Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces
Abstract
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m-Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikes.
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mathematics Article Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces Volodymyr Berezovski 1, Yevhen Cherevko 2,3 , Josef Mikeš 4,* and Lenka Rýparová 5 Citation: Berezovski, V.; Cherevko, Y.; Mikeš, J.; Rýparová, L. Canonical Almost Geodesic Mappings of the First Type of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces. Mathematics 2021,9, 437. https://doi.org/10.3390/math9040437 Academic Editor: Marian Ioan Munteanu Received: 26 January 2021 Accepted: 18 February 2021 Published: 22 February 2021 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Copyright: © 2021 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/). 1Department of Mathematics and Physics, Uman National University of Horticulture, 20300 Uman, Ukraine; [email protected] 2Department of Physics and Mathematics Sciences, Odesa National Academy of Food Technologies, 65039 Odesa, Ukraine; [email protected] 3Department of Economic Cybernetics and Information Technologies, Odesa National Economics University, 65082 Odesa, Ukraine 4Department of Algebra and Geometry, Faculty of Science, Palacky University, 771 46 Olomouc, Czech Republic 5Institute of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 602 00 Brno, Czech Republic; [email protected].cz *Correspondence: [email protected] Abstract: In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces, generalized 3-Ricci-symmetric spaces, and generalized m -Ricci-symmetric spaces. In either case the main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained results extend an amount of research produced by N.S. Sinyukov, V.E. Berezovski, J. Mikeš. Keywords: canonical almost geodesic mappings; Cauchy-type PDEs; space with affine connection; Ricci symmetric space MSC: 53B05, 35R01 1. Introduction In the sixties of the preceding century, N.S. Sinyukov [ 1 ] considered almost geodesic mappings of Riemannian and affinely connected spaces, that generalize the geodesic mappings in a natural way. The main results were presented in a monograph [ 2 ] and an expository article [ 3 ]. Geodesic mappings and their generalizations are studied in detail in monographs [4–6] and researches [7–9] by J.Mikeš et al. The theory goes back to the paper [ 10 ] 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 a remarkable fact that this problem is related to the study of equations of dynamics of mechanical systems. That direction is developing according to Petrov’s plan to build models of physical processes using mappings and transformations [ 11 ]. For example, in papers [ 12 , 13 ] is shown the possible physical application of almost geodesic mappings. N.S. Sinyukov specified three types of almost geodesic mappings π1 , π2 , π3 . The problem of completeness of classification had long remained unresolved. Berezovski and Mikeš [ 14 , 15 ] proved that for n> 5 other types of almost geodesic mappings except for π1 , π2 , and π3 do not exist. The authors have found conditions for the almost geodesic mappings π1 , π2 , π3 intersections. It is proved that if an almost geodesic mapping f is simultaneously π1 and π2 , then f is a mapping of affine connection spaces with a preserved Mathematics 2021,9, 437. https://doi.org/10.3390/math9040437 https://www.mdpi.com/journal/mathematics
Mathematics 2021,9, 437 2 of 12 linear complex of geodesic lines. If the mapping f is simultaneously π1 and π3 , then f is a mapping of affine connection spaces with a preserved quadratic complex of geodesic lines. The mappings preserving above mentioned complexes studied by V.M.Chernyshenko [ 16 ] and V.A.Dobrovolski [17]. The theory of almost geodesic mappings was developed by V.A.Aminova, A.M. Mukhamedov [18] , V.S.Sobchuk [ 19 , 20 ], N.Y.Yablonskaya [ 21 , 22 ], V.E.Berezovski, J. Mikeš [14,15,23–35] , O.Belova, J.Mikeš, K.Strambach [ 36 , 37 ], M.S.Stankoviˇc, Lj.S.Velimirovi´c, N.Vesi´c, M.Lj. Zlatanovi´c [38–44] et al. N.S.Sinyukov [ 3 ] proved that the main equations for canonical almost geodesic mappings of spaces with affine connections onto Ricci-symmetric spaces can be written as a closed system of partial differential equations of Cauchy type in covariant derivatives. It follows that the general solution of the system depends on a finite number of essential parameters. The results were extended by Berezovski and Mikeš [ 28 , 34 ] to the cases of canonical almost geodesic mappings of the first type of spaces with affine connections onto Riemannian spaces and canonical almost geodesic mappings of the first type of spaces with affine connections onto generalized Ricci-symmetric spaces. The paper is devoted to the study of canonical almost geodesic mappings of type π1 of spaces with affine connections onto generalized 2-Ricci-symmetric, generalized 3-Riccisymmetric, and generalized m -Ricci-symmetric spaces. The fundamental equations for the mappings are derived in the form of the closed systems of Cauchy-type PDEs. The maximum number of essential parameters on which general solutions of these systems depend was estimated. V.R.Kaigorodov [ 45 ] comprehensively studied generalized symmetric and recurrent spaces from the point of view of the General Theory of Relativity. This review contains 97 citations and is a through analysis of this issue. Let us note that in works [ 7 , 8 , 24 , 46 – 50 ] were studied geodesic and holomorphically projective mappings of above mentioned spaces. For the other mappings the main equations are obtained as closed systems of PDEs of Cauchy type, see [5,6,51–54]. 2. 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 [2,5,6]. Consider an n -dimensional space An with affine connection Γh ij without torsion. The space is referred to coordinates x1 , x2 , . . . , xn . We assume that all functions under consideration are sufficiently differentiable, and we suppose that n>2. A curve defined in a space with affine connection is called almost geodesic if there exists a two-dimensional (differentiable) distribution D parallel along the curve (relative to the affine connection) such that for any tangent vector of the curve its parallel translation along the curve belongs to the distribution D[5]. A mapping f:An→An is called almost geodesic if any geodesic curve of An is mapped under fonto an almost geodesic curve in An. Suppose, that a space An with affine connection Γh ij(x) admits a mapping f onto a space An with affine connection Γh ij(x) , and the spaces are referred to a common coordinate system x1,x2, . . . , xnwith respect to the mapping. The tensor Ph ij(x) = Γh ij(x)−Γh ij(x)(1) is called a deformation tensor of the connections Γh ij(x) and Γh ij(x) with respect to the mapping f . The symbols Γh ij(x) and Γh ij(x) are components of affine connections of the spaces Anand Anrespectively.
Mathematics 2021,9, 437 3 of 12 It is known [ 2 ] that in order that a mapping of a space An onto a space An to be almost geodesic, it is necessary and sufficient that the deformation tensor Ph ij(x) of the mapping f in the common coordinate system x1,x2, . . . , xnhas to satisfy the condition Ah αβγλαλβλγ=a·Ph αβλαλβ+b·λh, where λh is an arbitrary vector, a and b are certain functions of variables x1 , x2 , . . . , xn and λ1,λ2, . . . , λn. The tensor Ah ijk is defined as Ah ijk def =Ph ij,k+Pα ij Ph αk, where the comma “,” denotes the covariant derivative with respect to the connection of the space An. N. S. Sinyukov [ 2 ] distinguished three kinds of almost geodesic mappings, namely π1 , π2, and π3characterized, by following conditions for the deformation tensor. A mapping f:An→Anis called an almost geodesic of type π1, if the conditions Ah (ijk)=δh (iajk)+b(iPh jk), (2) are satisfied, where aij is a symmetric tensor, bi is a covariant vector and δh i is the Kronecker delta. We denote by the round parentheses a symmetrization without division with respect to the mentioned indices. If in the Equation (2) the condition bi≡ 0 holds, then the almost geodesic mapping π1 is called canonical. It is known [ 3 ] that any almost geodesic mapping π1 may be written as the composition of a canonical almost geodesic mapping of type π1 and a geodesic mapping. A mapping f:An→Anis called an almost geodesic of type π2, if the conditions Ph ij =δh (iψj)+Fh (iϕj)and Fh (i,j)+Fh αFα (iϕj)=δh (iµj)+Fh (iρj) holds, where ψi,ϕi,µi,ρiare covectors, Fh iis a tensor of type (1, 1). A mapping f:An→Anis called an almost geodesic of type π3, if the conditions Ph ij =δh (iψj)+θhaij and θh ,i=ρ·δh i+θhai holds, where θh is a torse forming vector, ψi , ai are covectors, aij is a symmetric tensor and ρis a function. As we have noted above, the mentioned types of mappings π1 , π2 and π3 can intersect. The classification completeness for spaces with affine connections of dimension n> 5 has been proved in [14,15]. 3. Ricci-Symmetric, Generalized Ricci-Symmetric, 2-Ricci-Symmetric and Generalized 2-Ricci-Symmetric Spaces A space An with an affine connection is called Ricci-symmetric if its Ricci tensor Rij satisfies the condition Rij;k=0. The symbol “;” denotes a covariant derivative with respect to the connection of the space An. In [ 3 ] N.S. Sinyukov considered canonical almost geodesic mappings of the first type of spaces with affine connections onto Riemannian Ricci-symmetric spaces. Taking into account the relations between the Riemannian tensors Rh ijk and Rh ijk of the spaces An and An respectively as follows from (1) , the Equations (2) for canonical almost geodesic mappings of the first type of spaces with affine connections were written in the form
Mathematics 2021,9, 437 4 of 12 3Ph ij,k+Ph kαPα ij=Rh (ij)k−Rh (ij)k+δh (kaij). (3) We give the following example. From the Equation (3) with a condition of preserving Riemannian tensor and aij =0, we obtain Ph ij,k=−Ph kαPα ij. In a flat space, the above equation is completely integrable. Therefore it has a solution with respect to Ph ij(x) for any initial condition Ph ij(x0) . If Phij(x0)6=δh (iψj)(x0) then the solution Ph ij(x0)generates an almost geodesic mapping An→¯ Anwhich is not geodesic. Considering (3) as a system of Cauchy type in An with respect to the functions Ph ij , one has found the integrability conditions of the system in the form Rh (ij)[k,l]=Rh (ij)[k,l]+δh (iajk),l−δh (iajl),k+3Pα ij Rh αkl −Ph α(jRα i)kl −Ph αkRh (ij)l−Rh (ij)l+δα (iajl)+Ph αlRh (ij)k−Rh (ij)k+δα (iajk), where the square brackets [ ] denote an antisymmetrization (or, alternation) without division with respect to the mentioned indices. Let us express in the left-hand side the covariant derivatives with respect to the connection of An in terms of the covariant derivatives with respect to the connection of An . Taking account of (1), we obtain Rh (ij)[k;l]=δh (iajk),l−δh (iajl),k+θh ijkl, (4) where θh ijkl =Rh (ij)[k,l]+3Pα ij Rh αkl −Ph α(jRα i)kl−Ph αkRh (ij)l−Rh (ij)l+δα (iajl) +Ph αlRh (ij)k−Rh (ij)k+δα (iajk)−Pα l(iRh |α|j)k−Pα l(iRh j)αk+Pα k(iRh |α|j)l+Pα k(iRh j)αl. Here the symbol |α|means that in applying symmetrization the index αis omitted. Using the Ricci identity, let us write the conditions (4) in the form Rh ilk;j+Rh jlk;i=δh (iajk),l−δh (iajl),k+θh ijkl. (5) Contracting (5) for h and k , we get the relations for covariant derivatives of the Ricci tensor Rij for the space An Ril;j+Rjl;i= (n+1)aij,l−al(i,j)+θα ijαl. (6) A space An with an affine connection is called generalized Ricci-symmetric if the Ricci tensor Rij for the space satisfies the condition Rij;k+Rkj;i=0. The concept of generalized Ricci-symmetric spaces was first introduced in [ 28 , 34 ]. The papers are devoted to the study of canonical almost geodesic mappings of spaces with affine connections onto the above mentioned spaces. It is proved that the main equations for the mappings can be obtained as a closed system of Cauchy-type differential equations in covariant derivatives. A space An with an affine connection is called 2-Ricci-symmetric if the Ricci tensor Rij for the space satisfies the condition Rij;km =0.
Mathematics 2021,9, 437 5 of 12 A space An with an affine connection is called generalized 2-Ricci-symmetric if its Ricci tensor Rij satisfies the condition Rij;km +Rkj;im =0. (7) 4. Canonical Almost Geodesic Mappings of type π1of Spaces with Affine Connections onto Generalized 2-Ricci-Symmetric Spaces The Equation (6) was obtained for a canonical almost geodesic mapping of type π1 of a space with an affine connection Anonto another space with an affine connection An. First we differentiate covariantly the Equation (6) with respect to the connection of the space An. Then taking account of (1), let us applicate the formulas Ril;j,k=Ril;jk +Rαl;jPα ik +Riα;jPα lk +Ril;αPα jk, Rij;k=Rij,k−RiαPα jk −RαjPα ik.(8) Finally, we get Ril;jk +Rjl;ik = (n+1)aij,lk −ali,jk −alj,ik +Cijlk, (9) where Cijlk =−θα ijαl,k−Rαl,j+Rjl,α−2RβlPβ αj−RαβPβ lj −RjβPβ lαPα ik −Riα,j+Rjα,i−2RβαPβ ij −RiβPβ αj−RjβPβ αiPα lk −Ril,α+Rαl,i−2RβlPβ αi−RiβPβ lα−RαβPβ li Pα jk. (10) Moreover, let us consider canonical almost geodesic mappings of type π1 of spaces An with affine connections onto generalized 2-Ricci-symmetric spaces An . Hence the Ricci tensor Rij for the space An satisfies the conditions. Then the Equation (9) may be written in the form (n+1)aij,lk −ali,jk −alj,ik =−Cijlk, (11) when Cijlk is defined by the formulas (10). Let us interchange the indices l and j in (11) and symmetrize in the indices i and j . Then we have ali,jk +alj,ik =−1 nC(i|l|j)k+2 naij,lk. (12) The Equation (11) by means of (12) can be written in the form n2+n−2 naij,lk =−Cijlk −1 nC(i|l|j)k. (13) In the following we have assumed that a space An with an affine connection is given. Then taking account of the structure of the tensor Cijlk which was determined by formula (10) , we see that the left hand side of the Equation (13) depends on the unknown functions Ph ij(x),aij(x),aij,k(x),Rh ijk(x),Rh ijk,m(x). Differentiate covariantly the conditions of integrability (4) with respect to the connection of the space An . Then express in the right-hand side the covariant derivatives with respect to the connection of An in terms of the covariant derivatives with respect to the connection of the space An. When we make use of the Ricci identity, we obtain Rh (ij)k;lm −Rh (ij)l;mk =δh (iajk),lm −δh (iajl),km +Th ijklm, (14)
Mathematics 2021,9, 437 6 of 12 where Th ijklm =Rh αmkRα (ij)l−Rα lmkRh (ij)α−Rα jmkRh (iα)l−Rα imkRh (jα)l−Ph mαδα (iajk),l −Pα mjδh (iaαk),l−Pα miδh (αajk),l−Pα mkδh (αaij),l−Pα mlδh (iajk),α−Ph mαδα (iajl),k +Pα miδh (αajl),k+Pα mjδh (iaαl),k+Pα mkδh (iajl),α−Pα mlδh (iajα),k −θh ijkl,m+Ph αmθα ijkl −Pα miθh αjkl −Pα mjθh iαkl −Pα mkθh ijαl−Pα mlθh ijkα. Alternating the equations (14) with respect to the indicies land m, we obtain Rh (ij)m;lk −Rh (ij)l;mk =δh (iajm),kl −δh (iajl),km +Th ijk[lm]+Rh (i|αk|Rα j)ml +Rh (ij)αRα kml −Rα (ij)kRh αml +Rh α(i|k|Rα j)ml +δh (αajk)Rα ilm +δh (αaik)Rα jlm +δh (iajα)Rα klm −δα (iajk)Rh αlm.(15) Taking account of the properties of a curvature tensor Rh ijk , we may write the conditions (15) in the form Rh iml;jk +Rh jml;ik =δh (iajl),km −δh (iajm),kl −Nh ijklm,(16) where Nh ijklm =Th ijk[lm]+Rα iml Rh (αj)k+Rα jmlRh (αi)k+Rα kml Rh (ij)α−Rh αml Rα (ij)k +δh (αajk)Rα ilm +δh (αaik)Rα jlm +δh (αaji)Rα klm −a(ijRh k)lm. Alternating the equations (16) with respect to the indicies jand k, we get Rh jml;ik −Rh kml;ij =δh (iajl),km −δh (iajm),kl −δh (iakl),jm +δh (iakm),jl −Nh i[jk]lm +Rh αml Rα ikj +Rh iαlRα mkj +Rh imαRα lkj −Rα iml Rh αkj.(17) Let us interchange iand kin (16) and subtract it from (17). Then we have 2Rh jml;ik =δh (iajl),km −δh (iajm),kl +δh (kajm),il +δh (iakm),jl −δh (iakl),jm −δh (kajl),im +Ωh ijklm,(18) where Ωh ijklm =Nh kjilm −Nh i[jk]lm +Rh αml Rα ikj +Rh iαlRα mkj +Rh imαRα lkj −Rα iml Rh αkj −Rα kml Rh αji +Rh αml Rα kji +Rh kαlRα mji +Rh kmαRα lji −Rα jmlRh αik +Rh αml Rα jik +Rh jαlRα mik +Rh jmαRα lik. Let us express in the left-hand side of the Equation (18) the covariant derivatives with respect to the connection of An in terms of the covariant derivatives with respect to the connection of An. We have 2Rh jml,ik =δh (iajl),km −δh (iajm),kl +δh (kajm),il +δh (iakm),jl −δh (iakl),jm −δh (kajl),im +Sh ijklm,(19)
Mathematics 2021,9, 437 7 of 12 where Sh ijklm =Ωh ijklm −2Rα jml,iPh αk−Rh αml,iPα jk −Rh jαl,iPα mk −Rh jmα,iPα lk −Rh jml,αPα ik +Rα jmlPβ αi−Rβ αml Pα ij −Rβ jαlPα im −Rβ jmαPα il Ph βk −Rα jmlPh αβ −Rh αml Pα βj−Rh jαlPα βm−Rh jmαPα βlPβ ik −Rα βml Ph αi−Rh αml Pα βi−Rh βαlPα im −Rh βmαPα il Pβ jk −Rα jβlPh αi−Rh αβlPα ji −Rh jαlPα βi−Rh jβαPα il Pβ km −Rα jmβPh αi−Rh αmβPα ji −Rh jαβPα mi −Rh jmαPα βiPβ kl. We introduce the tensors aijk and Rh ijkm defined by aij,k=aijk, (20) Rh ijk,m=Rh ijkm. (21) Taking account of (21), we may write the Equation (19) in the form 2Rh jmli,k=δh (iajl),km −δh (iajm),kl +δh (kajm),il +δh (iakm),jl −δh (iakl),jm −δh (kajl),im +Sh ijklm.(22) In the following we have assumed that in the left-hand side of the Equation (22) the second order covariant derivatives of the tensor aij are expressed according to (13) . Taking account of (20), the Equation (13) may be put in the form n2+n−2 naijl,k=−Cijlk −1 nC(i|l|j)k. (23) We have assumed that in the right-hand sides of the Equations (22) and (23) the covariant derivatives of tensors aij and Rh ijk with respect to the connection of the space An are expressed according to (20) and (21). Obviously, in the space An the Equations (3) and (20) – (23) form a closed system of PDEs of Cauchy type with respect to the functions Ph ij(x) , aij(x) , Rh ijk(x) , aijk(x) , Rh ijkm(x) . The functions must satisfy the algebraic conditions Ph ij(x) = Ph ji(x),aij(x) = aji(x), Rh i(jk)(x) = Rh (ijk)(x) = 0, Rh i(jk)l(x) = Rh (ijk)l(x) = 0. (24) Hence we have proved the following theorem. Theorem 1. In order that a space An with an affine connection admits a canonical almost geodesic mapping of type π1 onto a generalized 2-Ricci-symmetric space An , it is necessary and sufficient that the mixed system of differential equations of Cauchy type in covariant derivatives (3) and (20) – (24) have a solution with respect to the unknown functions Ph ij(x) , aij(x) , aijk(x) , Rh ijk(x) and Rh ijkm(x). Also we have obtained the corollary. Corollary 1. The family of all generalized 2-Ricci-symmetric spaces, which are images of a given space An with an affine connection with respect to canonical almost geodesic mappings of type π1 , depends on no more then 1 6n(2n−1)(n2−1) + 1 2n(1+n)2
Mathematics 2021,9, 437 8 of 12 essential parameters. 5. Canonical Almost Geodesic Mappings of Type π1of Spaces with Affine Connections onto Generalized 3-Ricci-Symmetric Spaces A space An with an affine connection is called generalized 3-Ricci-symmetric if the Ricci tensor Rij for the space satisfies the condition Rij;kml +Rkj;iml =0. (25) Taking account of (1) and (8), we obtain Rij;kl =Gijkl and Ril;jk,m=Ril;jkm +GαljkPα im +GiαjkPα lm +GilαkPα jm +GiljαPα km, (26) where Gijkl =Rij,kl −Riα,lPα jk −RiαPα jk,i−Rαj,lPα ik −RαjPα ik,l−Rαj,k−RβjPβ αk−RαβPβ jkPα il −Riα,k−RβαPβ ik −RiβPβ αkPα jl −Rij,α−RβjPβ iα−RiβPβ jαPα kl. Let us differentiate covariantly the Equation (9) with respect to the connection of the space An. Then taking account of (8) and (26), we get Ril;jkm +Rjl;ikm = (n+1)aij,lkm −ali,jkm −alj,ikm +Cijlk,m−Gαl(j|k|Pα i)m−G(i|α|j)kPα lm −G(i|lαk|Pα j)m−G(i|l|j)αPα km. (27) Let us consider canonical almost geodesic mappings of type π1 of spaces with affine connections An onto generalized 3-Ricci-symmetric spaces An . Hence the Ricci tensor Rij for the space An satisfies the conditions (25) . Then the Equation (27) could be written in such form as (n+1)aij,lkm −alj,ikm −ali,jkm =−Cijlkm, (28) where Cijlkm =Cijlk,m−Gαl(j|k|Pα i)m−G(i|α|j)kPα lm −G(i|lαk|Pα j)m−G(i|l|j)αPα km. (29) Let us interchange the indices l and j in (28) and symmetrize in the indices i and j . Then we have ali,jkm +alj,ikm =−1 nC(i|l|j)km +2 naij,lkm. (30) Because of (30) the Equation (28) may be written in the form n2+n−2 naij,lkm =−Cijlkm −1 nC(i|l|j)km. (31) The Equation (19) we obtain from the integrability conditions of (4) , is to hold for canonical almost geodesic mappings of type π1 of spaces with affine connections onto generalized 3-Ricci-symmetric spaces. In the following we have assumed that a space An with an affine connection is given. Then taking account of the structure of the tensor Cijlkm which was determined by formulas (4) , (18) and (29) , we see that the left hand side of Equation (31) depends on the unknown functions Ph ij(x),aij(x),aij,k(x),aij,km(x),Rh ijk(x),Rh ijk,m(x). Let us introduce the tensor aijkm defined by aijk,m=aijkm. (32)
Mathematics 2021,9, 437 9 of 12 Because of (32) we can write the Equation (31) in the form n2+n−2 naijlk,m=−Cijlkm −1 nC(i|l|j)km. (33) Obviously, in the space An the Equations (3) , (20) – (22) , (32) and (33) form a closed system of PDEs of Cauchy type with respect to the functions Ph ij(x) , aij(x) , Rh ijk(x) , Rh ijkm(x) , aijk(x),aijkm(x). Also the functions must satisfy the algebraic conditions (24). Hence we have proved the following theorem. Theorem 2. In order that a space An with an affine connection admits a canonical almost geodesic mapping of type π1 onto a generalized 3-Ricci-symmetric space An , it is necessary and sufficient that the mixed system of differential equations of Cauchy type in covariant derivatives (3) , (20) – (22) , (24) , (32) and (33) have a solution with respect to the unknown functions Ph ij(x) , aij(x) , Rh ijk(x) , Rh ijkm(x), aijk(x)and aijkm(x). Corollary 2. The family of all generalized 3-Ricci-symmetric spaces, which are images of a given space An with an affine connection with respect to canonical almost geodesic mappings of type π1 , depends on no more then 1 6n(2n−1)(n2−1) + 1 2n(1+n)(n2+n+1) essential parameters. 6. Canonical Almost Geodesic Mappings of Type π1of Spaces with Affine Connections onto Generalized m-Ricci-Symmetric Spaces A space An with an affine connection is called generalized m -Ricci-symmetric if the Ricci tensor Rij for the space satisfies the condition Rij;ρ1ρ2...ρm+Rρ1j;iρ2...ρm=0. (34) It is obvious that generalized 2-Ricci-symmetric spaces and generalized 3-Riccisymmetric spaces are special cases of generalized m -Ricci-symmetric spaces with m= 2 and m=3 respectively. If we put m= 1 in (34) we have a generalized Ricci-symmetric space. Let us differentiate covariantly (m− 3 ) times the Equation (27) with respect to the connection of the space An . Then express in the left-hand side the covariant derivatives with respect to the connection of the space An in terms of the covariant derivatives with respect to the connection of the space An, using the formula [24] (Rij;ρ1...ρτ−2ρτ−1),ρτ=Rij;ρ1...ρτ−2ρτ−1ρτ+Pα iρτRαj;ρ1...ρτ−2ρτ−1+ Pα jρτRiα;ρ1...ρτ−2ρτ−1+Pα ρ1ρτRij;α...ρτ−2ρτ−1+· · · +Pα ρτ−1ρτRij;ρ1...ρτ−2α. Transforming the left-hand side of (34), we obtain Rij;ρ1ρ2...ρm+Rρ1j;iρ2...ρm= (n+1)aiρ1,jρ2...ρm−ajρ1,iρ2...ρm−aij,ρ1ρ2...ρm+Ωijρ1ρ2...ρm, (35) where the tensor Ωijρ1ρ2...ρm depends on the unknown tensors Ph ij , Rh ijk , Rh ijk,m , aij , aij,ρ1 , . . . , aij,ρ1...ρm−1. Let us consider canonical almost geodesic mappings of type π1 of spaces with affine connections An onto generalized m -Ricci-symmetric spaces An . Hence the Ricci tensor Rij for the space An satisfies conditions (34) . Then Equation (35) could be written in such form as (n+1)aiρ1,jρ2...ρm−ajρ1,iρ2...ρm−aij,ρ1ρ2...ρm=−Ωijρ1ρ2...ρm. (36)