Novel Invariants for Almost Geodesic Mappings of the Third Type
aa r X i v : . [ m a t h . G M ] O c t NOVEL INVARIANTS FOR ALMOST GEODESICMAPPINGS OF THE THIRD TYPE
Dušan J. Simjanović ( a ) , Nenad O. Vesić ( b ) Abstract
Two kinds of invariance for geometrical objects under transformations are involved inthis paper. With respect to these kinds, we obtained novel invariants for almost geodesicmappings of the third type of a non-symmetric affine connection space in this paper. Ourresults are presented in two sections. In the Section 3, we obtained the invariants for theequitorsion almost geodesic mappings which do not have the property of reciprocity.
Key Words: mapping, invariance, almost geodesics,
Math. Subj. Classification:
Manu authors have obtained invariants for different mappings between symmetric and non-symmetric affine connection spaces. Some of them are J. Mikeš and his research group [1–3,5,13],N. S. Sinyukov [12], M. S. Stanković [11, 16, 17, 19], M. Z. Petrović [10, 11] and many others. An N -dimensional manifold M N equipped with a non-symmetric affine connection ∇ (cid:0) seeEisenhart [4] (cid:1) is the non-symmetric affine connection space GA N (cid:0) in Eisenhart’s sense (cid:1) .The affine connection coefficients of the space GA N are L ijk , L ijk L ikj for at least one pair ( j, k ) ∈ { , . . . , N } × { , . . . , N } . For this reason, the symmetric and anti-symmetric part of theaffine connection coefficients L ijk are defined as L ijk = 12 (cid:0) L ijk + L ikj (cid:1) and L ijk ∨ = 12 (cid:0) L ijk − L ikj (cid:1) , (1.1)respectively. ( a ) Metropolitan University, Tadeuša Košćuška 63, 11158 Belgrade, Faculty of Information Technology, BulevarSv. Cara Konstantina 80A, 18116 Niš, Serbia ( b ) Mathematical Institute of Serbian Academy of Sciences and Arts. This paper is financially supported bySerbian Minsitry of Education, Science and Technological Development through the Mathematical Institute ofSerbian Academy of Sciences and Arts M N equipped with the affine connection ∇ whose coefficients are L ijk is theassociated space A N (cid:0) of the space GA N (cid:1) .With respect to the affine connection ∇ , one kind of covariant derivative exists [1–3, 5, 12, 13] a ij | k = a ij,k + L iαk a αj − L αjk a iα , (1.2)for the partial derivative ∂/∂x k denoted by comma.In this case, it exists one Ricci-Type identity a ij | mn − a ij | nm = a αj R iαmn − a iα R αjmn , for thecurvature tensor R ijmn = L ijm,n − L ijn,m + L αjm L iαn − L αjn L iαm . (1.3)of the space A N .The geometrical object R ij = R αijα , (1.4)is the tensor of the Ricci-curvature.Based on the non-symmetric affine connection ∇ , S. M. Minčić defined four kinds of covariantderivatives [6–8] a ij | k = a ij,k + L iαk a αj − L αjk a iα , a ij | k = a ij,k + L ikα a αj − L αkj a iα ,a ij | k = a ij,k + L iαk a αj − L αkj a iα , a ij | k = a ij,k + L ikα a αj − L αjk a iα . (1.5)With respect to the identities of Ricci-type a ij | p m | q n − a ij | r n | s m , p, q, r, s ∈ { , . . . , } , a ij | k = a ij | k ,it is obtained the family of curvature tensors for the space GA N K ijmn = R ijmn + uL ijm ∨ | n + u ′ L ijn ∨ | m + vL αjm ∨ L iαn ∨ + v ′ L αjn ∨ L iαm ∨ + wL αmn ∨ L iαj ∨ , (1.6)for the curvature tensor R ijmn of the associated space A N and the coefficients [6–8] u , u ′ , v , v ′ , w . The corresponding family of the Ricci-curvatures K ij = K αijα is K ij = R ij + uL αij ∨ | α + u ′ L αiα ∨ | j + vL αij ∨ L βαβ ∨ − ( v ′ + w ) L αiβ ∨ L βjα ∨ . (1.7) A curve ℓ = ℓ ( t ) in the associated space A N whose tangential vector λ = dℓdt satisfies thesystem of differential equations [5, 12] dλ i dt + L iαβ λ α λ β = ζλ i , (1.8)for a scalar ζ , is the geodesic line of the space A N .2 curve ˜ ℓ = ˜ ℓ ( t ) in the associated space A N whose tangential vector ˜ λ = d ˜ λdt satisfies thesystem of equations [1–3, 5, 12, 13] ˜ λ i = a ( t )˜ λ i + b ( t )˜ λ i , ˜ λ i = ˜ λ i k α ˜ λ α , ˜ λ i = ˜ λ i (1) k α ˜ λ α , (1.9)for the functions a ( t ) , b ( t ) and the covariant derivative with respect to the affine connection ofthe space A N denoted by k , is the almost geodesic line of the space A N .A mapping f : A N → A N that any geodesic line of the space A N transforms to an almostgeodesic line of the space A N is the almost geodesic mapping of the space A N (cid:0) see [1–3,5,12,13] (cid:1) .Three types π , π , π of almost geodesic mappings are detected.Sinyukov proved that the inverse mapping of the almost geodesic one f : A N → A N of the type π is the almost geodesic mapping of the type π . Almost geodesicmappings of the third type whose inverse transformations are the almost geodesic mappings ofthe third type have the property of reciprocity (cid:0) see [12], page 191 (cid:1) .M. S. Stanković [14–16] generalized the Sinyukov’s work about almost geodesic mappings.The curve ˜ ℓ = ˜ ℓ ( t ) whose tangential vector ˜ λ = d ˜ ℓdt is the solution of the system [10, 11, 14–17, 19] ˜ λ k i (2) = a k ( t )˜ λ i + b k ( t )˜ λ k i (1) , ˜ λ k i (1) = ˜ λ i k k α ˜ λ α , ˜ λ ki (2) = ˜ λ k i (1) k k α ˜ λ α , (1.10) k ∈ { , } is the almost geodesic line of the k -th kind of the space GA N .The mapping f : GA N → GA N that any geodesic line of the space GA N transforms to thealmost geodesic line of a k -th type, k = 1 , , of the space GA N is the almost geodesic mappingof the space GA N .With respect to the Sinyukov’s work, M. S. Stanković [14–16] determined three types ofalmost geodesic lines and any of these types is divided into two subtypes. These subtypes ofalmost geodesic mappings are π r , π r , r = 1 , , .The almost geodesic mapping of a subtype π k , k = 1 , , has the property of reciprocity if itsinverse mapping is the almost geodesic mapping of the same subtype.We will obtain invariants for almost geodesic mappings of the third type of the space GA N which have or does not have the property of reciprocity below. Invariants for mappings between symmetric and non-symmetric affine connection spaces aresuch geometric objects whose values and forms do not change under the acting of the correspond-ing mapping. If an almost geodesic mapping f : GA N → GA N (cid:0) or f : A N → A N (cid:1) does not havethe property of reciprocity, the invariants of these mappings of the common values and formscan not be obtained. For this reason, to obtain the invariants for almost geodesic mappings withrespect to changes of the curvature tensors under almost geodesic mappings, authors assumethat these mappings satisfy the property of reciprocity.Two kinds of invariants are important in physics (cid:0) taken from the textbook Ð. Mušicki, B.Milić, Mathematical Foundations of Theoretical Physics With a Collection of Solved Problems [9],page 103 (cid:1) : 3
The covariant is the object whose form stays saved but whose value changes under the trans-formation of coordinates.-
The invariant is the object whose form changes but whose form stays saved under the trans-formation of coordinates.-
The total invariant is the object whose value and form stay saved under the transformation.Invariants for mappings between affine connection spaces which have been studied in differ-ential geometry are analogies to the total invariants from physics. Because all almost geodesicmappings do not have the property of reciprocity, we may obtain the geometrical objects whosevalues are preserved unlike their forms under these mappings. For this reason, we define twokinds of invariants for mappings between the affine connection spaces.
Definition 1.1.
Let f : GA N → GA N be a transformation and let U i ...i p j ...j q be a geometrical objectof the type ( p, q ) .- If the transformation f preserves the value of the object U i ...i p j ...j q but changes its form to V i ...i p j ...j q ,then the invariance for geometrical object U i ...i p j ...j q under the transformation f is valued .- If the transformation f preserves both the value and the form of the geometrical object U i ...i p j ...j q ,then the invariance for the geometrical object U i ...i p j ...j q under the transformation f is total . The transformations which are the main subject of the research in this paper are almostgeodesic mappings of the third type.
The transformation rules for affine connection coefficients with respect to the third typealmost geodesic mappings of symmetric and non-symmetric affine connection spaces are L ijk = L ijk + ψ j δ ik + ψ k δ ij and L ijk = L ijk + ψ j δ ik + ψ k δ ij + ξ ijk , (1.11)for the -form ψ i , the contravariant vector ϕ i and the tensors σ jk and ξ ijk symmetric and anti-symmetric in the covariant indices j and k , respectively.To generalize the Weyl projective tensor as an invariant for the third type almost geodesicmapping of the symmetric affine connection space A N , N. S. Sinyukov involved (cid:0) see [12], page193 (cid:1) the geometrical object q i such that q α ϕ α = e , e = ± . After some computations, Sinyukovgeneralized the Thomas projective parameter and the Weyl projective tensor as the invariantsfor the almost geodesic mapping f .M. S. Stanković (cid:0) see [16] (cid:1) continued Sinyukov’s research about invariants for almost geodesicmappings of the third type. With respect to the results presented in [16], N. O. Vesić, Lj. S.Velimirović and M. S. Stanković [19] obtained the family of invariants for equitorsion third typealmost geodesic mappings of a non-symmetric affine connection space.N. S. Sinyukov [12] obtained one generalization of the Weyl projective tensor as invariant forthe third type almost geodesic mapping of a symmetric affine connection space. M. S. Stanković[16] obtained one generalization of the Weyl projective tensor as invariant for the third type4lmost geodesic mapping of a non-symmetric affine connection space with respect to the changeof the curvature tensor of the corresponding associated space. N. O. Vesić, Lj. S. Velimirović, M.S. Stanković [19] obtained one family of invariants for the third type almost geodesic mappingsof a non-symmetric affine connection space which generalizes the Weyl projective tensor.In [18], it is obtained two invariants for mappings of an associated space analogue to theWeyl projective tensor (cid:0) called the invariants of the Weyl type (cid:1) . That motivated us to obtain theinvariants for almost geodesic mappings of the third type of a non-symmetric affine connectionspace with respect to the results obtained in [18].The formulae of invariants for mappings between symmetric and non-symmetric affine con-nection spaces are obtained in [18]. We will use these formulae to meet the main goals of thispaper. These goals are:1. To obtain the invariants for equitorsion almost geodesic mappings of a non-symmetric affineconnection space.2. To obtain the necessary and sufficient conditions for these invariants to be total. In [18], the invariants of mappings f : GA N → GA N are obtained. If the deformation tensor P ijk = L ijk − L ijk of the mapping f is P ijk = L ijk − L ijk = ω ijk − ω ijk + τ ijk − τ ijk , (2.1)for ω ijk = ω ikj , ω ijk = ω ikj , τ ijk = − τ ikj , τ ijk = − τ ikj , the basic associated invariants of Thomasand the Weyl type for the mapping f are e T ijk = L ijk − ω ijk , (2.2) f W ijmn = R ijmn − ω ijm | n + ω ijn | m + ω αjm ω iαn − ω αjn ω iαm . (2.3)To simplify the last formulae, the next geometrical object is used [18] L ijm | n = L ijm,n + L iαn L αjm − L αjn L iαm − L αmn L ijα . (2.4)In the case of ω ijk = δ ik ρ j + δ ij ρ k + σ ijk , (2.5)for σ ijk = σ ikj , the invariants for the mapping f given by the equations (2.2, 2.3) transform to e T ijk = L ijk − σ ijk − N + 1 (cid:16)(cid:0) L αjα − σ αjα (cid:1) δ ik + (cid:0) L αkα − σ αkα (cid:1) δ ij (cid:17) , (2.2’) f W ijmn = R ijmn − δ i [ m ρ j | n ] − δ ij ρ [ m | n ] − σ ij [ m | n ] − δ i [ m ρ j ρ n ] + δ i [ m ρ α σ αjn ] + σ αj [ m σ iαn ] , (2.6)The derived invariant of the Weyl type for the mapping f is [18]5 W ijmn = R ijmn + 1 N + 1 δ ij (cid:0) R [ mn ] + σ αα [ m | n ] (cid:1) + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j − σ ijm | n + σ ijn | m + σ αjm σ iαn − σ αjn σ iαm − N − δ im (cid:16) σ αα [ j | n ] + ( N + 1) (cid:0) σ αjn | α − σ αjα | n − σ αjn σ βαβ + σ αjβ σ βnα (cid:1)(cid:17) + 1 N − δ in (cid:16) σ αα [ j | m ] + ( N + 1) (cid:0) σ αjm | α − σ αjα | m − σ αjm σ βαβ + σ αjβ σ βmα (cid:1)(cid:17) . (2.3’)The basic invariant for the mapping f : GA N → GA N obtained with respect to the transfor-mation rule of the anti-symmetric part L ijk ∨ of the affine connection coefficient L ijk is ˆ T ijk = L ijk ∨ − τ ijk . (2.7)Let be ω i (1) .jk = L ijk , ω i (1) .jk = L ijk , ω i (2) .jk = ω ijk , ω i (2) .jk = ω ijk . The next equalities hold [18] ˆ T ijm k n − ˆ T ijm | n = P iαn ˆ T αjm − P αjn ˆ T iαm − P αmn ˆ T ijα , (2.8) L αjm ∨ L iαn ∨ − L αjm ∨ L iαn ∨ − L αjm ∨ τ iαn − L iαn ∨ τ αjm + τ αjm τ iαn + L αjm ∨ τ iαn + L iαn ∨ τ αjm − τ αjm τ iαn . (2.9)Because P ijk = ω i (1) .jk − ω ijk = ω i (2) .jk − ω i (2) .jk , and with respect to the equations (2.8, 2.9),it is obtained [18] θ i ( p ) .jmn = θ i ( p ) .jmn and Θ ijmn = Θ ijmn , where p = ( p , p , p ) , p , p , p ∈ { , } , where θ i ( p ) .jmn = L ijm ∨ | n − τ ijm | n − ω i ( p ) .αn ˆ T αjm + ω α ( p ) .jn ˆ T iαm + ω α ( p ) .mn ˆ T ijα , (2.10) Θ ijmn = L αjm ∨ L iαn ∨ − L αjm ∨ τ iαn − L iαn ∨ τ αjm + τ αjm τ iαn , (2.11)for the corresponding θ i ( p ) .jmn , Θ ijmn .The family of invariants for the mapping f with respect to the transformation of the family K ijmn of the curvature tensors for the space GA N is [18] W i ( p ) . ( p ) .jmn = f W ijmn + uθ i ( p ) .jmn + u ′ θ i ( p ) .jmn + v Θ ijmn + v ′ Θ ijnm + w Θ imnj , (2.12)for p = ( p , p , p ) , p = ( p , p , p ) , p ij ∈ { , } and the corresponding invariants θ i ( p ) .jmn , θ i ( p ) .jmn , Θ ijmn given by (2.10, 2.11).If the mapping f : GA N → GA N is equitorsion, the invariant ˆ T ijk given by (2.7) reduces to ˆ T i .jk = L ijk ∨ . (2.7’)6he family of invariants of the Weyl type for the equitorsion mapping f : GA N → GA N is W i . ( p ) . ( p ) .jmn = f W ijmn + uθ i . ( p ) .jmn + u ′ θ i . ( p ) .jmn , (2.13)where θ i . ( p ) .jmn = L ijm ∨ | n − ω i ( p ) .αn L αjm ∨ + ω α ( p ) .jn L iαm ∨ + ω α ( p ) .mn L ijα ∨ . (2.10’) Let f : GA N → GA N be an equitorsion almost geodesic mapping of the third type. Its basicequations are [16] L ijk = L ijk + ψ j δ ik + ψ k δ ij + 2 σ jk ϕ i ,ϕ i | j = ν j ϕ i + µ δ ij . (3.1)Let us rewrite the first of the last basic equations as L ijk = L ijk + ψ j δ ik + ψ k δ ij + D ijk , (3.1’)for the tensor D ijk , D ijk = D ikj = 2 σ jk ϕ i .In the case of the inverse mapping f − : GA N → GA N , it exists the tensor D ijk , D ijk = D ikj , D ijk = − D ijk , such that L ijk = L ijk − ψ j δ ik − ψ k δ ij − D ijk = L ijk − ψ j δ ik − ψ k δ ij + D ijk . (3.2)Hence, the equation (3.1’) transforms to L ijk = L ijk + ψ j δ ik + ψ k δ ij − (cid:0) D ijk − D ijk (cid:1) . (3.1”)After contracting the last equation by i and k , one gets ψ j = 1 N + 1 (cid:0) L αjα + 12 D αjα (cid:1) − N + 1 (cid:0) L αjα + 12 D αjα (cid:1) . (3.3)If substitute the expression (3.3) in the equation (3.1”) and use the expression D ijk = 2 σ jk ϕ i ,we will obtain ω ijk = 1 N + 1 δ ik (cid:0) L αjα + σ jα ϕ α (cid:1) + 1 N + 1 δ ij (cid:0) L αkα + σ kα ϕ α (cid:1) − σ jk ϕ i . (3.4)The second of the basic equations (3.1) is equivalent to ϕ i | j = − L iαj ∨ ϕ α + ν j ϕ i + µ δ ij . (3.5)After comparing the equations (2.5) and (3.5), one reads ρ j = 1 N + 1 (cid:0) L αjα + σ jα ϕ α (cid:1) and σ ijk = − σ jk ϕ i . (3.6)7ence, we get − σ ijm | n = (cid:0) σ jm ϕ i (cid:1) | n = σ jm | n ϕ i − σ jm L iαn ∨ ϕ α + σ jm ν n ϕ i + σ jm µ δ in , − σ αij | α = (cid:0) σ ij ϕ α ) | α = σ ij | α ϕ α − σ ij L βαβ ∨ ϕ α + σ ij ν α ϕ α + N µ σ ij , − σ ααi | j = (cid:0) σ αi ϕ α (cid:1) | j = (cid:0) σ iα ϕ α (cid:1) | j = σ αi | j ϕ α − σ βi L βαj ∨ ϕ α + σ αi ν j ϕ α + µ σ ij . (3.7)After substituting the expressions (3.4, 3.6, 3.7) in the equations (2.2’, 2.3’, 2.10’), we obtainthe next geometrical objects e T ijk = L ijk − N + 1 δ ik (cid:0) L αjα + σ jα ϕ α (cid:1) − N + 1 δ ij (cid:0) L αkα + σ kα ϕ α (cid:1) + σ jk ϕ i , (3.8) f W ijmn = R ijmn + 1 N + 1 δ ij (cid:16) R [ mn ] − (cid:0) σ [ mα | n ] − σ [ mβ L βαn ∨ ] + σ [ mα ν n ] (cid:1) ϕ α (cid:17) + σ j [ m | n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i − δ i [ m µ σ jn ] − N + 1 (cid:16) δ i [ m L αjα | n ] + (cid:0) δ i [ m σ jα | n ] − δ i [ m σ jβ L βαn ∨ ] + δ i [ m σ jα ν n ] (cid:1) ϕ α + δ i [ m µ σ jn ] (cid:17) + 1 N + 1 δ i [ m σ jn ] (cid:0) L βαβ + σ αβ ϕ β (cid:1) ϕ α − N + 1) (cid:0) L αjα + σ jα ϕ α (cid:1)(cid:0) δ i [ m L βn ] β + δ i [ m σ n ] β ϕ β (cid:1) , (3.9) f W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + σ j [ m | n ] ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i − N + 1 δ ij (cid:0) σ α [ m | n ] − σ β [ m L βαn ∨ ] + σ α [ m ν n ] (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] | α − δ i [ m σ jn ] L βαβ ∨ + δ i [ m σ jn ] ν α (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] σ αβ − δ i [ m σ jα σ n ] β (cid:1) ϕ α ϕ β − NN − (cid:0) δ i [ m σ αj | n ] − δ i [ m σ βj L βαn ∨ ] + δ i [ m σ αj ν n ] (cid:1) ϕ α − N − (cid:0) δ i [ m σ αn ] | j − δ i [ m σ βn ] L βαj ∨ + δ i [ m σ αn ] ν j (cid:1) ϕ α . (3.10)Let us express the invariant f W ijmn in the form f W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + δ ij X mn ] + δ i [ m Y jn ] + Z ijmn , (3.10’)for the corresponding tensors 8 ij = − N + 1 (cid:0) σ αi | j − σ βi L βαj ∨ + σ αi ν j (cid:1) ϕ α , (3.11) Y ij = 1 N − (cid:0) σ ij | α − σ ij L βαβ ∨ + σ ij ν α (cid:1) ϕ α + 1 N − (cid:0) σ ij σ αβ − σ iα σ jβ (cid:1) ϕ α ϕ β − NN − (cid:0) σ αi | j − σ βi L βαj ∨ + σ αi ν j (cid:1) ϕ α − N − (cid:0) σ αj | i − σ βj L βαi ∨ + σ αj ν i (cid:1) ϕ α , (3.12) Z ijmn = σ j [ m | n ] ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i . (3.13)After contracting the equality f W ijmn − f W ijmn by the indices i and j , one gets X mn ] − X mn ] = − N (cid:0) Y mn ] + Z ααmn (cid:1) + 1 N (cid:0) Y mn ] + Z ααmn (cid:1) , (3.14)where Y ij ] = − N + 1 (cid:0) σ α [ i | j ] − σ β [ i L βαj ∨ ] + σ α [ i ν j ] (cid:1) ϕ α . (3.15)With respect to the equations (3.12, 3.13, 3.14, 3.15) substituted into the equality f W ijmn − f W ijmn = R ijmn − R ijmn + 1 N + 1 δ ij (cid:0) R [ mn ] − R [ mn ] (cid:1) + NN − (cid:0) δ i [ m R jn ] − δ i [ m R jn ] (cid:1) + 1 N − (cid:0) δ i [ m R n ] j − δ i [ m R n ] j (cid:1) + δ ij (cid:0) X mn ] − X mn ] (cid:1) + (cid:0) δ i [ m Y jn ] − δ i [ m Y jn ] (cid:1) + Z ijmn − Z ijmn , (3.16)one gets ff W ijmn = ff W ijmn , where ff W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + σ j [ m | n ] ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i − NN + 1 δ ij (cid:0) σ α [ m | n ] − σ β [ m L βαn ∨ ] + σ α [ m ν n ] (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] | α − δ i [ m σ jn ] L βαβ ∨ + δ i [ m σ jn ] ν α (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] σ αβ − δ i [ m σ jα σ n ] β (cid:1) ϕ α ϕ β − NN − (cid:0) δ i [ m σ αj | n ] − δ i [ m σ βj L βαn ∨ ] + δ i [ m ν n ] (cid:1) ϕ α − N − (cid:0) δ i [ m σ αn ] | j − δ i [ m σ βn ] L βαj ∨ + δ i [ m σ αn ] ν j (cid:1) ϕ α . (3.17)9ased on the invariants f W ijmn and ff W ijmn , one concludes that the geometrical object (cid:0) σ α [ i | j ] − σ β [ i L βαj ∨ ] + σ α [ i ν j ] (cid:1) ϕ α is an invariant for the mapping f . Hence, the invariant f W ijmn given by (3.10) reduces to f W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + σ j [ m | n ] ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i + 1 N − (cid:0) δ i [ m σ jn ] | α − δ i [ m σ jn ] L βαβ ∨ + δ i [ m σ jn ] ν α (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] σ αβ − δ i [ m σ jα σ n ] β (cid:1) ϕ α ϕ β − NN − (cid:0) δ i [ m σ αj | n ] − δ i [ m σ βj L βαn ∨ ] + δ i [ m ν n ] (cid:1) ϕ α − N − (cid:0) δ i [ m σ αn ] | j − δ i [ m σ βn ] L βαj ∨ + δ i [ m σ αn ] ν j (cid:1) ϕ α . (3.18)If contracts the equality f W ijmn − f W ijmn (cid:0) equivalent to the equation (3.16) (cid:1) by the indices i and n and anti-symmetrizes the contracted equation by the indices j and m , one will obtain X jm ] − X jm ] = − N − (cid:0) Y jm ] − Y jm ] (cid:1) + 12 (cid:0) Z α [ jm ] α − Z α [ jm ] α (cid:1) . (3.19)After substituting the expression (3.19) in the equation (3.16), we obtain fff W ijmn = fff W ijmn , for fff W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + σ j [ m | n ] ϕ i − σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i + N − N + 1) δ ij (cid:0) σ α [ m | n ] − σ β [ m L βαn ∨ ] + σ α [ m ν n ] (cid:1) ϕ α − δ ij (cid:0) σ [ mα | n ] − σ [ mβ L βαn ∨ ] + σ [ mα ν n ] (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] | α − δ i [ m σ jn ] L βαβ ∨ + δ i [ m σ jn ] ν α (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] σ αβ − δ i [ m σ jα σ n ] β (cid:1) ϕ α ϕ β − NN − (cid:0) δ i [ m σ αj | n ] − δ i [ m σ βj L βαn ∨ ] + δ i [ m σ αj ν n ] (cid:1) ϕ α − N − (cid:0) δ i [ m σ αn ] | j − δ i [ m σ βn ] L βαj ∨ + δ i [ m σ αn ] ν j (cid:1) ϕ α . (3.20)After comparing the invariants fff W ijmn and f W ijmn for the mapping f , one gets that the invari-ant fff W ijmn given by (3.20) reduces to the invariant f W ijmn given by the equation (3.18).10ith respect to the transformation of the family (1.6) of the curvature tensors of the space GA N under the mapping f , and with respect to the equation (2.13), we obtain the next geomet-rical objects. W i . ( p ) . ( p ) .jmn = f W ijmn + uL ijm ∨ | n + u ′ L ijn ∨ | m − u (cid:0) ω i ( p ) .αn L αjm ∨ − ω α ( p ) .jn L iαm ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) − u ′ (cid:0) ω i ( p ) .αm L αjn ∨ − ω α ( p ) .jm L iαn ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) , (3.21) W i . ( p ) . ( p ) .jmn = f W ijmn + uL ijm ∨ | n + u ′ L ijn ∨ | m − u (cid:0) ω i ( p ) .αn L αjm ∨ − ω α ( p ) .jn L iαm ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) − u ′ (cid:0) ω i ( p ) .αm L αjn ∨ − ω α ( p ) .jm L iαn ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) , (3.22)for p , . . . , p ∈ { , } , ω i (1) .jk = L ijk and ω i (2) .jk = ω ijk , for the geometrical object ω ijk given inthe equation (3.4).It holds the next theorem. Theorem 3.1.
Let f : GA N → GA N be an equitorsion almost geodesic mapping of the type π .The geometrical object e T ijk given by (3.8) is the basic invariant of the Thomas type for themapping f . The invariance of this geometrical object is total.The geometrical object f W ijmn given by (3.9) is the basic associated invariant for the mapping f . The invariance of this geometrical object is valued. It is total if and only if the mapping f has the property of reciprocity.The geometrical object f W ijmn given by (3.18) is the associated derived invariant of the Weyltype for the mapping f . The invariance of this geometrical object is valued. It is total if and onlyif the mapping f has the property of reciprocity.The geometrical objects W ijmn , W ijmn , given by (3.21, 3.22) , are the invariants for the eq-uitorsion third type almost geodesic mapping f . The invariance of these geometrical objects arevalued. They are total if and only if the mapping f has the property of reciprocity. The basic equations for the almost geodesic mapping f : GA N → GA N of the type π are L ijk = L ijk + ψ j δ ik + ψ k δ ij + 2 σ jk ϕ i ,ϕ i | j = ν j ϕ i + µ δ ij . (3.23)The second of the basic equations (3.23) is equivalent to ϕ i | j = L iαj ∨ ϕ α + ν j ϕ i + µ δ ij . (3.24)Analogously as above, one obtains the following geometrical objects.11 T ijk = L ijk − N + 1 δ ik (cid:0) L αjα + σ jα ϕ α (cid:1) − N + 1 δ ij (cid:0) L αkα + σ kα ϕ α (cid:1) + σ jk ϕ i , (3.25) f W ijmn = R ijmn + 1 N + 1 δ ij (cid:16) R [ mn ] − (cid:0) σ [ mα | n ] + σ [ mβ L βαn ∨ ] + σ [ mα ν n ] (cid:1) ϕ α (cid:17) + σ j [ m | n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i + σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i − δ i [ m µ σ jn ] − N + 1 (cid:16) δ i [ m L αjα | n ] + (cid:0) δ i [ m σ jα | n ] + δ i [ m σ jβ L βαn ∨ ] + δ i [ m σ jα ν n ] (cid:1) ϕ α + δ i [ m µ σ jn ] (cid:17) + 1 N + 1 δ i [ m σ jn ] (cid:0) L βαβ + σ αβ ϕ β (cid:1) ϕ α − N + 1) (cid:0) L αjα + σ jα ϕ α (cid:1)(cid:0) δ i [ m L βn ] β + δ i [ m σ n ] β ϕ β (cid:1) , (3.26) f W ijmn = R ijmn + 1 N + 1 δ ij R [ mn ] + NN − δ i [ m R jn ] + 1 N − δ i [ m R n ] j + σ j [ m | n ] ϕ i + σ j [ m L iαn ∨ ] ϕ α + σ j [ m ν n ] ϕ i + σ j [ m σ αn ] ϕ α ϕ i + 1 N − (cid:0) δ i [ m σ jn ] | α + δ i [ m σ jn ] L βαβ ∨ + δ i [ m σ jn ] ν α (cid:1) ϕ α + 1 N − (cid:0) δ i [ m σ jn ] σ αβ − δ i [ m σ jα σ n ] β (cid:1) ϕ α ϕ β − NN − (cid:0) δ i [ m σ αj | n ] + δ i [ m σ βj L βαn ∨ ] + δ i [ m σ αj ν n ] (cid:1) ϕ α − N − (cid:0) δ i [ m σ αn ] | j + δ i [ m σ βn ] L βαj ∨ + δ i [ m σ αn ] ν j (cid:1) ϕ α , (3.27) W i . ( p ) . ( p ) .jmn = f W ijmn + uL ijm ∨ | n + u ′ L ijn ∨ | m − u (cid:0) ω i ( p ) .αn L αjm ∨ − ω α ( p ) .jn L iαm ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) − u ′ (cid:0) ω i ( p ) .αm L αjn ∨ − ω α ( p ) .jm L iαn ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) , (3.28) W i . ( p ) . ( p ) .jmn = f W ijmn + uL ijm ∨ | n + u ′ L ijn ∨ | m − u (cid:0) ω i ( p ) .αn L αjm ∨ − ω α ( p ) .jn L iαm ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) − u ′ (cid:0) ω i ( p ) .αm L αjn ∨ − ω α ( p ) .jm L iαn ∨ − ω α ( p ) .mn L ijα ∨ (cid:1) , (3.29)for p , . . . , p ∈ { , } , ω i (1) .jk = L ijk and ω i (2) .jk = ω ijk , for the geometrical object ω ijk given inthe equation (3.4).The next theorem holds. Theorem 3.2.
Let f : GA N → GA N be an equitorsion almost geodesic mapping of the type π .The geometrical object e T ijk given by (3.25) is the basic invariant of the Thomas type for themapping f . The invariance of this geometrical object is total.The geometrical object f W ijmn given by (3.26) is the basic associated invariant for the mapping . The invariance of this geometrical object is valued. It is total if and only if the mapping f has the property of reciprocity.The geometrical object f W ijmn given by (3.27) is the associated derived invariant of the Weyltype for the mapping f . The invariance of this geometrical object is valued. It is total if and onlyif the mapping f has the property of reciprocity.The geometrical objects W ijmn , W ijmn , given by (3.28, 3.29) , are the invariants for the eq-uitorsion third type almost geodesic mapping f . The invariance of these geometrical objects arevalued. They are total if and only if the mapping f has the property of reciprocity. We obtained novel invariants for the almost geodesic mappings of the third type of a non-symmetric affine connection space in this paper.In the Section 3, the invariants for equitorsion almost geodesic mappings are presented. Themethod used for obtaining of these invariants (cid:0) see [18] (cid:1) , simplified the corresponding methodpresented by Sinyukov [12] and used latter in [17, 19].The results obtained in this paper motivate the authors to continue their research aboutinvariants for almost geodesic mappings of non-symmetric affine connection spaces.
Acknowledgements
The authors thank to the anonymous referee who estimated the quality of this paper.
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