmetric if the Riemann tensor in it is absolutely parallel (P. A. Shirokov [36], É. Cartan [37], S. Helgason [38]).That is, symmetric spaces are characterized by the condition where R h ijk is the Riemann tensor of the space A n ; the sign " ; " denotes the covariant derivative with respect to the connection ∇ of the space A n .
Space A n is called 2-symmetric [27,39] if the conditions are met for the Riemann tensor Naturally, symmetric spaces are 2-symmetric spaces.Consider canonical F-planar mappings π(e) (e = ±1) of spaces with an affine connection onto *2-symmetric spaces A n , which are characterized by the Equations ( 2), and the affinor F h i satisfying the conditions (3) is defined in the space A n .We assume that the spaces A n and A n are related to the common coordinate system x 1 , x 2 , . . ., x n .Because then, given the Formula (1), we can write Based on the definition of the covariant derivative and taking into account the Formula (1), we have Differentiate ( 12) by x ρ 1 in the space A n .We obtain Comparing Equations ( 13) and ( 14), we have Taking account of ( 2) and ( 12), we might write (15) in the form where Given the structure of the tensor Θ h ijkm defined by the Formula ( 18), it is easy to see that the tensor Θ h ijkmρ 1 defined by the Formula ( 17) depends on the tensors F h k , R h ijk , ϕ k , as well as on covariant derivatives of the specified tensors by the connection ∇ of the space A n .In this case, the tensor F h k is considered to be given, and the conditions (3) are met for this tensor.
Let us introduce the tensor R h ijkm in the following way: Assume that the space A n is 2-symmetric.Then, for the Riemann tensor R h ijk of this space, the conditions (11) are met.Taking into account ( 19) from ( 16), we have where the tensor Θ h ijkmρ 1 is defined by the Formulas (17).
We assume that in (20) the tensors ϕ i,j , R h ijk,m are expressed in accordance with ( 10) and (19).
Obviously, Equations ( 10), ( 19) and (20) in this space A n represent a system of equations in covariant derivatives of the Cauchy type with respect to functions The functions R h ijk (x) and R h ijkl (x) must satisfy algebraic conditions that follow from the properties of the Riemannian tensor of A n : Thus, we proved the following Theorem.
Theorem 2. In order that an affine connection space A n admits a canonical F-planar mapping π(e) (e = ±1) onto a 2-symmetric space A n , it is necessary and sufficient that in the space A n a solution exists of a closed mixed system of Cauchy type equations in covariant derivatives (10), ( 19)-( 21) with respect to functions ϕ i (x), R h ijk (x) and R h ijkm (x).
Obviously, the general solution of the closed mixed system of Cauchy-type equations in covariant derivatives (10), ( 19)-( 21) depends on no more than The proof of Theorem 2 was actually done by us in the work [25] but in a different form.

Canonical F-Planar Mappings π(e) (e = ±1) of Spaces with Affine Connection onto m-Symmetric Spaces
The space of affine connection A n is called m-symmetric if the Riemann tensor R h ijk of this space satisfies the conditions The m-symmetric spaces are a natural generalization of symmetric and 2-symmetric spaces [39].
Based on the definition of the covariant derivative and taking into account the Formula (1), we have From the Formula (23) based on the Formulas (2) and ( 16), we obtain Differentiate ( 16) by x ρ 2 in the space A n .Taking into account the Formulas (24), we have We introduce the tensors R h ijkmρ 1 and Θ h ijkmρ 1 ρ 2 and assume Taking into account ( 26) and ( 27) from ( 25), we have Using the Equation ( 28), we covariantly differentiate (m − 2) times with respect to the connection of the space A n , and in the left part we proceed to the covariant derivative with respect to the connection of the space A n using the formula (R The Formula (30) is derived from (1).Suppose that the space A n is m-symmetric (m > 2).Then, taking into account ( 22) and ( 29) from the equation obtained in this way after substitutions and transformations, we have R h ijkρ 1 ...ρ m−2 ρ m−1 ,ρ m = Θ h ijkρ 1 ...ρ m−1 ρ m , (31) where Θ h ijkρ 1 ...ρ m−1 ρ m is some tensor depending on unknown tensors ϕ i , R h ijk , R h ijkρ 1 , . .., R h ijkρ 1 ...ρ m−1 , as well as on some well-known tensors.
Obviously, the general solution of the closed mixed system of the above mentioned equations depends on no more than 1/3 n 2 (n 2 − 1) (1 + n + n 2 + • • • + n m−1 ) + n essential parameters.