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

Citation: Berezovski, V.; Rýparová, L.; Cherevko, Y. Canonical F-Planar Mappings of Spaces with Affine Connection onto m-Symmetric Spaces. Mathematics 2023,11, 1246. https://doi.org/10.3390/ math11051246 Academic Editors: Graham Hall FRSE and Jean-Charles Pinoli Received: 26 January 2023 Revised: 23 February 2023 Accepted: 2 March 2023 Published: 4 March 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). mathematics Article Canonical F-Planar Mappings of Spaces with Affine Connection onto m-Symmetric Spaces Volodymyr Berezovski 1, Lenka Rýparová 2,3,* and Yevhen Cherevko 4 1Department of Mathematics and Physics, Uman National University of Horticulture, 20300 Uman, Ukraine 2Department of Algebra and Geometry, Faculty of Science, Palacky University in Olomouc, 771 46 Olomouc, Czech Republic 3Institute of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic 4 Department of Physical and Mathematical Sciences, Odesa National University of Technology, Kanatnaya 112, 65039 Odesa, Ukraine *Correspondence: lenka.rypar[email protected] Abstract: 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. Keywords: F -planar mapping; space with affine connection; symmetric space; 2-symmetric space; m-symmetric space MSC: 53B05; 35R01 1. 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. Many authors contributed to the development of the theory of geodesic mappings, including T. Thomas, H. Weyl, P.A. Shirokov, A.S. Solodovnikov, N.S. Sinyukov, A.V. Aminova, J. Mikeš, and others. The study of geodesic mappings raised questions many authors addressed and developed, i.e., V.F. Kagan, G. Vr˘anceanu, Ya.L. Shapiro, D.V. Vedenyapin, and others. The listed authors found special classes of (n−2)-projective spaces. A. Z. Petrov [ 2 ] introduced the concept of quasi-geodesic mappings. Special quasigeodesic mappings, in particular, are holomorphically projective mappings of Kaehler spaces, considered by T. Otsuki, Ya. Tashiro, M. Prvanovi´c, and J. Mikeš et al. 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, π1,π2, and π3. 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 π2 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 [5–7]. The above results are presented in a developed form in monographs and researchers, e.g., [8–16]. The theory of F -planar mappings is developed in many works, for example, [ 17 – 25 ]. Our work is devoted to the study of F -planar mappings onto m -symmetric spaces. In this Mathematics 2023,11, 1246. https://doi.org/10.3390/math11051246 https://www.mdpi.com/journal/mathematics Mathematics 2023,11, 1246 2 of 9 case, we find the fundamental equations in a new form. Analogous results were found for simpler cases in the theory of geodesic and almost geodesic mappings, for example, [24–33]. 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 Dru¸t˘a-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. 2. 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 An with torsion-free affine connection ∇ , assigned to the local coordinate system x1 , x2 , . . . , xn , in which the affinor structure F (i.e., a tensor field of type ( 1,1 ) ) is defined, for which in coordinates Fh i6=a·δh i , where δh i is the Kronecker symbol, ais some function. Definition 1. A curve ` defined by the equation `=`(t) is called F -planar if its tangent vector λ(t) = d`(t)/dt(6= 0 ) remains, under parallel translation along the curve ` , in the distribution generated by the vector functions λand Fλalong `. According to this definition, a curve ` is F -planar if and only if the following condition holds: ∇λ(t)λ(t) = ρ1(t)λ(t) + ρ2(t)Fλ(t), where ρ1(t)and ρ2(t)are some functions of the parameter t. The class of F -planar curves is wide enough. It includes geodesic (if F=ρId , where ρ is a function and Id is the identity operator, or a function ρ2≡ 0), quasi-geodesic, planar, and analytically planar curves. Let An and An be two spaces with torsion-free affine connections ∇ and ∇ , respectively. Let Fand Fbe affine structures defined on Anand An, respectively. Definition 2. The mapping π : An→An is called F -planar if any F -planar curve of space An is mapped onto an F-planar curve of space An. Let us recall what a deformation tensor is, see [ 9 , 15 , 35 ]. Consider the affine connection spaces Anand Anin a common F-planar coordinate system x1,x2, . . . , xn. The tensor Ph ij(x) = Γh ij(x)−Γh ij(x), (1) is called a tensor of the deformation of connections. Here, Γh ij(x) and Γh 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:An→An(n> 2 ) will be F -planar if and only if, for Mathematics 2023,11, 1246 3 of 9 the deformation tensor P in the coordinate system x1 , x2 , . . . , xn , the following equality holds: Ph ij =δh (iψj)+Fh (iϕj), where ψi(x) and ϕi(x) are some covectors, and the brackets mean symmetrization by the specified indices without division. F -planar mapping is called canonical if ψi 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 x1 , x2 , . . . , xn are characterized by the equations Ph ij =Fh (iϕj). (2) Suppose that the affinor F defines in the space An an e -structure [ 9 ] (p. 177), which satisfies the condition F2=eId, e=±1, in coordinates: Fh αFα i=eδh i. (3) In this case, F-planar mapping will be denoted π(e). 3. Properties of Vector ϕi It is known [ 9 ] that there is a dependence between the Riemann tensors of spaces An and An Rh ijk =Rh ijk +Ph ik,j−Ph ij,k+Pα ikPh αj−Pα ij Ph αk. (4) Given that the deformation tensor of connections (1) has the structure (2) , from the Formula (4) after transformations we obtain ϕi,jFh k+ϕk,jFh i−ϕi,kFh j−ϕj,kFh i=Bh ijk, (5) where Bh ijk =Rh ijk −Rh ijk −ϕiFh k,j−Fh j,k+eδh kϕj+ϕαFα kFh j−eδh jϕk−ϕαFα jFh k −ϕkFh i,j+ϕαFα iFh j+ϕjFh i,k+ϕαFα iFh k.(6) Note, that the right hand side of the Equation (5) does not depend on the derivatives of ϕi. Contracting (5) with the affinor Fm ρwith respect to the indices ρand h, we obtain δm kϕi,j+δm iϕk,j−δm jϕi,k−δm iϕj,k=eBα ijkFm α. (7) Next, we contract (7) with respect to the indices mand i. As a result, we find ϕk,j−ϕj,k=e n+1Bα βjkFβ α. (8) After contraction of (7) with respect to the indices mand k, we obtain nϕi,j−ϕj,i=eBα ijβFβ α. (9) The Equations (9) after taking into account (8) can be written as ϕi,j=e n−1Bα ijβ−1 n+1Bα βjiFβ α. (10) 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. Mathematics 2023,11, 1246 4 of 9 Theorem 1. The vector ϕi , participating in the Equations (2) of canonical F -planar mappings π(e), e =±1satisfies the conditions (10), where the tensor Bh ijk is defined by the Formulas (6). The right part of Equation (10) depends on the unknown tensor Rh ijk , the unknown vector ϕi, and the known affinor Fh iand its covariant derivative Fh i,kin An. 4. Canonical F-Planar Mappings π(e) (e=±1)of Spaces with Affine Connection onto 2-Symmetric Spaces Space An 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 Rh ijk;m=0, where Rh ijk is the Riemann tensor of the space An ; the sign “ ; ” denotes the covariant derivative with respect to the connection ∇of the space An. Space An is called 2-symmetric [ 27 , 39 ] if the conditions are met for the Riemann tensor Rh ijk Rh ijk;mρ1=0. (11) 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 An , which are characterized by the Equations (2) , and the affinor Fh i satisfying the conditions (3) is defined in the space An . We assume that the spaces Anand Anare related to the common coordinate system x1,x2, . . . , xn. Because Rh ijk;m=∂Rh ijk ∂xm+Γh mαRα ijk −Γα miRh αjk −Γα mjRh iαk−Γα mkRh ijα, then, given the Formula (1), we can write Rh ijk;m=Rh ijk,m+Ph mαRα ijk −Pα miRh αjk −Pα mjRh iαk−Pα mkRh ijα. (12) Based on the definition of the covariant derivative Rh ijk;m,ρ1=∂Rh ijk;m ∂xρ1+Γh αρ1Rα ijk;m−Γα iρ1Rh αjk;m−Γα jρ1Rh iαk;m −Γα kρ1Rh ijα;m−Γα mρ1Rh ijk;α, and taking into account the Formula (1), we have Rh ijk;m,ρ1=Rh ijk;mρ1−Ph αρ1Rα ijk;m+Pα iρ1Rh αjk;m+Pα jρ1Rh iαk;m +Pα kρ1Rh ijα;m+Pα mρ1Rh ijk;α.(13) Differentiate (12) by xρ1in the space An. We obtain Rh ijk;m,ρ1=Rh ijk,mρ1+Ph mα,ρ1Rα ijk +Ph mαRα ijk,ρ1−Pα mi,ρ1Rh αjk −Pα miRh αjk,ρ1 −Pα mj,ρ1Rh iαk−Pα mjRh iαk,ρ1−Pα mk,ρ1Rh ijα−Pα mkRh ijα,ρ1.(14) Mathematics 2023,11, 1246 5 of 9 Comparing Equations (13) and (14), we have Rh ijk,mρ1=Rh ijk;mρ1−Ph αρ1Rα ijk;m+Pα iρ1Rh αjk;m+Pα jρ1Rh iαk;m+Pα kρ1Rh ijα;m +Pα mρ1Rh ijk;α−Ph mα,ρ1Rα ijk −Ph mαRα ijk,ρ1+Pα mi,ρ1Rh αjk +Pα miRh αjk,ρ1 +Pα mj,ρ1Rh iαk+Pα mjRh iαk,ρ1+Pα mk,ρ1Rh ijα+Pα mkRh ijα,ρ1. (15) Taking account of (2) and (12), we might write (15) in the form Rh ijk,mρ1=Rh ijk;mρ1+Θh ijkmρ1, (16) where Θh ijkmρ1=−Fh (αϕρ1)Rα ijk,m+Θα ijkm+Fα (iϕρ1)Rh αjk,m+Θh αjkm +Fα (jϕρ1)Rh iαk,m+Θh iαkm+Fα (kϕρ1)Rh ijα,m+Θh ijαm +Fα (mϕρ1)Rh ijk,α+Θh ijkα−Fh (mϕα),ρ1Rα ijk −Fh (mϕα)Rα ijk,ρ1 +Fα (mϕi),ρ1Rh αjk +Fα (mϕi)Rh αjk,ρ1+Fα (mϕj),ρ1Rh iαk +Fα (mϕj)Rh iαk,ρ1+Fα (mϕk),ρ1Rh ijα+Fα (mϕk)Rh ijα,ρ1, (17) Θh ijkm =Fh (mϕα)Rα ijk −Fα (mϕi)Rh αjk −Fα (mϕj)Rh iαk−Fα (mϕk)Rh ijα. (18) Given the structure of the tensor Θh ijkm defined by the Formula (18) , it is easy to see that the tensor Θh ijkmρ1 defined by the Formula (17) depends on the tensors Fh k , Rh ijk , ϕk , as well as on covariant derivatives of the specified tensors by the connection ∇ of the space An . In this case, the tensor Fh k is considered to be given, and the conditions (3) are met for this tensor. Let us introduce the tensor Rh ijkm in the following way: Rh ijk,m=Rh ijkm, (19) Assume that the space An is 2-symmetric. Then, for the Riemann tensor Rh ijk of this space, the conditions (11) are met. Taking into account (19) from (16), we have Rh ijkm,ρ1=Θh ijkmρ1, (20) where the tensor Θh ijkmρ1is defined by the Formulas (17). We assume that in (20) the tensors ϕi,j , Rh ijk,m are expressed in accordance with (10) and (19). Obviously, Equations (10) , (19) and (20) in this space An represent a system of equations in covariant derivatives of the Cauchy type with respect to functions ϕi(x) , Rh ijk(x) , Rh ijkm(x). The functions Rh ijk(x) and Rh ijkl(x) must satisfy algebraic conditions that follow from the properties of the Riemannian tensor of An: Rh i(jk)=0, Rh (ijk)=0, Rh i(jk)l=0, Rh (ijk)l=0. (21) Thus, we proved the following Theorem. Theorem 2. In order that an affine connection space An admits a canonical F -planar mapping π(e) (e=± 1 ) onto a 2-symmetric space An , it is necessary and sufficient that in the space An Mathematics 2023,11, 1246 6 of 9 a solution exists of a closed mixed system of Cauchy type equations in covariant derivatives (10) , (19)–(21)with respect to functions ϕi(x), Rh ijk(x)and Rh 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 1/3 n2(n3+n2−n−1) + n essential parameters. The proof of Theorem 2was actually done by us in the work [ 25 ] but in a different form. 5. Canonical F-Planar Mappings π(e) (e=±1)of Spaces with Affine Connection onto m-Symmetric Spaces The space of affine connection An is called m -symmetric if the Riemann tensor Rh ijk of this space satisfies the conditions Rh ijk;ρ1ρ2...ρm=0. (22) The m -symmetric spaces are a natural generalization of symmetric and 2-symmetric spaces [39]. Based on the definition of the covariant derivative Rh ijk;mρ1,ρ2=∂Rh ijk;mρ1 ∂xρ2+Γh αρ2Rα ijk;mρ1−Γα iρ2Rh αjk;mρ1−Γα jρ2Rh iαk;mρ1 −Γα kρ2Rh ijα;mρ1−Γα mρ2Rh ijk;αρ1−Γα ρ1ρ2Rh ijk;mα, and taking into account the Formula (1), we have Rh ijk;mρ1,ρ2=Rh ijk;mρ1ρ2−Ph αρ2Rα ijk;mρ1+Pα iρ2Rh αjk;mρ1+Pα jρ2Rh iαk;mρ1 +Pα kρ2Rh ijα;mρ1+Pα mρ2Rh ijk;αρ1+Pα ρ1ρ2Rh ijk;mα.(23) From the Formula (23) based on the Formulas (2) and (16), we obtain Rh ijk;mρ1,ρ2=Rh ijk;mρ1ρ2−Fh (αϕρ2)Rα ijk,mρ1−Θα ijkmρ1+Fα (iϕρ2)Rh αjk,mρ1−Θh αjkmρ1 +Fα (jϕρ2)Rh iαk,mρ1−Θh iαkmρ1+Fα (kϕρ2)Rh ijα,mρ1−Θh ijαmρ1 +Fα (mϕρ2)Rh ijk,αρ1−Θh ijkαρ1+Fα (ρ1ϕρ2)Rh ijk,mα−Θh ijkmα. (24) Differentiate (16) by xρ2 in the space An . Taking into account the Formulas (24) , we have Rh ijk,mρ1ρ2=Rh ijk;mρ1ρ2−Fh (αϕρ2)Rα ijk,mρ1−Θα ijkmρ1+Fα (iϕρ2)Rh αjk,mρ1−Θh αjkmρ1 +Fα (jϕρ2)Rh iαk,mρ1−Θh iαkmρ1+Fα (kϕρ2)Rh ijα,mρ1−Θh ijαmρ1 +Fα (mϕρ2)Rh ijk,αρ1−Θh ijkαρ1+Fα (ρ1ϕρ2)Rh ijk,mα−Θh ijkmα+Θh ijkmρ1,ρ2. (25) We introduce the tensors Rh ijkmρ1and Θh ijkmρ1ρ2and assume Rh ijkm,ρ1=Rh ijkmρ1, (26) Mathematics 2023,11, 1246 7 of 9 Θh ijkmρ1ρ2=−Fh (αϕρ2)Rα ijk,mρ1−Θα ijkmρ1+Fα (iϕρ2)Rh αjk,mρ1−Θh αjkmρ1 +Fα (jϕρ2)Rh iαk,mρ1−Θh iαkmρ1+Fα (kϕρ2)Rh ijα,mρ1−Θh ijαmρ1 +Fα (mϕρ2)Rh ijk,αρ1−Θh ijkαρ1+Fα (ρ1ϕρ2)Rh ijk,mα−Θh ijkmα+Θh ijkmρ1,ρ2. (27) Taking into account (26) and (27) from (25), we have Rh ijkmρ1,ρ2=Rh ijk;mρ1ρ2+Θh ijkmρ1ρ2. (28) Let us introduce tensors Rh ijkρ1ρ2ρ3,. . ., Rh ijkρ1ρ2ρ3...ρm−2ρm−1, and let us put Rh ijkρ1ρ2,ρ3=Rh ijkρ1ρ2ρ3, . . . Rh ijkρ1ρ2ρ3...ρm−2,ρm−1=Rh ijkρ1ρ2ρ3...ρm−2ρm−1. (29) Using the Equation (28) , we covariantly differentiate (m− 2 ) times with respect to the connection of the space An , and in the left part we proceed to the covariant derivative with respect to the connection of the space Anusing the formula (Rh ijk;ρ1...ρτ−2ρτ−1),ρτ=Rh ijk;ρ1...ρτ−2ρτ−1ρτ−Ph αρτRα ijk;ρ1...ρτ−2ρτ−1+Pα iρτRh αjk;ρ1...ρτ−2ρτ−1 +Pα jρτRh iαk;ρ1...ρτ−2ρτ−1+Pα kρτRh ijα;ρ1...ρτ−2ρτ−1+Pα ρ1ρτRh ijk;α...ρτ−2ρτ−1 +· · · +Pα ρτ−1ρτRh ijk;ρ1...ρτ−2α. (30) The Formula (30) is derived from (1). Suppose that the space An 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 Rh ijkρ1...ρm−2ρm−1,ρm=Θh ijkρ1...ρm−1ρm, (31) where Θh ijkρ1...ρm−1ρm is some tensor depending on unknown tensors ϕi , Rh ijk , Rh ijkρ1 , . . . , Rh ijkρ1...ρm−1, as well as on some well-known tensors. Obviously, Equations (10) , (19) , (20) , (26) , (29) and (31) form a closed system of Cauchy type equations with respect to functions ϕi(x) , Rh ijk(x) , Rh ijkρ1(x) , . . . , Rh ijkρ1...ρm−1(x) ; moreover, the conditions of an algebraic nature (21) must be fulfilled and Rh i(jk)l1...lρ=0 and Rh (ijk)l1...lρ=0, ρ=2, 3, . . . , m−1. (32) Thus, we proved the following theorem. Theorem 3. In order that an affine connection space An admits a canonical F -planar mapping π(e) (e=± 1 ) onto an m -symmetric space An , it is necessary and sufficient that a solution of a closed mixed system of Cauchy-type equations in covariant derivatives (10) , (19) – (21) , (26) , (29) , (31),(32)exists with respect to unknown functions ϕi(x), Rh ijk(x), Rh ijkρ1(x),. . ., Rh ijkρ1...ρm−1(x). Obviously, the general solution of the closed mixed system of the above mentioned equations depends on no more than 1/3 n2(n2−1) (1+n+n2+· · · +nm−1) + n essential parameters. Mathematics 2023,11, 1246 8 of 9 6. Conclusions The paper deals with F -planar mappings of affine connection spaces with affinor e -structure onto m -symmetric spaces. We have found the fundamental equations of the considered mapping, which are in Cauchy form. Therefore, the general solution to this problem depends on the finite number of real parameters. The number of these parameters was also calculated. Author Contributions: Investigation, V.B., L.R. and Y.C. All of the authors contributed equally and significantly to the writing of this article. All authors have read and agreed to the published version of the manuscript Funding: The second author was supported by Palacky University in Olomouc the grant IGA PˇrF 2023010 and by the grant of Faculty of Civil Engineering, Brno University of Technology (research project No. FAST-S-22-7867). Data Availability Statement: No new data were created or analyzed in this study. Data sharing is not applicable to this article. 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