Finsler connection preserving the two-vector angle under the indicatrix-inhomogeneous treatment
aa r X i v : . [ m a t h . DG ] S e p Finsler connection preserving the two-vector angle underthe indicatrix-inhomogeneous treatment
G.S. Asanov
Division of Theoretical Physics, Moscow State University119992 Moscow, Russia ( e-mail: [email protected] ) bstract The Finsler spaces in which the tangent Riemannian spaces are conformally flatprove to be characterized by the condition that the indicatrix is a space of constantcurvature. In such spaces the Finslerian normalized two-vector angle can be explicatedfrom the respective two-vector angle of the associated Riemannian space. Therefore theway is opening to propose explicitly the connection preserving the angle even at theindicatrix-inhomogeneous level, that is, when the indicatrix curvature value C Ind. is permit-ted to be an arbitrary smooth function of the indicatrix position point x . The connectionobtained is metrical with the deflection part which is proportional to the gradient of thefunction H ( x ) entering the equality C Ind. ≡ H . Also the connection is covariant-constant.When the transitivity of covariant derivative is used, from the commutators of covariantderivatives the associated curvature tensor is found. Various useful representations havebeen developed. The Finsleroid space has been explicitly outlined.
Motivation and Introduction
In the Finsler space the tangent bundle
T M over the base manifold M is geometrizedby means of the Finsler metric function F ( x, y ), such that at each point x ∈ M the tangentvectors y ∈ T x M are used, where T x M is the tangent space supported by the x .The embedded position of the indicatrix I x ⊂ T x M in the tangent Riemannianspace R { x } = { T x M, g { x } ( y ) } (where g { x } ( y ) denotes the Finslerian metric tensor with x considered fixed and y used as being the variable) induces the Riemannian metric on theindicatrix through the well-known method (see, e.g., Section 5.8 in [1]) and in this sensemakes the indicatrix a Riemannian space. Therefore, the geodesics can be introduced onthe indicatrix by applying the conventional Riemannian methods.In any (sufficiently smooth) Finsler space the two-vector angle α { x } ( y , y ) can locallybe determined with the help of the indicatrix geodesic arc, which provokes the importantquestion whether the Finsler geometry can be profoundly settled down by developing andapplying the connection which preserves the angle.In general, the angle α { x } ( y , y ) is complicated and cannot be determined in anexplicit form, except for rare Finsler metric functions. The lucky example is given by theFinsler space F N which is characterized by the condition that the indicatrix is a space ofconstant curvature. In the space F N the angle α { x } ( y , y ) can be found in the explicitand simple form. Namely, it is possible to prove that (under attractive conditions) in anydimension N ≥
3, the tangent Riemannian space R { x } is conformally flat if and only if theindicatrix is a space of constant curvature. The respective transformation y = C ( x, ¯ y )is positively homogeneous of the degree which we shall denote by H . The remarkableequality C Ind. ≡ H arises, where C Ind. = C Ind. ( x ) denotes the value of the curvature of theindicatrix I x ⊂ T x M .Under this transformation y = C ( x, ¯ y ) each tangent Riemannian space R { x } is con-formally changed to become a Euclidean space E { x } . The distribution of the last spaces E { x } over the base manifold M composes the associated Riemannian space , which wedenote by R N = ( M, S ), where S = p a mn ( x ) y m y n is the Riemannian metric constructedfrom the metric tensor a mn ( x ) of the space E { x } . We are entitled to induce the angle α Riem { x } conventionally defined in the Riemannian space R N into the Finsler space F N , obtainingsimply α { x } ( y , y ) = (cid:0) /H ( x ) (cid:1) α Riem { x } (¯ y , ¯ y ) . To explicate the coefficients N mn of nonlinear connection from the Finsler angle α = α { x } ( y , y ), we should successfully propose the preservation equation. The nearestpossibility is to formulate the equation d i α = 0 in accordance with the formulas (I.1.12)and (I.1.15), applying the separable operator d i indicated in (I.1.11).This possibility has been realized in the preceding work [10,11]. Namely, in that workthe separable preservation equation d i α = 0 has been solved in the Finsler space F N underthe assumption that C Ind. = const, and whence H = const. The explicit coefficients N mn have been obtained.In general the indicatrix curvature value C Ind. may depend on the points x ∈ M which support the indicatrix. We call the space F N indicatrix-homogeneous, if the valueis a constant, whence H = const. If the dependence C Ind. = C Ind. ( x ) does hold, we say thatthe space F N is indicatrix-inhomogeneous, in which case H i = 0, where H i = ∂H/∂x i .The representations obtained in the previous work [10,11] are the ( H i → N mn .This conclusion can be drawn from the implications which are derivable by the helpof the coincidence-limit method (see Section 3.2 in [12]) which extracts the informationfrom behavior of Riemannian geodesics. To this end we should use the distance function E = E ( x, y , y ) with E = (1 / α . Namely, evaluating various partial derivatives ofthe function with respect to y and y and finding the coincidence limits when y → y ,we can obtain a valuable information on the derivatives of the Finsler metric tensor ofthe Finsler space. Performing the required evaluations on the level of the second-orderpartial derivatives ∂ /∂y m ∂y n , and, then, applying the operation y → y to the resultantexpressions, it is possible to arrive at the following general conclusion: In any Finslerspace the vanishing assumption d i α = 0 of the separable type entails the equality D i h mn = 2 F h mn d i F. If we additionally postulate d i F = 0 , we obtain D i h mn = 0 and, therefore, themetricity D i g mn = 0 which is formulated with the covariant derivative D arisen fromthe deflectionless connection. We can apply the derivative ∂ /∂y l ∂y h to the equality D i g mn = 0, which leads after simple evaluations to the vanishing D i S nkjm = 0 . Here, the S nkjm is the tensor which describes the curvature of indicatrix (see Section 5.8 in [1]).Clearly, the vanishing D i S nkjm = 0 can be realized in but rare particular cases of theFinsler space. The vanishing is realized in the indicatrix-homogeneous case of the space F N , and cannot be realized in the indicatrix-inhomogeneous space F N .Therefore, the account for the dependence H = H ( x ) in the Finsler space F N isneither straightforward nor trivial task.These important (and rather unexpected?) implications enforce us to look for morecapable ideas to formulate the preservation of angle. The attractive idea is to substitutethe normalized angle α { H ( x ) }{ x } ( y , y ) = H ( x ) α { x } ( y , y ) (see (I.1.26)) with the initial angle α { x } ( y , y ) in the separable preservation law, according to (I.1.27). The law obtained is ofthe recurrent-type (I.1.28), namely d i α + (1 /H ) H i α = 0 . It appears that this preservationis reconciled with the indicatrix-inhomogeneous Finsler space F N at any scalar H = H ( x ).The reason thereto is the following assertion obtainable by the help of the coincidence-limitmethod: In any Finsler space the vanishing assumption d i α + (1 /H ) H i α = 0 entails theequality D i h mn = 2 F h mn d i F − H H i h mn . In these patterns the vanishing d i F = 0 yields the equality D i g mn = − (2 /H ) H i h mn , whichentails the extension of the previous vanishing D i S nkjm = 0 such that the right-handpart of the extension is just the expression which is obtained when the characteristicrepresentation of the tensor S nkjm of the space F N under study is inserted under theaction of the covariant derivative D i . Thus, the recurrent-type equation d i α + (1 /H ) H i α = 0 of preservation of the angle isreconciled with the indicatrix-inhomogeneous Finsler space F N at any scalar H = H ( x )(see Proposition I.1.2 in Section I.1), and therefore is accepted in the present work toapply. We solve the equation with respect to the coefficients N mn ( x, y ).The N mn ( x, y ) thus appeared to read (I.2.15) can naturally be interpreted as the coefficients of the non-linear connection produced by the angle in the space F N studied onthe general indicatrix-inhomogeneous level. Because of the conformal flatness of the tangent Riemannian spaces R { x } , the Finslerspace F N involves the associated Riemannian space R N and, therefore, the Rieman-nian connection coefficients L mij = a mij + S mij (shown in (I.1.14)) in which the enteredChristoffel symbols a mij are to be constructed from the Riemannian metric tensor a mn ( x )of the space R N ; the notation S mij is the torsion tensor.With the knowledge of the coefficients N mn ( x, y ), we can straightforwardly evaluatethe derivative coefficients N kim and express the Finslerian connection coefficients T kim through the Riemannian connection coefficients L mij = L mij ( x ) and the function H = H ( x ) (by the help of the formulas (I.1.33) and (I.2.18)).The coefficients T kim involve the deflection tensor ∆ kim = − N kim − T kim whichis non-vanishing as far as H i = 0, namely ∆ kim = (1 /H ) H i h km . There arises the co-variant derivative T , which properties are listed in (I.1.37)-(I.1.40). In distinction fromthe connection developed in the indicatrix-homogeneous case, the T -connection obtainedis no more deflectionless. Nevertheless, the T -connection is metrical and the equality N mj = − T mji y i holds.In this way, the metrical non-linear Finsler connection F N = { N mi , T mij } is inducedin the space F N from the metrical linear connection R L = { L mj , L mij } evidenced in theRiemannian space R N , where L mj = − L mji y i . The involved function H = H ( x ) may bean arbitrary smooth function of x .The Finsler connection F N = { N mi , T mij } can be understood to be a result ofan appropriate nonlinear deformation of the connection R L = { L mj , L mij } . It is thetransformation y = C ( x, ¯ y ) that represents the deformation said.In other words, in the Finsler space F N we evidence the phenomenon that themetrical non-linear angle-preserving connection is the C -deformation of the metrical linearconnection applicable in the Riemannian space R N : F N = C · R L. We shall show that the C -deformation is T -covariant constant: T · C = 0 . Also, thecovariant derivative T is the manifestation of the transitivity of the connection under thistransformation, in short, T = C · ∇ , where ∇ is the covariant derivative applicable in theRiemannian space R N .In the Riemannian geometry we have merely H = 1. In the Finsler space F N , thescalar H ( x ) plays the role of the parameter which changes the indicatrix curvature value.Varying the scalar H ( x ) evokes the changes in the Finsler space F N .In the theory of Finsler spaces the notion of connection was studied on the basisof various convenient sets of axioms (see [1-5] and references therein). Regarding thesignificance of the angle notion, the important step was made in [6] were in processes ofstudying implications of the two-vector angle defined by area, the theorem was provedwhich states that a diffeomorphism between two Finsler spaces is an isometry iff it keepsthe angle. This Tam´assy’s theorem clearly substantiates the idea to develop the Finslerconnection from the Finsler two-vector angle, possibly on the analogy of the Riemanniangeometry.To meet new methods of applications, the interesting chain of linear connections wasintroduced and studied in [3]. It was emphasized that in the Riemannian geometry wehave naturally the metrical and linear connection applicable on the tangent bundle of thevariables x, y . Like to the constructions developed in the preceding work [10,11] dealtwith the indicatrix-homogeneous case, in the present indicatrix-inhomogeneous study ofthe space F N the export of this connection generates the required Finsler connection.By performing the comparison between the commutators of the obtained Finslercovariant derivative T and the commutators of the underlined Riemannian covariantderivative ∇ , not assuming H = const so that H ( x ) is permitted to be an arbitrarysmooth function of x , the associated curvature tensor ρ knij can straightforwardly bederived.The Finsleroid case of the space F N provides us with the example when the keytransformation y = C ( x, ¯ y ) is known explicitly. Therefore, we can straightforwardlyapply the developed indicatrix-inhomogeneous theory taking the metric function of theFinsleroid type. The explicit representations for the respective Finsleroid coefficients N mn , as well as for the entailed derivative coefficients N kim and N kimn , are found. Thuswe have got prepared the connection F N in the Finsleroid space at our disposal with anarbitrary input scalar H ( x ).Below we are interested in spaces of the dimension N ≥
3. The two-dimensionalcase has been studied in [8,9].
Chapter I. The Outline
I.1. Basic representations
For a given function Finsler metric function F = F ( x, y ) we can construct the co-variant tangent vector ˆ y = { y i } and the Finslerian metric tensor { g ij } in the conventionalway: y i := (1 / ∂F /∂y i and g ij := ∂y i /∂y j . The contravariant tensor { g ij } is definedby the reciprocity conditions g ij g jk = δ ki , where δ stands for the Kronecker symbol. Theindices i, j, . . . refer to local admissible coordinates { x i } on the base manifold M . Weshall also use the tensor C ijk = (1 / ∂g ij /∂y k . By l we shall denote the unit vectors,namely, l = y/F ( x, y ), such that F ( x, l ) = 1.Let U x be a simply connected and geodesically complete region on the indicatrix I x supported by a point x ∈ M . Any point pair u , u ∈ U x can be joined by the respectivearc A { x } ( l , l ) ⊂ I x of the Riemannian geodesic line drawn on U x . By identifying thelength of the arc with the angle notion we arrive at the geodesic-arc angle α { x } ( y , y ),where y , y ∈ T x M are two vectors issuing from the origin 0 ∈ T x M and possessing theproperty that their direction rays 0 y and 0 y intersect the indicatrix at the point pair u , u ∈ U x . We obtain α { x } ( y , y ) = ||A { x } ( l , l ) || . (I.1.1)The coefficients N ki = N ki ( x, y ) are required to construct the operator d i := ∂∂x i + N ki ∂∂y k . (I.1.2)These coefficients are assumed naturally to be positively homogeneous of degree 1 withrespect to the vector argument y .The derivative coefficients N knm = ∂N kn ∂y m , N knmj = ∂N knm ∂y j (I.1.3)possess the identities N knm y m = N kn , N knmj y m = N knmj y j = 0 , N knmj = N knjm . (I.1.4)The coefficients are used to construct the covariant derivatives D k F := d k F, D k l m := d k l m − N mkn l n , D k l m := d k l m + N hkm l h , (I.1.5)and D k g mn := d k g mn + N hkm g hn + N hkn g mh . (I.1.6)The identities ∂ D k F∂y m = D k l m , ∂ D k l m ∂y n = D k g mn + l h N hkmn (I.1.7)are obviously valid, together with ∂ D i g mn ∂y j = 2 D i C mnj + N timj g tn + N tinj g mt , (I.1.8)where D i C mnj := d i C mnj + N tij C mnt + N tim C tnj + N tin C mtj . (I.1.9)In addition to the Finsler metric tensor g mn , we shall use also the tensor h mn = g mn − l m l n (I.1.10)which possesses the property h mn y n = 0 . The covariant derivative of this tensor will beconstructed in the manner similar to (1.6), namely D k h mn := d k h mn + N hkm h hn + N hkn h mh . To deal with the two-vector angle α = α { x } ( y , y ), we merely extend the operator d i in the separable way , namely d i = ∂∂x i + N ki ( x, y ) ∂∂y k + N ki ( x, y ) ∂∂y k , y , y ∈ T x M, (I.1.11)and introduce the covariant derivative D i α according to D i α = d i α. (I.1.12)In the Riemannian geometry we have the separable operator d Riem i = ∂∂x i + L ki ( x, y ) ∂∂y k + L ki ( x, y ) ∂∂y k , y , y ∈ T x M, (I.1.13)with the linear coefficients L ki ( x, y ) = − L kij ( x ) y j and L ki ( x, y ) = − L kij ( x ) y j obtainedfrom the Riemannian connection coefficients L mij = a mij + S mij , (I.1.14)where a mij = a mij ( x ) stands for the Christoffel symbols constructed from the Riemannianmetric tensor a mn ( x ) and S mij = S mij ( x ) is an arbitrary torsion tensor : S mij = − S jim with S mij = a mh S hij . When applied to the Riemannian two-vector angle α Riem { x } ( y , y ) = a mn ( x ) y m y n /S S , where S = p a mn ( x ) y m y n and S = p a mn ( x ) y m y n , the operatorreveals the fundamental vanishing property d Riem i α Riem { x } ( y , y ) = 0 , y , y ∈ T x M. By analogy, one may assume that the Finsler coefficients N ki fulfill the separableangle-preservation equation D i α = 0 (I.1.15)to try developing the theory in which the properties D k F = 0 , D k l m = 0 , D k l m = 0 , (I.1.16)together with the metricity D k g mn = 0 (I.1.17)hold fine. This metricity, taken in conjunction with the identities indicated in (1.7), justentails the vanishing l h N hkmn = 0 . (I.1.18)The following valuable implication can be deduced from angle by applying thecoincidence-limit method exposed in Section 3.2 in [12]: In any Finsler space the vanishingassumption d i α = 0 of the separable type entails the equality D i h mn = 2 F h mn d i F (I.1.19)(take below the formula (1.29), keeping H = const). If we additionally postulate d i F = 0 , we obtain D i h mn = 0 and, therefore, D i g mn = 0 .Thus, starting with the separable angle-preservation equation leads to the followingimplication: P RESERV AT ION OF AN GLE AN D LEN GT H = ⇒ M ET RICIT Y, (I.1.20)that is, the two conditions d i α = 0 and d i F = 0 entail D i g mn = 0.When D i g mn = 0 , the identity (1.8) communicates the validity of the vanishing2 D i C mnj + N timj g tn + N tinj g mt = 0 , (I.1.21)which in turn entails that, because the tensor C mnj is totally symmetric, the tensor N nimj := N timj g tn must be totally symmetric with respect to the subscripts n, m, j : N nimj = N minj = N jimn = N nijm , (I.1.22)and whence N kimn = −D i C kmn , (I.1.23)where D i C kmn := d i C kmn − N kit C tmn + N tim C ktn + N tin C kmt . With the representation (1.23), the vanishing (1.18) can be regarded as a direct implicationof the identity y k C knj = 0 shown by the tensor C knj .Thus, in any Finsler space the two conditions d i α = 0 and d i F = 0 entail therepresentation (1.23) for the coefficients N kimn .By differentiating these coefficients with respect to y j and making the interchangeof the indices m, j , and also noting that ∂N kimn /∂y j − ∂N kijn /∂y m = 0 and ∂C kmn ∂y j − ∂C kjn ∂y m = − (cid:0) C hnm C khj − C hnj C khm (cid:1) , from (1.23) we can arrive at the following vanishing after a short evaluation: D i S nkjm = 0 , where S nkij = (cid:0) C hnj C khi − C hni C khj (cid:1) F . (I.1.24)However, there are no reasons to trust that the separable form (1.15) for the anglepreservation is applicable in general to any Finsler space. For it might happen thatthe equation (1.15) doesn’t permit any solution with respect to the coefficients N ki = N ki ( x, y ). Indeed, the formula (1.24) tells us that the following proposition is true. Proposition I.1.1.
One is entitled to hope to determine the coefficients N mn of aFinsler space from the separable equation d i α = 0 supplemented by the condition d i F = 0 if only the Finsler space possesses the property D i S nkjm = 0 . Clearly, the vanishing D i S nkjm = 0 can be realized in but rare particular cases ofthe Finsler space.In this connection it can be of help to introduce a characteristic indicatrix scale R ( x )in each tangent space to normalize the angle. If the volume V I x of the Finslerian indicatrix I x ⊂ T x M is finite, it is attractive to obtain the scale by the help of the equality V I x = C ( R ( x )) N − , C = const . (I.1.25)In this case the R ( x ) has the geometrical meaning of the radius of the indicatrix supportedby p. x .In this respect, there is the deep qualitative distinction of the Finsler geometry fromthe Riemannian geometry. Namely, in the latter geometry we have simply V I x = const,and whence R = const. The new reality that the value of V I x may vary from point topoint of the background manifold M arises in the Finsler geometry, in which case the R may be a function of x .The R ( x ) thus appeared proposes naturally the scale factor in the tangent Rieman-nian space R { x } supported by the point x .This motivation suggests the idea to replace the above angle α { x } ( y , y ) by the normalized angle α { H ( x ) }{ x } ( y , y ) := H ( x ) α { x } ( y , y ) , y , y ∈ T x M, (I.1.26)where we have introduced the scalar H ( x ) = 1 /R ( x ) , to use the preservation equation d i α { H ( x ) }{ x } ( y , y ) = 0 (I.1.27)instead of d i α { x } ( y , y ) = 0 formulated in (1.15). The preservation law (1.27) can bewritten in the recurrent form d i α + 1 H H i α = 0 . (I.1.28)The d i is the operator (1.11) and H i = ∂H/∂x i . Since the angle α { x } ( y , y ) is measured by the indicatrix arc length, it seems quitenatural to normalize the angle by means of the characteristic scale factor, according to(1.26).To elucidate patterns, it proves being of great help to apply the coincidence-limitmethod (see Section 3.2 in [12]). Namely, with the function E = (1 / α the recurrentpreservation d i α + (1 /H ) H i α = 0 proposed by (1.28) entails the following E -equation ∂E∂x i + N k i ∂E∂y k + N k i ∂E∂y k = − H H i E, where N k i = N ki ( x, y ) and N k i = N ki ( x, y ) . Evaluating various partial derivatives ofthis E -equation with respect to y and y and finding the coincidence limits when y → y ,we can obtain a valuable information of the tensors of the Finsler space. Performingthe required evaluations on the level of the second-order partial derivatives ∂ /∂y m ∂y n ,and, then, applying the operation y → y to the resultant expressions, it is possibleto arrive at the general conclusion that in any Finsler space the vanishing assumption d i α + (1 /H ) H i α = 0 entails the equality D i h mn = 2 F h mn d i F − H H i h mn . (I.1.29)The formula (1.29) has been derived in Appendix E in all detail by performingrequired long substitutions (see (E.37) in Appendix E).By differentiating the equality (1.29) with respect to y j , it is possible to obtain thecoefficients N kimn . In this way, when the vanishing d i F = 0 is also keeping valid, simpledirect evaluations yield the representation N kimn = 2 H H i F l k h mn − D i C kmn , (I.1.30)which extends the previous (1.23). The symmetry (1.22) is now replaced by N timj g tn − H H i F h mj l n = N timn g tj − H H i F h mn l j . Instead of the vanishing (1.18) we obtain
F N kinm l k = 2 H H i h mn . (I.1.31)The vanishing D i S nkjm = 0 indicated in (1.24) is now extended, namely the above repre-sentation (1.30) straightforwardly entails the equality D i S nkjm = − H H i (cid:0) h kj h mn − h km h jn (cid:1) . From (1.29) we can conclude that when d i F = 0 we have D i g mn = − H H i h mn (I.1.32)at an arbitrary smooth function H = H ( x ).The equality (1.32) suggests us to introduce the total connection coefficients T kim = − N kim − H H i h km , (I.1.33)so that the deflection tensor ∆ kim def = − N kim − T kim (I.1.34)is non-vanishing as far as H i = 0, namely∆ kim = 1 H H i h km . (I.1.35)It follows that T kim y m = − N kim y m ≡ − N ki , l k T kim = − l k N kim . (I.1.36)There arises the total covariant derivative T i , showing the properties T i F = 0 , T i l m = 0 , T i l m = 0 , (I.1.37)and the metricity T i g nm = 0 , (I.1.38)where T i F def = d i F, T i l m def = d i l m − T him l h , T i l m def = d i l m + T mih l h , (I.1.39)0and T i g nm def = d i g nm − T him g hn − T hin g hm . (I.1.40)In all the previous formulas started with (1.26), the H ( x ) was an arbitrary smoothscalar not related anyhow to the indicatrix curvature, the constancy of the indicatrixcurvature was not implied, and the Finsler space was arbitrary.If the indicatrix of a Finsler space is a space of constant curvature at any point x ∈ M , we say that the Finsler space is the F N - space , where N ≥ F N is motivated by the following important observa-tions. Given an arbitrary Finsler space of any dimension N ≥
3. The tangent Riemannianspace R { x } ⊂ T x M is conformally flat if and only if the indicatrix I x ⊂ T x M is a space ofconstant curvature, assuming naturally that the involved conformal multiplier is homo-geneous with respect to the argument y . The dependence of the conformal multiplier onthe variable y is presented by the power of the Finsler metric function. The remarkableequality C Ind. ≡ H ensues. These observations form the content of Proposition II.2.1(formulated and proved in Section II.2 of Chapter II), which extends Proposition 2.1 ofthe preceding work [10,11] in the following essential aspect.In [10,11], the assumption was made that the respective conformal multiplier is of thepower dependence on the Finsler metric function, in accordance with the representationsindicated in the formula (II.2.3) of Section II.2. In proving Proposition II.2.1 in SectionII.2, we outline the reasoning line which explains that the representations are actually thedirect consequences of the property that the indicatrices are spaces of constant curvature.We say that the Finsler space F N is indicatrix-homogeneous if C Ind. = const. In thiscase, the deflectionless connection has been derived from the separable angle-preservationequation in the preceding work [10,11].Alternatively, the Finsler space F N is said to be indicatrix-inhomogeneous if C Ind. = C Ind. ( x ). On this level, because of the equality C Ind. ≡ H , we have H = H ( x ) and H i = 0.On the indicatrix-inhomogeneous level of study of the Finsler space F N with d i F = 0the separable preservation law for the angle is impossible to introduce . Indeed, the lawentails the metricity D i g mn = 0 of the deflectionless type (see (1.17) and the definition(1.6)), together with the representation (1.23) for the coefficients N kimn and the vanishing D i S nkjm = 0 , where S nkij = (cid:0) C hnj C khi − C hni C khj (cid:1) F (see (1.24)). It is known that theindicatrix is a space of constant curvature if and only if the last tensor fulfills the equality S nkij = C ( h nj h ki − h ni h kj ) with the factor C which is independent of y , in which case C Ind. = 1 − C (see Section 5.8 in [1]). In the Finsler space F N , we have C Ind. = H . Thetwo vanishings D i g mn = 0 and D i F = 0 entail D i h mn = 0. Whence from D i S nkjm = 0 itfollows that H i = 0.If, however, we start with recurrent preservation law supplemented by the vanishingcondition D i F = 0, then from (1.29) we have D i h mn = − (2 /H ) H i h mn . Applying thecovariant derivative D i to the tensor S nkij = C ( h nj h ki − h ni h kj ) and taking into accountthat C = 1 − H , after short evaluations we now arrive at the equality D i S nkjm = − H H i (cid:0) h kj h mn − h km h jn (cid:1) Proposition I.1.2.
The recurrent-type preservation (1.28) of the angle, that is, d i α + (1 /H ) H i α = 0 , is reconciled with the indicatrix-inhomogeneous Finsler space F N at any scalar H = H ( x ) obtainable from the identification C Ind. = H . The observations motivate us to go to the preservation law (1.27) which is not sep-arable from the standpoint of the indicatrix-arc angle α { x } ( y , y ), whenever H = const. In so doing, the coefficients N mn of the Finsler space F N are obtained to read (I.2.16)in Section I.2. They don’t involve explicitly the gradients H n . If, however, we expand thepartial derivatives ∂/∂x n which enter the right-hand part of (I.2.16), the coefficients willbreak down into two parts: N mn = N I mn + ˘ N mn , ˘ N mn = ˘ N m H n . (I.1.41)Here, the first part N I mn are the coefficients of the indicatrix-homogeneous case (givenby the formula (2.30) in [10], and by the formula (2.36) in [11]) in which the constant H has been merely replaced by arbitrary H ( x ), and the vector field ˘ N m does not involveany gradient of H ( x ). We may say that the coefficients N mn are of the linear dependence on the gradient H n .The entailed coefficients N kmn are given by the representation (II.3.32) of Chapter IIwhich is applicable to any indicatrix-inhomogeneous Finsler space F N . It is also possibleto evaluate explicitly the derivative coefficients N kmni = ∂N kmn /∂y i . The required evalu-ations lead straightforwardly to the validity of the representation (1.30) in the F N -spacewith an arbitrary smooth function H ( x ), provided the vanishing d n F = 0 is assumed (seeProposition II.3.5 in Chapter II).Having evaluated the coefficients N kmn , we obtain from (1.33) the total connectioncoefficients T kim thereby solving the problem of finding the connection in the F N -spaceat the indicatrix-inhomogeneous level. The coefficients T kim involve the deflection ten-sor ∆ kim indicated in (1.34) and (1.35). There arises the covariant derivative T , whichproperties are listed in (1.36)-(1.40).Section I.2 gives a brief summary of Chapter II.The formula (I.2.16) indicates the representation of the coefficients N mn which isvalid for an arbitrary Finsler space of the type F N . The representation involves the vectorfield U i which realizes the key transformation y = C ( x, ¯ y ) indicated in (I.2.1). Given aparticular Finsler space of the type F N , the formula (I.2.16) yields the coefficients N mn in a completely explicit way when the respective field U i is known.The Finsleroid case to which Section I.3 is devoted provides us with such an example,for the required field U i is explicitly given, namely by means of the representation (I.3.20)(which was earlier found in Section 6 of [7]). Therefore, we can straightforwardly applythe developed theory of the F N -space to the metric function of the Finsleroid type.The expansion (1.41) for the respective Finsleroid coefficients N mn has been evaluated.The explicit representation of the entailed derivative coefficients N kim is indicated. Therespective validity of the representations (1.29) and (1.30) of the tensors D i h mn and N kimn on the indicatrix-inhomogeneous level of study of the Finsleroid space has been verifiedby direct evaluations presented in detail.Several Appendices are added in which numerous fragments of the underlined eval-uations have been displayed.2 I.2. Indicatrix of constant curvature
Let M be the base manifold, such that F N = ( M, F ), where F = F ( x, y ) is theFinsler metric function and N ≥ F N - space . Denote by C Ind. the value of curvature of the indicatrix supported by the point x ∈ M . If C Ind. is a constant over the manifold M , we say that the space F N is of the indicatrix-homogeneous case.In general, the value C Ind. may vary from point to point of M , in which case we saythat the space F N is of the indicatrix-inhomogeneous type. The possibility is character-ized by a function C Ind. = C Ind. ( x ) such that the derivative ∂ C Ind. /∂x i does not vanishidentically.In such spaces, the transformation y = C ( x, ¯ y ) , y, ¯ y ∈ T x M, (I.2.1)can be proposed which maps the tangent vectors y ∈ T x M into the tangent vectors ofthe same tangent space T x M , subject to the following conditions. The transformation isnon-linear with respect to ¯ y . Non-singularity and sufficient smoothness are implied. Also,the transformation is positively homogeneous of a degree H ( x ) regarding dependence ontangent vectors y . Each tangent Riemannian space R { x } = { T x M, g { x } ( y ) } is conformallytransformed to Euclidean space, to be denoted by E { x } . The distribution of the last spaces E { x } over the base manifold M composes the associated Riemannian space , which wedenote by R N = ( M, S ), where S = p a mn ( x ) y m y n is the Riemannian metric constructedfrom the metric tensor a mn ( x ) of the space E { x } .Under these conditions, the scalar H ( x ) can be taken from the identification C Ind. ≡ H . (I.2.2)The equality S ( x, ¯ y ) = ( F ( x, y )) H ( x ) (I.2.3)arises (see (II.2.10)), which validates the indicatrix correspondence to the Euclideansphere; S ( x, ¯ y ) = p a mn ( x )¯ y m ¯ y n . The relevant conformal multiplier p is constructedfrom the Finsler metric function F , according to p = 1 H F − H . (I.2.4)We take 1 > H > H being a straightforward task.If f ( x, y ) is the involved conformal multiplier in the tangent Riemannian space R { x } ,then the equality g { x } ( y ) = f ( x, y ) u { x } ( y )should introduce the tensor u { x } ( y ) which associated Riemannian curvature tensor van-ishes identically. The function f ( x, y ) is assumed naturally to be homogeneous withrespect to the argument y . Denoting the homogeneity degree of f ( x, y ) by means of2 a ( x ), we just conclude that the difference 1 − a is exactly the homogeneity degree of thetransformation (2.1) considered, that is, H = 1 − a. F N -space if and only if theindicatrix of the Finsler space is a space of constant curvature. The dependence of themultiplier f on the variable y is presented by the power of the Finsler metric function F (see Proposition II.2.1 in Section 2 of Chapter II).The respective two-vector angle α { x } ( y , y ) proves to be obtainable from the angle α Riem { x } ( y , y ) operative in the Riemannian space, namely the simple equality α { x } ( y , y ) = 1 H ( x ) α Riem { x } (¯ y , ¯ y ) (I.2.5)(see (II.2.51)-(II.2.52)) is valid.We locally represent the transformation (2.1) by means of the functions y i = y i ( x, t ) , t n ≡ ¯ y n . (I.2.6)The homogeneity entails y i ( x, kt ) = k /H y i ( x, t ) with k > ∀ t, together with y in t n =(1 /H ) y i , where y in = ∂y i /∂t n .The definition U i def = (1 /S )¯ y i ≡ (1 /F H )¯ y i (I.2.7)introduces the normalized vector, which is obviously unit: U i U i = 1 and U i = a ij U j . The zero-degree homogeneity U i ( x, ky ) = U i ( x, y ) with k > ∀ t holds, entailing theidentity U in y n = 0 with U in := ∂U i ∂y n = 1 F H t in − F HU i l n , (I.2.8)where t in = ∂t i /∂y n . It follows that F H U hs y kh = h ks , F H U ik y kt = δ it − U i U t , U i U in = 0 . (I.2.9)The vanishing U i (cid:18) ∂U i ∂x n + L ikn U k (cid:19) = 0 (I.2.10)holds obviously, where L ink are the Riemannian connection coefficients (I.1.14).The representation (2.5) of the angle takes on the simple form α { x } ( y , y ) = 1 H ( x ) arccos λ, with λ = a mn ( x ) U m U n , (I.2.11)where U m = U m ( x, y ) and U m = U m ( x, y ) . When the recurrent preservation d i α + (1 /H ) H i α = 0 proposed by (1.28) is appliedto the angle given in (2.11), we obtain simply d i λ = 0 , (I.2.12)where d i is the separable operator (1.11). That is, the recurrent preservation law formu-lated for the Finsler F N -space angle α { x } given by (2.11) is tantamount to the separablepreservation law for the Euclidean angle α Riem { x } = arccos λ , whence to the separable preser-vation law (2.12).4The form of the right-hand part in the formula λ = a mn ( x ) U m U n is such that thelaw (2.11) is obviously equivalent to the vanishing D n U i = 0 (I.2.13)for the field U i = U i ( x, y ), where we introduced the covariant derivative D n U i := d n U i + L ink U k . (I.2.14)Since d n U i = ∂U i ∂x n + N kn U ik , we arrive at the conclusion that in the F N -space, the coefficients N mn can unambigu-ously be found from the equation d n (cid:0) H ( x ) α { x } ( y , y ) (cid:1) = 0 to be given explicitly by therepresentation N mn = − y mi F H (cid:18) HF U i ∂F∂x n + ∂U i ∂x n + (cid:0) a ink + S ink (cid:1) U k (cid:19) + l m d n F (I.2.15)(see (II.3.12) in Chapter II). Here, a ink are the Riemannian Christoffel symbols; S ink = S ink ( x ) is an arbitrary torsion tensor, that is, the tensor possessing the skew-symmetryproperty S ink = − S kni , where S ink = a ij S jnk .Whenever d n F = 0, the representation (2.15) takes on the form N mn = − l m ∂F∂x n − y mi F H (cid:18) ∂U i ∂x n + (cid:0) a ink + S ink (cid:1) U k (cid:19) (I.2.16)(see (II.1.19) in Chapter II). These coefficients N mn present the general solution to thecouple equations d n (cid:0) H ( x ) α { x } ( y , y ) (cid:1) = 0 and d n F = 0, so that no problem of uniquenessof connection coefficients may be questioned. The entrance of the torsion tensor S ikn isthe only freedom, in complete analogy to the connection coefficients of the Riemannianspace.The evaluations performed in Section II.3 of Chapter II have arrived also at therepresentation N mn = d Riem n y m ( x, t ) + 1 H H n y m ln F (I.2.17)(see (II.3.29) in Chapter II) which is alternative to (2.16); here, y m = y m ( x, t ) are thefunctions (2.6).The representations (2.15)-(2.17) involve the gradient H n and are applicable to anyindicatrix-inhomogeneous Finsler space F N .The coefficients N kmn can be evaluated from (2.16) to read N kmn = − F h kn ∂F∂x m − l k ∂l n ∂x m − C kns N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) (I.2.18)5(see Proposition II.3.4 in Chapter II). With these coefficients, the validity of the repre-sentation (1.30) for the entailed coefficients N kimn can straightforwardly be verified (seeProposition II.3.5 in Chapter II).The space F N is obtainable from the Riemannian space R N by means of the defor-mation y = C ( x, ¯ y ) (see (II.2.1) in Chapter II) which can be presented by the deformationtensor C im := p ¯ y im , (I.2.19)so that g mn = C im C jn a ij (I.2.20)and the zero-degree homogeneity C im ( x, ky ) = C im ( x, y ) , k > , ∀ y, (I.2.21)holds, together with the identity C im ( x, y ) y m = ( F ( x, y )) − H ¯ y i (I.2.22)(see (II.2.24)-(II.2.27)). In Section II.4 we show that the C -deformation is T -covariantconstant: T · C = 0 , (I.2.23)where T designates the covariant derivative introduced by the help of the formulas (I.1.33)-(I.1.40) (see Proposition II.4.1).Also, the covariant derivative T is the manifestation of the transitivity of the con-nection under the C -transformation, in short, T = C · ∇ , (I.2.24)where ∇ is the covariant derivative applicable in the background Riemannian space R N (see Proposition II.4.2). In other words, in the Finsler space F N the metrical non-linearangle-preserving connection is the C -export of the metrical linear connection (II.1.2)applicable in the space R N .In Section II.5 we perform the attentive comparison between the commutators ofthe involved Finsler covariant derivative T and the commutators of the underlined Rie-mannian covariant derivative ∇ , not assuming H = const, such that H ( x ) can be anarbitrary smooth function of x . In this way, we derive the associated curvature tensor ρ knij . Important properties of the tensor are elucidated. I.3. Reduction to the Finsleroid space
In the Finsleroid case, we make the notation change H ( x ) → h ( x ).The scalar g ( x ) obtained through h ( x ) = r − g ( x )4 , with − < g ( x ) < , (I.3.1)6plays the role of the characteristic parameter.It follows that g i = − hg h i , (I.3.2)where g i = ∂g/∂x i and h i = ∂h/∂x i .We assume that in addition to a Riemannian metric p a ij ( x ) y i y j the manifold M admits a non-vanishing 1-form b = b i ( x ) y i of the unit length: a ij ( x ) b i ( x ) b j ( x ) = 1 , (I.3.3)where b i ( x ) = a ij ( x ) b j ( x ) . The tensor a ij ( x ) is reciprocal to a ij ( x ), so that a ij a jn = δ ni ,where δ ni stands for the Kronecker symbol. We need also the quadratic form B = b + gbq + q ≡ (cid:18) b + 12 gq (cid:19) + h q , (I.3.4)where q = √ r mn y m y n with r mn = a mn − b m b n , (I.3.5)so that a ij ( x ) y i y j = b + q . (I.3.6)We shall also use the scalar χ = 1 h (cid:16) − arctan G Lhb (cid:17) , if b ≥ χ = 1 h (cid:16) π − arctan G Lhb (cid:17) , if b ≤ , (I.3.7)with the function L = q + ( g/ b fulfilling the identity L + h b = B. (I.3.8)The definition range 0 ≤ χ ≤ h π is of value to describe all the tangent space. The normalization in (3.7) is such that χ (cid:12)(cid:12) y = b = 0 . (I.3.9)The quantity (3.7) can conveniently be written as χ = 1 h f (I.3.10)with the function f = arccos A ( x, y ) p B ( x, y ) (I.3.11)ranging as follows: 0 ≤ f ≤ π. (I.3.12)7The Finsleroid-axis vector b i relates to the value f = 0, and the opposed vector − b i relatesto the value f = π : f = 0 ∼ y = b ; f = π ∼ y = − b. (I.3.13)With these ingredients, we construct the Finsler metric function K = √ B J, with J = e − gχ . (I.3.14)The normalization is such that K ( x, b ( x )) = 1 (I.3.15)(notice that q = 0 at y i = b i ). The positive (not absolute) homogeneity holds: K ( x, γy ) = γK ( x, y ) for any γ > x, y ).Under these conditions, we call K ( x, y ) the F F
P Dg -Finsleroid metric function , ob-taining the
F F
P Dg - Finsler space
F F
P Dg := { M ; a ij ( x ); b i ( x ); g ( x ); K ( x, y ) } . (I.3.16) Definition . Within any tangent space T x M , the metric function K ( x, y ) producesthe F F
P Dg -Finsleroid
F F
P Dg ; { x } := { y ∈ F F P Dg ; { x } : y ∈ T x M, K ( x, y ) ≤ } . (I.3.17) Definition . The
F F
P Dg -Indicatrix IF P Dg ; { x } ⊂ T x M is the boundary of the F F
P Dg -Finsleroid, that is, IF P Dg { x } := { y ∈ IF P Dg { x } : y ∈ T x M, K ( x, y ) = 1 } . (I.3.18) Definition . The scalar g ( x ) is called the Finsleroid charge . The 1-form b = b i ( x ) y i is called the Finsleroid–axis form .The entailed components y i := (1 / ∂K /∂y i ) of the covariant tangent vector ˆ y = { y i } can be found in the simple form y i = ( u i + gqb i ) J , (I.3.19)where u i = a ij y j .Let us elucidate the structure of the coefficients N km in the Finsleroid case proper.From (6.26) of [7] it follows that the quantity U i = (1 /K h )¯ y i can explicitly be given by U i = (cid:20) hv i + (cid:18) b + 12 gq (cid:19) b i (cid:21) √ B , (I.3.20)where v i = y i − bb i . So we have ∂U i ∂g = − g h v i √ B + 12 qb i √ B − B U i qb, ∂U i ∂g = − g h U i + g h ( b + 12 gq ) b i √ B + 12 qb i √ B − B U i qb. Since K h y mi U i = 1 h y m (a consequence of the homogeneity involved) and K h y mi b i = " b m + 1 B h (cid:18) b + 12 gq (cid:19) − b − gq ! y m √ B (see (D.12) in [7]), we can straightforwardly evaluate the contraction K h y mi ∂U i ∂g = − g h h y m − B h qby m + g h ( b + 12 gq ) b m + g h B h (cid:18) b + 12 gq (cid:19) y m − g h B ( b + 12 gq )( b + gq ) y m + 12 q " b m + 1 B h (cid:18) b + 12 gq (cid:19) − b − gq ! y m . Using the equality (cid:18) b + 12 gq (cid:19) = B − h q (see (3.4)) leads to the representation K h y mi ∂U i ∂g = g h (cid:18) b + 12 gq (cid:19) b m − g B h q y m − g h B (cid:18) b + 12 gq (cid:19) ( b + gq ) y m + 12 q " b m + 1 B h gq − b − gq ! y m , which can be simplified as follows: K h y mi ∂U i ∂g = g h (cid:18) b + 12 gq (cid:19) b m − g h B (cid:18) b + 12 gq (cid:19) ( b + gq ) y m + 12 h q " b m (cid:18) − g (cid:19) − B ( b + gq ) y m (cid:18) − g (cid:19) = 12 h (cid:18) q + 12 gb (cid:19) b m − h B (cid:18) q + 12 gb (cid:19) ( b + gq ) y m , so that K h y mi ∂U i ∂g = 12 h B (cid:18) q + 12 gb (cid:19) [ Bb m − ( b + gq ) y m ] . A m = N g qK h q b m − ( b + gq ) v m i ≡ KC mnn (see (A.27) in [7]), we come to K h y mi ∂U i ∂g = 1 h qB (cid:18) q + 12 gb (cid:19) KN g A m . (I.3.21)Therefore, in the Finsleroid case the coefficients N ki proposed by (I.2.16) are thesum N ki = N I ki + ˘ N ki , ˘ N ki = ˘ N k g i , (I.3.22)where ˘ N k = − h qB (cid:18) q + 12 gb (cid:19) KN g A k −
12 ¯
M y k (I.3.23)with ¯ M coming from ∂K ∂g = ¯ M K . (I.3.24)The torsion tensor S kij = S kij ( x ) has been neglected. The N I ki are the coefficients (6.48)of [7] ( they can also be found in [10,11]), namely, N I ki = "(cid:18) b − h (cid:18) b + 12 gq (cid:19)(cid:19) η kj + (cid:18) q v k (cid:18) b − h ( b + gq ) (cid:19) + (cid:18) h − (cid:19) b k (cid:19) y j ∇ i b j − a kij y j . (I.3.25)They don’t involve the gradient g i . The tensor η kn = a kn − b k b n − q v k v n (I.3.26)enters the representation. This tensor obeys the nullification y k η kn = b k η kn = 0 . (I.3.27)The designation ∇ i stands for the Riemannian covariant derivative constructed with thehelp of the Riemannian Christoffel symbols a kij = a kij ( x ).The N I ki are the coefficients N ki obtained when the condition h = const whichspecifies the indicatrix-homogeneous case is postulated.For the coefficients ˘ N kim = ∂ ˘ N ki ∂y m N kim = 1 h g i q B (cid:18) g bq − h (cid:19) N g A m l k + 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h km + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m A k − g i ¯ M h km + 1 K l m ˘ N ki (I.3.28)is obtained (see Appendix A).Using (3.28) we find straightforwardly that y k ∂ ˘ N ki ∂y m ∂y n = 2 h h i h mn . (I.3.29)For the coefficients ˘ N kimn = ∂ ˘ N kim ∂y n the representation ˘ N kimn = − g h g i K h mn l k − gh g i K A kmn (I.3.30)can explicitly be derived (see Appendix A); A kmn = KC kmn .The full coefficients read N kimn = 2 h h i K l k h mn − K D i A kmn (I.3.31)(see Appendix A). Thus in the Finsleroid case proper we have straightforwardly verifiedthe validity of the representation (1.30). Chapter II. Phenomenon of indicatrix of constant curvaturewith C Ind. = C Ind. ( x ) II.1. Motivation
In any dimension N ≥ F geometrizes the tangent bundle T M over the base manifold M such that at each point x ∈ M the tangent space T x M is endowed with the curvature tensor constructed from the respective Finslerian metrictensor g { x } ( y ) by means of the conventional rule of the Riemannian geometry considering y to be the variable argument. There arises the Riemannian space R { x } = { T x M, g { x } ( y ) } supported by the point x ∈ M such that T x M plays the role of the base manifold for thespace. We call R { x } the tangent Riemannian space .Given an N -dimensional Riemannian space R N = ( M, S ), where S denotes theRiemannian metric function, one may endeavor to obtain a Finsler space F N = ( M, F )1by applying an appropriate transformation C to tangent spaces. The base manifold M iskeeping the same for both the spaces, R N and F N .We assume that the transformation C is restrictive , in the sense that no point x ∈ M is shifted under the transformation, so that in each tangent space T x M the deformationmaps tangent vectors y ∈ T x M into the tangent vectors of the same T x M : y = C ( x, ¯ y ) , y, ¯ y ∈ T x M. (II.1.1)In general, this transformation is non-linear with respect to ¯ y . Non-singularity and suffi-cient smoothness are always implied.We may evidence in the Riemannian space R N the metrical linear Riemannian con-nection R L , which in terms of local coordinates { x i } introduced in M is given by R L = { L mj , L mij } : L mj = − L mji y i , L mij = a mij + S mij , (II.1.2)where a mij = a mij ( x ) stands for the Christoffel symbols constructed from the Rieman-nian metric tensor a mn ( x ) of the space R N and S mij = S mij ( x ) is an arbitrary torsiontensor : S mij = − S jim with S mij = a mh S hij . The respective covariant derivative ∇ can beintroduced in the natural way. Namely, considering the (1,1)-type tensor W nm ( x, y ) on thetangent bundle associated to the space R N , we can take the definition ∇ i W nm = d Riem i W nm + L nhi W hm − L hmi W nh , (II.1.3)which involves the action of the operator d Riem i = ∂∂x i + L ki ∂∂y k . (II.1.4)In the tangent Riemannian space R { x } we can construct from the metric tensor g ij = g ij ( x, y ) the curvature tensor b R { x } = { b R nmij ( x, y ) } by the help of the ordinaryRiemannian method, regarding { y i } as variables. Namely, we obtain the representation b R nmij = ∂C mni ∂y j − ∂C mnj ∂y i + C hni C mhj − C hnj C mhi . Since ∂C mni /∂y j − ∂C mnj /∂y i ≡ − (cid:0) C hni C mhj − C hnj C mhi (cid:1) , we have simply b R nmij = 1 F S nmij , where S nmij = (cid:0) C hnj C mhi − C hni C mhj (cid:1) F . The tensor S nmij describes the curvature of indicatrix (see Section 5.8 in [1]).We need the metrical non-linear Finsler connection F N , such that F N = { N mi , T mij } : N mi = N mi ( x, y ) , T mij = T mij ( x, y ) , (II.1.5)where the objects N mi ( x, y ) and T mij ( x, y ) are to depend on the variable y in an essentiallynon-linear way. The adjective “metrical” means that the action of the entailed covariantderivative on the Finsler metric function, and also on the Finsler metric tensor, yields2identically zero. The coefficients N mi and T mij are assumed to be positively homogeneousregarding the dependence on vectors y , respectively of degree 1 and degree 0.In the Riemannian limit of the Finsler space, the spaces R { x } are Euclidean spacesand the tensor g { x } ( y ) is independent of y . The conformally flat structure of the spaces R { x } can naturally be taken to treat as the next level of generality of the Finsler space.Can the metrical connection preserving the two-vector angle be introduced on that level?The deformation of the Riemannian space to the Finsler space proves to be theconvenient method of consideration to apply. Namely, when the Riemannian space canbe deformated to the Finsler space characterized by the conformally flat structure of thespaces R { x } the positive and clear answer to the above question can be arrived at. Therespective conformal multiplier is shown to be a power of the Finsler metric function.We shall evidence the phenomenon that the used non-linear deformation F N = C · R L (II.1.6)of the Riemannian connection yields the Finsler connection F N which preserves the Fins-lerian two-vector angle α { x } ( y , y ). II.2. Key observations
Below, any dimension N ≥ M be an N -dimensional C ∞ differentiable manifold, T x M denote the tangentspace to M at a point x ∈ M , and y ∈ T x M \ T M a Riemannian metric S . Denote by R N = ( M, S ) theobtained N -dimensional Riemannian space. Let additionally a Finsler metric function F be introduced on this T M , yielding a Finsler space F N = ( M, F ). We shall study theFinsler space F N can be specified according to the following definition.INPUT DEFINITION. The Finsler space F N under consideration is the deformatedRiemannian space R N : F N = C · R N , (II.2.1)specified by the condition that in each tangent space T x M the metric tensor g { x } ( y )produced by the Finsler metric is the C -transformation of the tensor which is conformal to the Euclidean metric tensor entailed by the Riemannian metric of the space R N . Itis assumed that the applied C -transformations (1.1) do not influence any point x ∈ M of the base manifold M and that they are sufficiently smooth and invertible. It is alsonatural to require that the C -transformations (1.1) send unit vectors to unit vectors: I F { x } = C · S { x } . (II.2.2)Additionally, we subject the C -transformation to the condition of positive homogeneitywith respect to tangent vectors y , denoting the degree of homogeneity by H .If f ( x, y ) is the involved conformal multiplier in the tangent Riemannian space R { x } ,then the equality g { x } ( y ) = f ( x, y ) u { x } ( y ) should introduce the tensor u { x } ( y ) which asso-ciated Riemannian curvature tensor vanishes identically. The function f ( x, y ) is assumednaturally to be homogeneous with respect to the argument y . Denoting the homogeneitydegree of f ( x, y ) by means of 2 a ( x ), we just conclude that the difference 1 − a is exactlythe homogeneity degree of the transformation (2.1) considered, that is, H = 1 − a. The following proposition is valid.3
Proposition II.2.1.
A Finsler space is the F N -space if and only if the indicatrix ofthe Finsler space is a space of constant curvature. The dependence of the multiplier f onthe variable y proves to be presented by the power of the Finsler metric function F , suchthat g { x } ( y ) = p u { x } ( y ) , p = c ( x ) ( F ( x, y )) a ( x ) , c ( x ) > . (II.2.3) The equality C Ind. = H ensues. The proposition is of the local meaning in both the base manifold and the tangentspace.
Proof . Given an arbitrary Finsler space of any dimension N ≥
3. The tangentRiemannian space R { x } is conformally flat if and only if the indicatrix is a space ofconstant curvature. Indeed, in dimensions N ≥ W ijmn vanishes identically. By evaluating the tensorand considering the direct implications of the contraction vanishing W ijmn l n l j = 0 , weimmediately obtain the representation S nmij = C ( h nj h mi − h ni h mj ) which is characteristicof the constancy of the indicatrix curvature. In the dimension N ≥
3, the conformalflatness of the space R { x } is tantamount to the identical vanishing of the respective Cotton-York tensor. Considering the vanishing attentively leads again to the representation S nmij = C ( x )( h nj h mi − h ni h mj ). These observations prove the first part of PropositionII.2.1. All the involved computations are explicitly represented in Appendix B.To get the required conclusions concerning the form of the respective conformalmultiplier we can start with the tensor u ij = z ( x, y )( c ( x )) − F − a ( x ) g ij , where z is a testsmooth positive function homogeneous of the degree zero with respect to the argument y . We evaluate the respective curvature tensor e R { x } and assume e R { x } = 0 to determinethe tensor S nmij = (cid:0) C hnj C mhi − C hni C mhj (cid:1) F . After that, we consider the implicationsof the vanishing S nmij l m l j = 0 and arrive at the representation S nmij = a (2 − a )( h nj h mi − h ni h mj ) + F z (cid:0) z h g hs z s (cid:1) ( h nj h mi − h ni h mj )+ a − z (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni ) + l n ( z i h mj − z j h mi ) − l m ( z i h jn − z j h in ) (cid:17) F, where z k = ∂z/∂y k . The tensor S nmij must obviously possess the property S nmij l i = 0 . Therefore, we must fulfill the equation ( a − (cid:0) z n h mj − z m h nj (cid:1) = 0 . Because of a = 1,we can take only z n = 0, which means that the function z is independent of y . Withoutany loss of generality we can take z = 1. Thus we have proved the second part inProposition II.2.1. From the above representation of the tensor S nmij we just obtain C Ind. = 1 − a (2 − a ) ≡ (1 − a ) . Since the difference 1 − a is equal to H , the identification C Ind. = H is valid. All the computations which are required to trace the validity of theformulas exposed can be found in Appendix C. Proposition II.2.1 is valid. To have theequality S ( x, ¯ y ) = ( F ( x, y )) H ( x ) , we make the choice c = 1 /H .Let the C -transformation (I.2.1) proposed in Chapter I be assigned locally by meansof the differentiable functions ¯ y m = ¯ y m ( x, y ) , (II.2.4)subject to the required homogeneity¯ y m ( x, ky ) = k H ¯ y m ( x, y ) , k > , ∀ y. (II.2.5)4This entails the identity ¯ y mk y k = H ¯ y m , (II.2.6)where ¯ y mk = ∂ ¯ y m /∂y k . Fulfilling (2.1) means locally g mn ( x, y ) = c ij ( x, ¯ y )¯ y im ¯ y jn , c ij ( x, ¯ y ) = ( p ( x, y )) a ij ( x ) . (II.2.7)If we contract this tensor by y m y n and use the homogeneity identity (2.6), we obtain theequality p ( x, y ) = 1 H ( x ) F ( x, y ) S ( x, ¯ y ) . (II.2.8)On every punctured tangent space T x M \
0, the Finsler metric function F is assumedto be positive, and also positively homogeneous of degree 1: F ( x, ky ) = kF ( x, y ) , k > , ∀ y. The entailed Finsler metric tensor is positively homogeneous of degree 0. Therefore, tocomply the representation (2.7) with the stipulation (2.3), we must put H = 1 − a. (II.2.9)With this observation, comparing (2.3) with (2.8) yields the equality c S = 1 H F H . (II.2.10)To comply with the indicatrix correspondence (2.2), we should put c = 1 /H , which leadsto the equality S = F H indicated in (I.2.3).Denote by y i = y i ( x, t ) , t n ≡ ¯ y n , (II.2.11)the inverse transformation, so that y i ( x, kt ) = k /H y i ( x, t ) , k > , ∀ t, and y in t n = 1 H y i , (II.2.12)where y in = ∂y i /∂t n . The inverse to (2.7) reads: g kh y km y hn = c mn . (II.2.13)The following useful relations can readily be arrived at: y m y mn = F HS t n ≡ H F − H ) t n , t n = a nh t h , (II.2.14)and y m y mnl t lj + g mj y mn = 2 (cid:18) H − (cid:19) F − H y j t n + 1 H F − H ) a nh t hj , t lj = ¯ y lj and y mnl = ∂y mn /∂y l . Alternatively, t h t hn = HS F y n ≡ HF H − y n (II.2.15)and t h t hnu y ui + a hi t hn = 2( H − F − t i y n + HF H − g nu y ui , (II.2.16)where t hnu = ∂t hn /∂y u . We may also write t h t hni = H (1 − H ) F H − ( g ni − l n l i ) . (II.2.17)From (2.13) it follows that g nm y mi = p t jn a ij , y khp t pn = − y kp t pnv y vh . Differentiating (2.7) with respect to y k yields the following representation for thetensor C mnk = (1 / ∂g mn /∂y k :2 C mnk = (1 − H ) 2 F l k g mn + p ( t imk t jn + t im t jnk ) a ij . (II.2.18)Contracting this tensor by y n results in the equality p t imk t j a ij = (cid:18) H − (cid:19) ( h km − l k l m ) , (II.2.19)where the vanishing C mnk y n = 0 and the homogeneity identity (2.6) have been taken intoaccount.Symmetry of the tensor C mnk demands(1 − H ) 2 F ( l k g mn − l m g kn ) + p ( t im t jnk − t ik t jnm ) a ij = 0 , (II.2.20)so that we may alternatively write C mnk = (1 − H ) 1 F ( l k g mn + l n g mk − l m g nk ) + p t im t jnk a ij . (II.2.21)Contracting the last tensor by g nk yields2 C m = (1 − H ) 2 F l m + g nk p ( t ink t jm + t in t jmk ) a ij ≡ C mnk g nk , from which it ensues that2 C m = (1 − H ) 2 F l m + 2 g nk p t ink t jm a ij + g nk p ( t in t jmk − t im t jnk ) a ij , or 2 C m = (1 − H ) 2 F l m + 2 g nk p t ink t jm a ij − (1 − H ) g nk F ( l m g nk − l n g mk ) a ij . It is also convenient to use the representation6
F C m = − ( N − − H ) l m + F g nk p t ink t jm a ij . (II.2.22)Since y mi = p t jn a ij g nm , we can write F C m = − ( N − − H ) l m + F g nk t ink y mi . (II.2.23)The space F N is obtainable from the Riemannian space R N by means of the defor-mation which, owing to (2.7), can be presented by the conformal deformation tensor C im := p ¯ y im , (II.2.24)so that g mn = C im C jn a ij . (II.2.25)The zero-degree homogeneity C im ( x, ky ) = C im ( x, y ) , k > , ∀ y, (II.2.26)holds, together with C im ( x, y ) y m = ( F ( x, y )) − H ¯ y i . (II.2.27)The indicatrix correspondence (2.2) is a direct implication of the equality S = F H .We may apply the transformation (1.1) to the unit vectors: l = C · L : l i = y i ( x, L ); L = C − · l : L i = t i ( x, l ) , (II.2.28)where l i = y i /F ( x, y ) and L i = t i /S ( x, t ) are components of the respective Finslerian andRiemannian unit vectors, which possess the properties F ( x, l ) = 1 and S ( x, L ) = 1. Wehave L m = t m ( x, l ). On the other hand, from (2.7) it just follows that g mn ( x, l ) = 1 H a ij ( x ) t im ( x, l ) t jn ( x, l ) , (II.2.29)so that under the transformation (2.28) we have g mn ( x, l ) dl m dl n = 1 H a ij ( x ) dL i dL j . (II.2.30) Note.
The deformation performed by the formulas (2.24) and (2.25) is unholonomic ,in the sense that ∂C im ∂y n − ∂C in ∂y m = 0 . (II.2.31)The vanishing appears if only the factor p = F − H /H is independent of the vectors y ,that is, when H = 1 (which is the Riemannian case proper). Regarding the y -dependence,the tensor C im is homogeneous of degree zero, in accordance with (2.26). If we divide thetensor by p , we obtain from (2.24) the tensor ¯ y im which is the derivative tensor, namely¯ y im = ∂ ¯ y i /∂y m . However, such a property cannot be addressed to the tensor C im . It isthe reason why we start with the stipulation that the underlined transformation (whichis downloaded locally by the formulas (2.4)-(2.7)) be homogeneous of the degree H withrespect to the variable y . By proceeding in this way, it proves possible to come to the7conformal representation (2.30) of g mn ( x, l ) dl m dl n which is of the key significance to obtainthe angle and the connection coefficients.No support vector enters the right-hand part of (2.30). Therefore, any two nonzerotangent vectors y , y ∈ T x M in a fixed tangent space T x M form the F N -space angle α { x } ( y , y ) = 1 H ( x ) arccos λ, (II.2.32)where the scalar λ = a mn ( x ) t m t n S S , with t m = t m ( x, y ) and t m = t m ( x, y ) , (II.2.33)is of the entire Riemannian meaning in the space R N ; the notation S = p a mn ( x ) t m t n and S = p a mn ( x ) t m t n has been used.From (2.33) it follows that ∂λ∂x i = a mn,i t m t n S S + 1 S S a mn (cid:18) ∂t m ∂x i t n + t m ∂t n ∂x i (cid:19) − λ (cid:20) S S (cid:18) a mn,i t m t n + 2 a mn ∂t m ∂x i t n (cid:19) + 1 S S (cid:18) a mn,i t m t n + 2 a mn ∂t m ∂x i t n (cid:19)(cid:21) , where a mn,i = ∂a mn /∂x i , and ∂λ∂y k = (cid:20) a mn t n S S − a mn t n S S λ (cid:21) t m k , ∂λ∂y k = (cid:20) a mn t n S S − a mn t n S S λ (cid:21) t m k . When the recurrent preservation d i α + (1 /H ) H i α = 0proposed by (I.1.28) is applied to the angle given in (2.32), we obtain simply d i λ = 0 , (II.2.34)where d i is the separable operator (I.1.11). That is, the recurrent preservation law formu-lated for the Finsler F N -space angle (2.32) is tantamount to the separable preservationlaw (2.34) for the Euclidean angle arccos λ .We note also that A k ∂λ∂y k = F g nh t inh y ki ∂λ∂y k = F g nh t inh (cid:20) t i S S − t i S S λ (cid:21) . II.3. Derivation and properties of the coefficients N mn in the F N -space Let us start from (2.11) and introduce the vector U i = U i ( x, y ) according to U i def = 1 S t i ≡ F H t i , (II.3.1)8which is obviously unit: U i U i = 1 , U i = a ij U j . (II.3.2)The zero-degree homogeneity U i ( x, ky ) = U i ( x, y ) , k > , ∀ y, (II.3.3)holds, entailing the identity U in y n = 0 , (II.3.4)where U in := ∂U i ∂y n = 1 F H t in − F HU i l n . (II.3.5)From (2.14) it follows that F H U hs y kh = h ks , F H U ik y kt = δ it − U i U t , U i U in = 0 . (II.3.6)The vanishing U i (cid:18) ∂U i ∂x n + L ikn U k (cid:19) = 0 (II.3.7)holds obviously, where L ikn = L ikn ( x ) are the Riemannian connection coefficients ap-peared in (1.2).The representation (2.33) takes on the simple form λ = a mn ( x ) U m U n , (II.3.8)with U m = U m ( x, y ) , U m = U m ( x, y ) . (II.3.9)The form of the rght-hand part in the formula (3.8) which represents the scalar λ is such that the preservation law d i λ = 0 written in (2.34) is obviously equivalent to thevanishing D n U i = 0 (II.3.10)for the field U i = U i ( x, y ), with the covariant derivative D n U i := d n U i + L ink U k . (II.3.11)Since d n U i = ∂U i ∂x n + N kn U ik , we obtain the representation N mn = − y mi F H (cid:18) HF U i ∂F∂x n + ∂U i ∂x n + L ink U k (cid:19) + l m d n F (II.3.12)which was indicated in (I.2.15).9We have arrived at the following proposition. Proposition II.3.1.
Given an arbitrary smooth function H ( x ) , the angle preserva-tion equation d n α + (1 /H ) H n α = 0 in the F N -space entails the representation (3.12) forthe coefficients N mn . By differentiating (3.10) with respect to y m we may conclude that the covariantderivative D n U im := d n U im − D hnm U ih + L inl U lm , D hnm = − N hnm , (II.3.13)vanishes identically: D n U im = 0 . (II.3.14)Below, we shall assume that d n F = 0 . Using t i = F H U i together with D n t i := d n t i + L ikn t k , (II.3.15)from (3.10) we find D n t i = t i H n ln F. (II.3.16)Differentiating (3.16) with respect to y m leads to the conclusion that the covariantderivative D n t im := d n t im − D hnm t ih + L inl t lm (II.3.17)possesses the property D n t im = (cid:18) t im ln F + t i l m F (cid:19) H n . (II.3.18)With p = (1 /H ) F − H from (3.18) we get D n ( pt im ) = pt i l m F H n − H H n pt im , (II.3.19)so that, D n ( pt im ) = − pH H n h km t ik . (II.3.20)Consider (2.7): g mn ( x, y ) = p t im t jn a ij ( x ) . We obtain D k g mn = p (cid:18) t i l m F − H t im (cid:19) H k t jn a ij + p t im (cid:18) t j l n F − H t jn (cid:19) H k a ij . Using t h t hn = HF H − y n (see (2.34)) leads to D k g mn = − H H k g mn + p l m H k HF H − l n + p t im t j l n F H k a ij . In this way we arrive at the following result after the direct evaluations performed.0
Proposition II.3.2.
Given an arbitrary smooth function H ( x ) in the F N -space,the angle preservation d n α + (1 /H ) H n α = 0 taken in conjunction with the preservation d n F = 0 of the metric function entails that the covariant derivative of the metric tensorreads D k g mn = − H H k h mn . (II.3.21)Now, we contract (3.19) by y mk , getting y mk D n t im = y mk d n t im − D hnm y mk t ih + L ink = (cid:18) δ ik ln F + t i t k HF H (cid:19) H n . Since y nk t kj = δ nj , the previous identity can be transformed to t im d n y mk + D hnm y mk t ih − L ink = − (cid:18) δ ik ln F + t i t k HF H (cid:19) H n . Contract this equality by y ji , obtaining the equality d n y jk + D jnm y mk − L ink y ji = − (cid:18) y jk ln F + y j t k H F H (cid:19) H n , (II.3.22)which can be written simply as d n (cid:18) p y jk (cid:19) + T jnm y mk p − L ink y ji p = 0 , (II.3.23)where T jnm are the coefficients introduced in (I.1.33). Taking into account the represen-tations (3.15)-(3.16) together with the identity t m ∂ p y jk ∂t m = 0ensuing from the homogeneity, the equality (3.23) becomes d Riem n (cid:18) p y jk (cid:19) + T jnm y mk p − L ink y ji p = 0 . (II.3.24)We have used the Riemannian operator d Riem n introduced in (II.1.4).We know that d Riem n t k + L knm t m = 0 . (II.3.25)Therefore, contracting (3.24) by t k yields d Riem n (cid:18) Hp y j (cid:19) − N j n Hp = 0 . (II.3.26)1Here we have Hp = F − H = S /H S , (II.3.27)so that d Riem n Hp = 1 Hp H H n ln S ≡ Hp H H n ln F. (II.3.28)We arrive at the following proposition. Proposition II.3.3.
With an arbitrary smooth function H ( x ), in the F N - spacewith d n F = 0 the representation N mn = d Riem n y m ( x, t ) + 1 H H n y m ln F (II.3.29) written by the help of the Riemannian operator d Riem n is valid. The derivative coefficients N kmn = ∂N km /∂y n can straightforwardly be evaluatedfrom the coefficients N km written in (I.2.16). We obtain N kmn = − F h kn ∂F∂x m − l k ∂l n ∂x m − (cid:18) y khp t pn + HF y kh l n (cid:19) F H (cid:18) ∂U h ∂x m + L hms U s (cid:19) − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) . Owing to (3.10)), we have ∂U h ∂x m + L hms U s = − N sm U hs , so that using (3.6) we observe that the coefficients N kmn are equal to − F h kn ∂F∂x m − l k ∂l n ∂x m + U hs y khp t pn F H N sm + HF y kh l n F H N sm U hs − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) , or N kmn = − F h kn ∂F∂x m − l k ∂l n ∂x m − h vs y kj t jnv N sm + HF h ks l n N sm − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) , where the relation F H U hs y khp t pn = − F H U hs y kp t pnv y vh = − h vs y kp t pnv has been used.From (2.21) we have p t im t jnk a ij = C mnk − (1 − H ) 1 F ( l k g mn + l n g mk − l m g nk ) , g mv y vj t jnk = C mnk − (1 − H ) 1 F ( l k g mn + l n g mk − l m g nk ) . We obtain y kj t jnv = C knv − (1 − H ) 1 F ( l v δ kn + l n δ kv − l k g nv ) (II.3.30)and h vs y kj t jnv = C kns − (1 − H ) 1 F ( l n h ks − l k h ns ) . (II.3.31)In this way we come to the representation N kmn = − F h kn ∂F∂x m − l k ∂l n ∂x m − (cid:18) C kns − (1 − H ) 1 F ( l n h ks − l k h ns ) (cid:19) N sm + HF h ks l n N sm − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) . The eventual result reads N kmn = − F h kn ∂F∂x m − l k ∂l n ∂x m − C kns N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm − y kh F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) . (II.3.32)Thus we can formulate the following assertion. Proposition II.3.4.
With an arbitrary smooth function H ( x ), in the F N - spacewith d n F = 0 the coefficients N kmn can be given by means of the explicit representationwritten in (3.32).We are also able to evaluate the entailed coefficients N kmni = ∂N kmn /∂y i . Therequired evaluations which have been presented in detail in Appendix D lead straightfor-wardly to the representation N kmni = 2 H H m F l k h ni − D m C kni . (II.3.33)Thus we can formulate the following assertion. Proposition II.3.5.
Given an arbitrary smooth function H ( x ), in the F N - spacewith d n F = 0 the coefficients N kmni admit the simple representation (3.33) in terms ofthe covariant derivative of the tensor C kni . II.4. Properties of covariant derivative
The equality (3.20) can be written in the form T i ( pt mn ) = 0 (II.4.1)3with T i ( pt mn ) = d i ( pt mn ) − T hin pt mh + L mil pt ln , (II.4.2)where T hin are the coefficients (I.1.24). If we contract the last vanishing by y n and notethat T i y n = 0 (see (I.1.37)), we get T i ( Hpt m ) = 0 , (II.4.3)where T i ( Hpt m ) = d i ( Hpt m ) + L mik Hpt k . (II.4.4)We may write T i ( Hpt m ) = t m d i ( Hp ) + Hp D i t m , (II.4.5)where D i t m = d i t m + L mik t k .Owing to the equality C mn = pt mn (see (2.24)), from (4.1) we are entitled to formulatethe following proposition. Proposition II.4.1.
The C -deformation is T -covariant constant: T · C = 0 . (II.4.6)In terms of local coordinates the previous vanishing reads T n C mk = 0 , (II.4.7)where T n C mk = d n C mk − T hnk C mh + L mnl C lk . (II.4.8)The reciprocal coefficients e C nm = 1 p y nm (II.4.9)fulfills the similar vanishing T n e C mk = 0 , (II.4.10)where T n e C mk = d Riem n e C mk + T mnh e C hk − L ink e C mi (II.4.11)(see (3.23)).Let us realize the action of the C -transformation (2.1)-(2.2) on tensors by the helpof the deformaton { w ( x, y ) } = C · { W ( x, t ) } , (II.4.12)assuming that the tensors { w ( x, y ) } are positively homogeneous of degree 0 with respect4to the variable y , and that the tensors { W ( x, t ) } are positively homogeneous of degree 0with respect to the variable t . Namely, in the scalar case we use the identification w ( x, y ) = W ( x, t ) , (II.4.13)obtaining merely d i w = d Riem i W (II.4.14)(because of the vanishing D i U j = 0 indicated in (3.10)-(3.11)), where d Riem i is the operatordefined by (II.1.4). Given a tensor w n ( x, y ) of the type (0,1) we use the transformation w n = C mn W m . (II.4.15)The metrical linear connection R L introduced by (1.2) may be used to define the covariantderivative ∇ in R N according to the conventional rule: ∇ i W m = ∂W m ∂x i + L ki ∂W m ∂t k − L him W h , L kj = − L kij t i , (II.4.16)which can be written shortly ∇ i W m = d Riem i W m − L him W h . (II.4.17)We have ∇ i S = 0 , ∇ i t j = 0 , ∇ i a mn = 0 . (II.4.18)By virtue of the nullification T i C mn = 0 shown in (4.7), we obtain the transitivityproperty T i w n = C mn ∇ i W m (II.4.19)for the covariant derivatives.The method can be repeated in case of the covariant vectors w n ( x, y ) and W n ( x, t ),namely we write w n = e C nm W m , (II.4.20)obtaining T i w n = e C nm ∇ i W m , (II.4.21)where the reciprocal coefficients e C nm = (1 /p ) y nm defined by (4.9) have been arisen.The method can also be extended to more general tensors in a direct manner. For ex-ample, considering the (1,1)-type tensors { w nm ( x, y ) , W nm ( x, t ) } of the zero-degree positivehomogeneity with respect to the variables y and t , we can use the covariant derivative ∇ i W nm = ∂W nm ∂x i + L ki ∂W nm ∂t k + L nhi W hm − L hmi W nh ≡ d Riem i W nm + L nhi W hm − L hmi W nh (II.4.22)and the deformation w nm = e C nh C km W hk (II.4.23)to obtain the transitivity property T i w nm = e C nh C km ∇ i W hk (II.4.24)for the covariant derivatives T i and ∇ i .5Now we may formulate the following proposition. Proposition II.4.2.
The covariant derivative T is the manifestation of the transi-tivity of the connection under the C -transformation. In short, T = C · ∇ . (II.4.25) II.5. Entailed curvature tensor
Henceforth, the torsion tensor S mij (entered the initial connection (1.2)) is not ac-counted for.Given a tensor w nk = w nk ( x, y ) of the tensorial type (1,1), commuting the covariantderivative T i w nk := d i w nk + T nih w hk − T hik w nh (II.5.1)yields the equality[ T i T j − T j T i ] w nk = M hij ∂w nk ∂y h − E khij w nh + E hnij w hk (II.5.2)with the tensors M nij := d i N nj − d j N ni (II.5.3)and E knij := d i T njk − d j T nik + T mjk T nim − T mik T njm . (II.5.4)By applying the commutation rule (5.2) to the particular choices { F, y n , y k , g nk } andnoting the vanishing {T i F = T i y n = T i y k = T i g nk = 0 } , we obtain the identities y n M nij = 0 , y k E knij = − M nij , y n E knij = M kij , (II.5.5)and E mnij + E nmij = 2 C mnh M hij with C mnh = 12 ∂g mn ∂y h . (II.5.6)It proves pertinent to replace in the commutator (5.2) the partial derivative ∂w nk /∂y h by the definition S h w nk := ∂w nk ∂y h + C nhs w sk − C mhk w nm (II.5.7)which has the meaning of the covariant derivative in the tangent space supported by thepoint x ∈ M . In particular, S h g nk := ∂g nk ∂y h − C mhn g mk − C mhk g nm = 0 . With the curvature tensor ρ knij := E knij − M hij C nhk , (II.5.8)the commutator (5.2) takes on the form( T i T j − T j T i ) w nk = M hij S h w nk − ρ khij w nh + ρ hnij w hk . (II.5.9)6We denote ρ knij = g mn ρ kmij . The skew-symmetry ρ mnij = − ρ nmij (II.5.10)holds (cf. (5.6)).The equalities y k ρ knij = − M nij , y n ρ knij = M kij , (II.5.11)obviously hold.Let us evaluate the tensor M nij , using the coefficients N nj = − l n ∂F∂x j − y nh F H (cid:18) ∂U h ∂x j + L hkj U k (cid:19) indicated in (I.2.16).We directly obtain M nij = − F N ni ∂F∂x j + 1 F N nj ∂F∂x i − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i − d i (cid:0) y nh F H (cid:1) (cid:18) ∂U h ∂x j + L hkj U k (cid:19) + d j (cid:0) y nh F H (cid:1) (cid:18) ∂U h ∂x i + L hki U k (cid:19) − y nh F H (cid:20)(cid:18) ∂L hkj ∂x i − ∂L hki ∂x j (cid:19) U k + L hkj ∂U k ∂x i − L hki ∂U k ∂x j (cid:21) − y nh F H (cid:20) N si (cid:18) ∂U hs ∂x j + L hkj U ks (cid:19) − N sj (cid:18) ∂U hs ∂x i + L hki U ks (cid:19)(cid:21) . Using here the Riemannian curvature tensor a khij = ∂L hkj ∂x i − ∂L hki ∂x j + L ukj L hui − L uki L huj (II.5.12)leads to M nij = − F N ni ∂F∂x j + 1 F N nj ∂F∂x i − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i − d i (cid:0) y nh F H (cid:1) (cid:18) ∂U h ∂x j + L hkj U k (cid:19) + d j (cid:0) y nh F H (cid:1) (cid:18) ∂U h ∂x i + L hki U k (cid:19) − y nh F H (cid:20)(cid:0) a khij − L ukj L hui + L uki L huj (cid:1) U k + L hkj ∂U k ∂x i − L hki ∂U k ∂x j (cid:21) − y nh F H (cid:2) N si (cid:0) d j U hs + L hkj U ks (cid:1) − N sj (cid:0) d i U hs + L hki U ks (cid:1)(cid:3) . d i U h + L his U s = 0 (see (3.10)-(3.11)), getting M nij = − F N ni ∂F∂x j + 1 F N nj ∂F∂x i − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i + d i (cid:0) y nh F H (cid:1) N tj U ht − d j (cid:0) y nh F H (cid:1) N ti U ht − y nh F H (cid:0) a khij U k − L hkj N ti U kt + L hki N tj U kt (cid:1) − y nh F H (cid:2) N ti (cid:0) d j U ht + L hkj U kt (cid:1) − N tj (cid:0) d i U ht + L hki U kt (cid:1)(cid:3) . Taking into account the equality F H U ht y kh = h kt (see (3.6)), we arrive at the representation M nij = − F N ni ∂F∂x j + 1 F N nj ∂F∂x i − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i − y nh F H a khij U k − N ti d j h nt + N tj d j h nt , which can readily be simplified to read M nij = 1 F N ni N sj l s − F N nj N si l s − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i − y nh F H a khij U k + N si d j ( l n l s ) − N sj d j ( l n l s )= − l n N si ∂l s ∂x j + l n N sj ∂l s ∂x i − y nh F H a khij U k + l n N si d j l s − l n N sj d j l s . The eventual result is M nij = − y nt t h a htij . (II.5.13)Next, we use the equaity T i T j w nm = e C nh C km ∇ i ∇ j W hk (see (4.24)) to consider the relation[ T i T j − T j T i ] w nm = e C nh C km [ ∇ i ∇ j − ∇ j ∇ i ] W hk . ∇ i ∇ j − ∇ j ∇ i ] W nk = − t s a shij ∂W nk ∂t h − a khij W nh + a hnij W hk (II.5.14)the Riemannian curvature tensor a shij is constructed in accordance with the ordinary rule(5.12). Whence, M hij ∂w nm ∂y h − E mhij w nh + E hnij w hm = e C nh C km (cid:20) − t s a srij ∂W hk ∂t r − a krij W hr + a rhij W rk (cid:21) . Using here the equality W hk = C hn e C mk w nm (taken from (4.23)) leads to M hij ∂w nm ∂y h − E mhij w nh + E hnij w hm = − t s a shij ∂w nm ∂t h + W hk t s a srij ∂y nh t km ∂t r − t km a krij y hr w nh + y ns a rsij t rh w hm Now we use here the representation (5.13) obtained for the tensor M hij . We are leftwith − E mhij w nh + E hnij w hm = t hu y vk w uv t s a srij ∂y nh t km ∂t r − t km a krij y hr w nh + y ns a rsij t rh w hm . In this way we obtain the explicit representation E knij = y nh t hkm M mij + y nm a hmij t hk . (II.5.15)From (5.8) and (5.15) it follows that ρ knij = (cid:0) y nh t hkm − C nmk (cid:1) M mij + y nm a hmij t hk . Inserting here the tensor C nmk taken from (2.21) and noting the vanishing l m M mij = 0(see (5.5)), we get ρ knij = (cid:18) y nh t hkm − (1 − H ) 1 F ( l k δ nm + l n g mk ) − p t lm t hrk a lh g nr (cid:19) M mij + y nm a hmij t hk . Let us lower here the index n and use the equality g nm y mi = p t jn a ij (see the formulasbelow (2.17)). This yields ρ knij = (cid:18) p t ln t hkm a lh − (1 − H ) 1 F ( l k g mn + l n g mk ) − p t lm t hnk a lh (cid:19) M mij + p a ml t ln a hmij t hk . Next, we use here the skew-symmetry relation (2.20), obtaining ρ knij = (cid:18) (1 − H ) 2 F ( l n g mk − l m g kn ) − (1 − H ) 1 F ( l k g mn + l n g mk ) (cid:19) M mij + p a ml t ln a hmij t hk , ρ knij = − (1 − H ) 1 F ( l k M nij − l n M kij ) + p a hlij t hk t ln , (II.5.16)where a hlij = a lr a hrij . Finally, we return the index n to the upper position, arriving at ρ knij = − (1 − H ) 1 F ( l k δ nm − l n g mk ) M mij + y nm a hmij t hk . (II.5.17)The totally contravariant components ρ knij := g pk a mi a nj ρ pnmn read ρ knij = − (1 − H ) 1 F ( l k M nij − l n M kij ) + 1 p y kh y nr a hrij , (II.5.18)where a hrij = a hl a mi a nj a lrmn and M mij := a hi a nj M mhn . Similarly, we can conclude from (5.13) that the tensor M nij := g nm M mij can begiven by means of the representation M nij = − p t h t mn a hmij . (II.5.19)Squaring yields M nij M nij = p t l a lnij t h a hnij . (II.5.20)Now we square the ρ -tensor: ρ knij ρ knij = (1 − H ) F M nij M nij − − H ) 1 F ( l k M nij − l n M kij ) p a hlij t hk t ln + a knij a knij = (1 − H ) F M nij M nij − − H ) H F p ( a hlij t h t ln M nij − a hlij t hk t l M kij ) + a knij a knij , or ρ knij ρ knij = (1 − H ) p F t l a lnij t h a hnij +2(1 − H ) Hp F ( a hlij t h t r a rlij − a hlij t l t r a rhij )+ a knij a knij , which is ρ knij ρ knij = a knij a knij + 2 S (cid:18) H − (cid:19) t l a lnij t h a hnij . (II.5.21)Because of the nullifications T i (cid:18) p y nm (cid:19) = 0 , T i ( Hpt m ) = 0(see (4.3) and (4.10)), from (5.13) it follows that T l M nij = − y nt t h (cid:18) ∇ l − H H l (cid:19) a htij . (II.5.22)0From (5.17) we can conclude that T l ρ knij = (1 − H ) 1 F ( l k δ nm − l n g mk ) y mt t h (cid:18) ∇ l − H H l (cid:19) a htij + y nm t hk ∇ l a hmij + H l F ( l k δ nm − l n g mk ) M mij . (II.5.23)The covariant derivatives T k M nij = d k M nij + T nkt M tij − a ski M nsj − a skj M nis (II.5.24)and T l ρ knij = d l ρ knij + T nlt ρ ktij − T tlk ρ tnij − a sli ρ knsj − a slj ρ knis (II.5.25)have been used. Appendix A: Evaluations for Finsleroid connection coefficients with g = g ( x )Below we present various important evaluations which underlined the considerationperformed in Section I.3 of Chapter I.We shall use the relations ∂ bq∂y n = 2 BN Kgq A n (A.1)and ∂ q B∂y n = − q B q B (cid:20) BN Kq A n + 2 2 BN Kgq A n bq (cid:21) = − q B N K (cid:18) g bq (cid:19) A n , so that ∂ q B∂y n = − qB gN K (2 b + gq ) A n . (A.2)Moreover, ∂ bqB∂y n = − qB gN K (2 b + gq ) A n bq + q B BN Kgq A n and ∂ bqB∂y n = − bB (2 b + gq ) 2 gN K A n + 2 gN K A n . (A.3)1We shall also meet the convenience to apply the identity(2 b + gq ) (cid:18) q + 12 gb (cid:19) = 2 h bq + gB. (A.4)The equality ∂ ¯ M∂y n = 2 q B gN K A n (A.5)can be obtained from the relation ∂y n ∂g = ¯ M y n + 12 K ∂ ¯ M∂y n . (A.6)It follows that ∂g mn ∂g = ¯ M g mn + 2 q B gN K A m y n + y m q B gN K A n − qB gN (2 b + gq ) A m gN A n + q B gN (cid:20) − A m l n + 2 N A m A n − gN bq H mn (cid:21) , which is ∂g mn ∂g = ¯ M g mn + q B gN ( A m l n + A n l m ) − bqB gN gN A m A n − bqB h mn , (A.7)entailing ∂h mn ∂g = − bqB gN gN A m A n − bqB h mn . (A.8)We can also obtain ∂A mnj ∂g = 32 ¯ M A mnj + (cid:18) g − bqB (cid:19) A mnj − gbqB gN ( A m h nj + A n h mj + A j h mn ) − gbqB gN gN gN A j A m A n , or ∂A mnj ∂g = 32 ¯ M A mnj + (cid:18) g − bqB (cid:19) A mnj − gbqB gN gN gN A j A m A n . (A.9)Evaluations frequently involve the vector m i = (2 /N g ) A i which possesses the prop-erties g ij m i m j = 1 , y i m i = 0 . m i = K q ( b i − bK y i ) . (A.10)The equality K ∂m i ∂y n = − m n l i + gm n m i − bq H in (A.11)holds, where H in = h in − m i m n .The contravariant components m i can be taken from (A.27) of [7]: m i = 1 qK h q b i − ( b + gq ) v i i , (A.12)entailing K ∂m i ∂y n = − m n l i − gm i m n − q ( b + gq ) H in . (A.13)With the representation A ijk = 1 N (cid:20) A i h jk + A j h ik + A k h ij − N g A i A j A k (cid:21) (A.14)(see (A.8) in [7]), we find that ∂A m A k ∂y n = 1 K A k (cid:20) − A n l m − N g bq H mn (cid:21) + 1 K A m (cid:20) − A n l k − N g q ( b + gq ) H kn (cid:21) . (A.15)Recollecting the scalar h ( x ) = p − ( g ( x ) / G = g/h ,we get ∂h∂g = − G, ∂G∂g = 1 h , (A.16)so that ∂K ∂g = ¯ M K , (A.17)where ¯ M = bqB − h f + 12 GhB ( q + 12 gbq ) , or ¯ M = − h f + 12 GhB q + 1 h B bq. (A.18)The function K ( x, y ) is given by the formulas K ( x, y ) = p B ( x, y ) J ( x, y ) , J ( x, y ) = e − G ( x ) f ( x,y ) , (A.19)3entailing ∂ ln J∂y n = 1 N C n and ∂ ¯ M∂y n = 1 h gN K A n − gh qB gN K (2 b + gq ) A n + 1 h (cid:18) − bB (2 b + gq ) 2 gN K A n + 2 gN K A n (cid:19) , or ∂ ¯ M∂y n = (cid:18) − gq (cid:18) b + 12 gq (cid:19) − b (2 b + gq ) + 2 B (cid:19) h B gN K A n which is equivalent to (A.5).Starting with (I.3.22)-(I.3.23), we get˘ N kim = 1 h g i q B (cid:18) g bq (cid:19) N g A m l k + 1 h g i q B (cid:18) g bq (cid:19) q ( b + gq ) H km − h g i (cid:20) N N g A m A k + l m q B (cid:18) g bq (cid:19) N g A k − q ( b + gq ) 2 N g A m q B (cid:18) g bq (cid:19) N g A k (cid:21) − g i (cid:18) q B N g A m l k + 12 ¯ M δ km (cid:19) = 1 h g i q B (cid:18) g bq − h (cid:19) N g A m l k + 1 h g i B (cid:18) q + 12 gb (cid:19) ( b + gq ) h km − h g i B (cid:20) gB − (2 b + 2 gq ) (cid:18) q + 12 gb (cid:19)(cid:21) N g N g A m A k − g i ¯ M h km + 1 K l m ˘ N ki . Using here the equality (A.4) leads to˘ N kim = 1 h g i q B (cid:18) g bq − h (cid:19) N g A m l k + 1 h g i B (cid:18) q + 12 gb (cid:19) ( b + gq ) h km + 1 h g i B (cid:20) h bq + gq (cid:18) q + 12 gb (cid:19)(cid:21) N g N g A m A k − g i ¯ M h km + 1 K l m ˘ N ki . Eventually we obtain4˘ N kim = 1 h g i q B (cid:18) g bq − h (cid:19) N g A m l k + 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h km + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m A k − g i ¯ M h km + 1 K l m ˘ N ki . (A.20)Thus the representation (I.3.28) is valid.Next, we find that˘ N kimn = ∂ ˘ N kim ∂y n = − h g i qB gN K (2 b + gq ) A n (cid:18) g bq − h (cid:19) N g A m l k + 1 h g i g N Kg A n N g A m l k + 1 h g i q B (cid:18) g bq − h (cid:19) N g K (cid:20) − A n l m + 2 N A n A m − bq N g H mn (cid:21) l k + 1 h g i q B (cid:18) g bq − h (cid:19) N g A m K h kn − h g i qB gN K (2 b + gq ) A n (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h km + 1 h g i q B g BN Kgq A n (cid:18) bq + g (cid:19) h km + 1 h g i q B (cid:18) g bq (cid:19) BN Kgq A n h km − h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) K ( l k h mn + l m h kn ) − h g i qB gN K (2 b + gq ) A n (cid:18) bq + 12 g (cid:19) N g A m A k + 1 h g i q B BN Kgq A n N g N g A m A k − h g i q B (cid:18) bq + 12 g (cid:19) N g K (cid:20) A k (cid:18) A n l m + N g bq H mn (cid:19) + A m (cid:18) A n l k + N g q ( b + gq ) H kn (cid:19)(cid:21) − g i q B gN K A n h km + 12 g i ¯ M K ( l k h mn + l m h kn )+ 1 K h mn (cid:20) − h g i qB (cid:18) q + 12 gb (cid:19) KN g A k − g i ¯ M y k (cid:21) + 1 K l m h g i q B (cid:18) g bq − h (cid:19) N g A n l k + 1 K l m (cid:20) h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h kn + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g A n A k − g i ¯ M h kn (cid:21) . Simplifying yields˘ N kimn = − h g i bqB A n (cid:18) g bq − h (cid:19) N g K A m l k + 1 h g i g N Kg A n N g A m l k − h g i bq B (cid:18) g bq − h (cid:19) K h mn l k + 1 h g i bq B (cid:18) g bq − h (cid:19) K N g A m A n l k + 1 h g i q B (cid:18) g bq − h (cid:19) N g A m K h kn − h g i B (2 b + gq ) (cid:18) q + 12 gb (cid:19) (cid:18) bq + g (cid:19) gN K A n h km + 1 h g i g N Kg A n (cid:18) bq + g (cid:19) h km + 1 h g i (cid:18) g bq (cid:19) N Kg A n h km − h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) K l k h mn − h g i q B (2 b + gq ) (cid:18) bq + 12 g (cid:19) N g N g N g K A n A m A k + 1 h g i N Kg A n N g N g A m A k − h g i q B (cid:18) bq + 12 g (cid:19) N g N g K (cid:20) A k N g bq H mn + A m A n l k (cid:21) − h g i q B (cid:18) bq + 12 g (cid:19) N g K ( b + gq ) A m H kn − g i q B gN K A n h km − K h mn h g i qB (cid:18) q + 12 gb (cid:19) KN g A k , or ˘ N kimn = 1 h g i g N Kg A n N g A m l k − h g i bq B (cid:18) g bq − h (cid:19) K h mn l k − h g i bq B (cid:18) g bq − h (cid:19) K N g N g A m A n l k + 1 h g i q B (cid:18) g bq − h (cid:19) N g A m K h kn − h g i B (2 h bq + gB ) (cid:18) bq + g (cid:19) gN K A n h km + 1 h g i g N Kg A n (cid:18) bq + g (cid:19) h km + 1 h g i (cid:18) g bq (cid:19) N Kg A n h km − h g i q B (cid:18) g bq (cid:19) g K l k h mn − h g i bq B (cid:18) bq + 12 g (cid:19) N g N g N g K A n A m A k + 1 h g i N Kg A n N g N g A m A k − h g i q B (cid:18) bq + 12 g (cid:19) N g K A k bq H mn − h g i q B (cid:18) bq + 12 g (cid:19) N g N g K A m A n l k − h g i q B (cid:18) bq + 12 g (cid:19) N g K ( b + gq ) A m h kn − g i q B gN K A n h km − K h mn h g i qB (cid:18) q + 12 gb (cid:19) KN g A k . Additional reductions are possible, leading to7˘ N kimn = 1 h g i g N Kg A n N g A m l k − g h g i K h mn l k − h g i bq B (cid:18) g bq − h (cid:19) K N g N g A m A n l k + 1 h g i q B (cid:18) g bq − h (cid:19) N g A m K h kn − h g i (cid:18) h + g (cid:18) bq + g (cid:19)(cid:19) gN K A n h km + 1 h g i g N Kg A n (cid:18) bq + g (cid:19) h km + 1 h g i (cid:18) g bq (cid:19) N Kg A n h km + 1 h g i N Kg A n N g N g A m A k − h g i bq B (cid:18) bq + 12 g (cid:19) N g K A k h mn − h g i q B g N g N g K A m A n l k − h g i q B (cid:18) bq + 12 g (cid:19) N g K ( b + gq ) A m h kn − K h mn h g i qB (cid:18) q + 12 gb (cid:19) KN g A k , or simply˘ N kimn = − g h g i K h mn l k + 1 h g i B (cid:18) − q + 12 gbq + 12 g q (cid:19) N g K h kn A m − h g i (cid:18) g g bq (cid:19) gN K A n h km + 1 h g i g N K A n h km + 1 h g i (cid:18) g bq (cid:19) N Kg A n h km + 1 h g i N Kg A n N g N g A m A k − h g i B (cid:18) b + 12 gq (cid:19) ( b + gq ) 2 N g K A m h kn − h g i N g K h mn A k . (A.21)8Using here the representation (A.14) of the tensor A ijk , we are coming to y k ∂ ˘ N ki ∂y m ∂y n = 2 h h i h mn . (A.22)Let us verify the validity of the equality D i h nm = − h h i h nm . (A.23)To this end we find 2 ˘ N ki C kmn = − h g i qB (cid:18) q + 12 gb (cid:19) N g A k A kmn = − h g i qB (cid:18) q + 12 gb (cid:19) N g A k N (cid:20) A n h nk + A n h mk + A k h mn − N g A m A n A k (cid:21) , so that 2 ˘ N ki C kmn = − h g i qB (cid:18) q + 12 gb (cid:19) N g N (cid:20) A m A n + N g h mn (cid:21) . (A.24)We can also observe that g i ∂g mn ∂g + 2 ˘ N ki C kmn = g i ¯ M g mn + g i q B gN ( A m l n + A n l m ) − g i bqB N g A m A n − g i bqB h mn − h g i qB (cid:18) q + 12 gb (cid:19) (cid:20) N g N A m A n + g h mn (cid:21) . Simultaneously, g kn ˘ N kim = 1 h g i q B (cid:18) g bq − h (cid:19) N g A m l n + 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h nm + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m A n − g i ¯ M h mn + 1 K l m (cid:20) − h g i qB (cid:18) q + 12 gb (cid:19) KN g A n − g i ¯ M y n (cid:21) . In this way we obtain g i ∂g mn ∂g + 2 ˘ N ki C kmn + g km ˘ N kin + g kn ˘ N kim g i q B gN ( A m l n + A n l m ) − g i bqB gN gN A m A n − g i bqB h mn − h g i qB (cid:18) q + 12 gb (cid:19) (cid:20) N g N A m A n + g h mn (cid:21) + 1 h g i q B (cid:18) g bq − h (cid:19) N g ( A m l n + A n l m ) + 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h nm + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m A n − h g i qB (cid:18) q + 12 gb (cid:19) N g ( l m A n + l n A m ) . Reducing similar terms leads to g i ∂g mn ∂g + 2 ˘ N ki C kmn + g km ˘ N kin + g kn ˘ N kim = − g i bqB gN gN A m A n − g i bqB h mn − h g i qB (cid:18) q + 12 gb (cid:19) (cid:20) N g N A m A n + g h mn (cid:21) + 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) h nm + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m A n = − g i bqB h mn + 1 h g i B (2 h bq + gB ) h nm = 12 h gg i h nm . We get D i g nm = 12 h gg i h nm . (A.25)Thus the equality (A.23) is valid.Now we want to verify the validity of the equality (A.9). Differentiating (A.14) withrespect to y j yields 2 ∂C mnj ∂g = 2 ¯ M C mnj + 2 q B gN K A j g mn − qB gN K (2 b + gq ) A j gN ( A m l n + A n l m ) + 1 K q B gN ( A m h nj + A n h mj )0+ q B gN K (cid:20)(cid:18) − A j l m + 2 N A m A j − bq N g H mj (cid:19) l n + (cid:18) − A j l n + 2 N A n A j − bq N g H nj (cid:19) l m (cid:21) + (cid:18) bB (2 b + gq ) 2 gN K A j − gN K A j (cid:19) gN gN A m A n − bqB g N K (cid:20)(cid:18) − A j l m + 2 N A m A j − bq N g H mj (cid:19) A n + (cid:18) − A j l n + 2 N A n A j − bq N g H nj (cid:19) A m (cid:21) + (cid:18) bB (2 b + gq ) 2 gN K A j − gN K A j (cid:19) h mn + bqB K ( l m h jn + l n h mj ) − bqB C mnj . We may reduce as follows:2 ∂C mnj ∂g = 2 ¯ M C mnj + 2 q B gN K A j h mn + 1 K q B gN ( A m h nj + A n h mj )+ (cid:18) B − q + b B gN K − gN K (cid:19) gN gN A j A m A n − bqB gN gN K h(cid:18) N A m A j − bq N g h mj + bq N g A m A j (cid:19) A n + (cid:18) N A n A j − bq N g h nj + bq N g A n A j (cid:19) A m i + (cid:18) B − q + b B gN K − gN K (cid:19) A j h mn − bqB C mnj = 2 ¯ M C mnj + 1
K B − gbqB gN ( A m h nj + A n h mj + A j h mn ) − bqB C mnj + b − q B gN K gN gN A j A m A n − bqB gN gN K (cid:18) N A m A j + bq N g A m A j (cid:19) A n = 2 ¯ M K A mnj + 1
K B − gbqB gN ( A m h nj + A n h mj + A j h mn )1 − B + gbqB gN K gN gN A j A m A n − bqB C mnj = 2 ¯ M K A mnj + 2 K (cid:18) g − bqB (cid:19) A mnj − K gbqB gN ( A m h nj + A n h mj + A j h mn ) − gbqB gN K gN gN A j A m A n . Thus (A.9) is valid.Next, we evaluate the term˘ N ki ∂A jmn ∂y k = − h g i qB (cid:18) q + 12 gb (cid:19) KN g A k ∂A jmn ∂y k . With the representation ∂A ijk ∂y n = 1 K N ( A jkn A i + A ikn A j + A ijn A k ) − K ( l j A kni + l i A knj + l k A ijn )+ 1 K N N ( H jk A i A n + H ik A j A n + H ij A k A n ) − gb Kq ( H jk H in + H ik H jn + H ji H kn )(A.26)we obtain KA n ∂A ijk ∂y n = 2 N ( A jkn A i + A ikn A j + A ijn A k ) A n − ( l j A kni + l i A knj + l k A ijn ) A n + g H jk A i + H ik A j + H ij A k ) . Using here the equality A jkn A n = 1 N (cid:18) N g N g h jk + A j A k (cid:19) leads to KA n ∂A ijk ∂y n = 2 N N (cid:18) N g N g h jk A i + h ik A j + h ij A k ) + 3 A i A j A k (cid:19) − N N g N g h jk l i + h ik l j + h ij l k ) − N ( A j A k l i + A i A k l j + A i A j l k )2+ g h jk A i + h ik A j + h ij A k ) − g N g N g A i A j A k . So we may write KA n ∂A ijk ∂y n = − N N g N g h jk l i + h ik l j + h ij l k ) − N ( A j A k l i + A i A k l j + A i A j l k )+ g ( h jk A i + h ik A j + h ij A k ) . (A.27)We also need the term A jkn ˘ N kim = 1 h g i q B (cid:18) g bq (cid:19) (cid:18) bq + g (cid:19) A jmn + 1 h g i q B (cid:18) bq + 12 g (cid:19) N g N g A m N (cid:18) N g N g h jn + A j A n (cid:19) − g i ¯ M A jmn − l m h g i qB (cid:18) q + 12 gb (cid:19) N g N (cid:18) A j A n + N g h jn (cid:19) . Summing all the addends yields g i ∂A jmn ∂g + ˘ N ki ∂A jmn ∂y k + A jkm ˘ N kin + A jkn ˘ N kim + A mnk ˘ N kij = g i (cid:18) g − bqB (cid:19) A mnj − g i bqB N gN gN A j A m A n − h g i qB (cid:18) q + 12 gb (cid:19) N g " − N g h jm l n + h mn l j + h nj l m ) − N ( A j A m l n + A m A j l n + A n A j l m )+ g ( h mn A j + h mj A n + h nj A m ) +3 1 h g i B (cid:18) q + 12 gb (cid:19) ( b + gq ) A jmn + 1 h g i q B (cid:18) bq + 12 g (cid:19) N ( h mn A j + h mj A n + h nj A m ) + 3 h g i q B (cid:18) bq + 12 g (cid:19) N g A m A j A n − h g i qB (cid:18) q + 12 gb (cid:19) N g N ( A j A m l n + A m A j l n + A n A j l m ) − h g i qB (cid:18) q + 12 gb (cid:19) g h jm l n + h mn l j + h nj l m ) , or g i ∂A jmn ∂g + ˘ N ki ∂A jmn ∂y k + A jkm ˘ N kin + A jkn ˘ N kim + A mnk ˘ N kij = g i (cid:18) g − bqB (cid:19) A mnj − g i bqB N gN gN A j A m A n − h g i g qB (cid:18) q + 12 gb (cid:19) N g ( h mn A j + h mj A n + h nj A m )+ 1 h g i B (cid:20) h bq + 3 gB − b (cid:18) q + 32 gb (cid:19)(cid:21) A jmn + 1 h g i gq B (cid:16) b + g q (cid:17) N g ( h mn A j + h mj A n + h nj A m ) + 3 h g i gq B (cid:16) b + g q (cid:17) N g A m A j A n = g i g A jmn + 1 h g i B (cid:20) h bq + 3 gB − b (cid:18) q + 32 gb (cid:19)(cid:21) A jmn + 1 h g i q B (cid:18) b − h b − gq − g b (cid:19) A jmn = g i g A jmn + 34 h g i A jmn . By the help of such evaluations we eventually obtain g i ∂A jmn ∂g + ˘ N ki ∂A jmn ∂y k + A jkm ˘ N kin + A jkn ˘ N kim + A mnk ˘ N kij = g i gh A jmn + 1 h g i A jmn , (A.28)which shows that the representation˘ N kimn = − g h g i K h mn l k − gh g i K A kmn (A.29)indicated in (I.3.30) is valid.4The full coefficients N kimn = N I kimn + ˘ N kimn can be obtained on taking into ac-count the components (A.29) together with the representation N I kimn = − (1 /K ) D i A kmn obtainable in the ( g = const)-case (see [10,11]). The result reads N kimn = 2 h h i K l k h mn − K D i A kmn . (A.30)Thus the representation indicated in (I.3.31) is also valid. Appendix B: Conformal property of the tangent Riemannian space
Given an arbitrary Finsler space of any dimension N ≥
3. At any fixed point x ,the Riemannian curvature tensor b R { x } = { b R nmij ( x, y ) } of the tangent Riemannian space R { x } is given by means of the components b R nmij = 1 F S nmij , (B.1)where S nmij = (cid:0) C hnj C mhi − C hni C mhj (cid:1) F . (B.2)Let us construct the Weyl tensor W ijmn in the space R { x } , so that F W ijmn = S ijmn − N − S im g jn + S jn g im − S in g jm − S jm g in ) + 1( N − N −
2) ˘ S ( g im g jn − g in g jm ) , (B.3)where S ijmn = g jh S ijmn , S im = g jn S ijmn and ˘ S = g im S im . Contracting the tensor twotimes by the unit vector l n = (1 /F ) y n yields directly( N − F W ijmn l n l j = − S im + 1 N − Sh im , where h im = g im − (1 /F ) y i y m . Therefore, in any dimension N ≥ W ijmn =0 is tantamount to the representation S nmij = C ( h nj h mi − h ni h mj ) . (B.4)It is known (see Section 5.8 in [1]) that the indicatrix is a space of constant curvatureif and only if the tensor (B.2) fulfills the representation (B.4), in which case C = C ( x )(that is, the factor C is independent of y ). The respective indicatrix curvature value C Ind. is given by C Ind. = 1 − C. (B.5)Next, in the dimension N = 3 the tensor W ijmn vanishes identically and, therefore,the equality S ijmn = L ( h im h jn − h in h jm ) with L = 12 ˘ S (B.6)5holds, where L may depend on y . Taking S im = Lh im , we should examine the tensor C im := 1 F (cid:18) S im −
14 ˘ Sg im (cid:19) (B.7)of the Cotton-York type. Let us use the Riemannian covariant derivative S operative inthe space R { x } under consideration. Denoting L n = ∂L/∂y n and taking into account thevanishing S n g im = 0, we have S n C im − S m C in = 1 F (cid:18) L n h im − L m h in − L F ( l n h mi − l m h ni ) −
12 ( L n g im − L m g in ) (cid:19) + L F ( l n h im − l m h in ) , which is S n C im − S m C in = 1 F (cid:18) L n (cid:18) h im − g im (cid:19) − L m (cid:18) h in − g in (cid:19)(cid:19) , so that S n C im − S m C in = 0 (B.8)holds iff L n = 0, that is when ˘ S = ˘ S ( x ) . The vanishing (B.8) means the conformal flatnessof the three-dimensional space R { x } .Thus we are entitled to set forth the validity of the following proposition. Proposition . Given an arbitrary Finsler space of any dimension N ≥ . Thetangent Riemannian space R { x } is conformally flat if and only if the indicatrix is a spaceof constant curvature. The question arises: What is the form of the conformal multiplier of the space R { x } under study? See the next appendix. Appendix C: Multiplier for the tangent Riemannian space
To find the form of the conformal multiplier of the space R { x } under study, we canstart with the conformal tensor u ij = z ( x, y )( c ( x )) F a ( x ) g ij (C.1)(cf. (II.2.3)), where z is a test smooth positive function.Denoting u ijk = ∂u ij /∂y k , we get( c ) u ijk = 1 F a (cid:16) − z aF l k + z k (cid:17) g ij + 2 z F a C ijk , C ijk = (1 / ∂g ij /∂y k and z k = ∂z/∂y k .Constructing the coefficients Z ijk := 12 ( u kji + u iki − u ijk )leads to F a ( c ) Z ijk = (cid:18) − zaF l i + 12 z i (cid:19) g kj + (cid:18) − zaF l j + 12 z j (cid:19) g ik − (cid:18) − zaF l k + 12 z k (cid:19) g ij + zC ijk . Since the components u ij reciprocal to the components (C.1) are of the form u ij = (1 /z ) F a g ij ( c ) , the coefficients Z mij = u mh Z ijh read merely Z mij = (cid:18) − aF l i + 12 z z i (cid:19) δ mj + (cid:18) − aF l j + 12 z z j (cid:19) δ mi − (cid:18) − aF l m + 12 z g mk z k (cid:19) g ij + C mij . (C.2)We straightforwardly obtain ∂Z mni ∂y j = aF l j ( l n δ mi + l i δ mn − l m g ni ) − aF (cid:20) h ij δ mn + h nj δ mi − h mj g in − (cid:18) l m − z Fa g mk z k (cid:19) F C inj (cid:21) + ∂C mni ∂y j + 12 " (ln z ) nj δ mi + (ln z ) ij δ mn − ∂ (cid:0) g mk (ln z ) k (cid:1) ∂y j g ni and ∂Z mni ∂y j − ∂Z mnj ∂y i = aF [ l n ( l j δ mi − l i δ mj ) − l m ( l j g ni − l i g nj )] − aF [( h nj δ mi − h ni δ mj ) − ( h mj g in − h mi g jn )] + ∂C mni ∂y j − ∂C mnj ∂y i + 12 (cid:0) (ln z ) nj δ mi − (ln z ) ni δ mj (cid:1) − ∂ (cid:0) g mk (ln z ) k (cid:1) ∂y j g ni − ∂ (cid:0) g mk (ln z ) k (cid:1) ∂y i g nj ! , so that ∂Z mni ∂y j − ∂Z mnj ∂y i = − aF ( h nj h mi − h ni h mj ) + ∂C mni ∂y j − ∂C mnj ∂y i
7+ 12 (cid:0) (ln z ) nj δ mi − (ln z ) ni δ mj (cid:1) − g mh [(ln z ) hj g ni − (ln z ) hi g nj ]+(ln z ) h (cid:2) C mhj g ni − C mhi g nj (cid:3) . (C.3)Also, Z hni Z mhj − Z hnj Z mhi = − aF h − aF [ l n ( l i δ mj − l j δ mi ) + l m ( l j g ni − l i g nj )] + ( l i C mnj − l j C mni ) i − aF l i δ hn " z z h δ mj − z g mk z k g hj − [ ij ]+ 12 z z i δ hn "(cid:18) − aF l h + 12 z z h (cid:19) δ mj − (cid:18) − aF l m + 12 z g mk z k (cid:19) g hj + C mhj − [ ij ] − (cid:16) aF (cid:17) ( g in δ mj − g jn δ mi )+ aF l h g in " z z h δ mj + 12 z z j δ mh − z g mk z k g hj − [ ij ] − z g hk z k g in "(cid:18) − aF l h + 12 z z h (cid:19) δ mj + (cid:18) − aF l j + 12 z z j (cid:19) δ mh − (cid:18) − aF l m + 12 z g mk z k (cid:19) g hj − [ ij ] − z g hk z k ( g in C mhj − g jn C mhi ) − aF l j C min + C hin " z z h δ mj + 12 z z j δ mh − z g mk z k g hj − [ ij ]+ C hni C mhj − C hnj C mhi , or Z hni Z mhj − Z hnj Z mhi = − (cid:16) aF (cid:17) ( h in h mj − h jn h mi ) + C hni C mhj − C hnj C mhi − aF l i " z z n δ mj − z g mk z k g nj + 12 z z i (cid:18) − aF l n + 12 z z n (cid:19) δ mj − [ ij ]+ aF l h g in " z z h δ mj − z g mk z k g hj − [ ij ] − z g hs z s g in "(cid:18) − aF l h + 12 z z h (cid:19) δ mj − aF l j δ mh + aF l m g hj − z g mk z k g hj − [ ij ]+ 12 z z h (cid:2) − C hmj g in + C hmi g jn + C hin δ mj − C hjn δ mi (cid:3) . In this way we come to Z hni Z mhj − Z hnj Z mhi = − (cid:16) aF (cid:17) ( h in h mj − h jn h mi ) + C hni C mhj − C hnj C mhi − aF z z n ( l i δ mj − l j δ mi ) + aF z g mk z k ( l i g nj − l j g ni )+ 12 z (cid:18) − aF l n + 12 z z n (cid:19) ( z i δ mj − z j δ mi )+ aF z ( l s z s )( g in δ mj − g jn δ mi ) − z (cid:0) z h g hs z s (cid:1) ( g in δ mj − g jn δ mi )+ 12 z (cid:18) − aF l m + 12 z g mk z k (cid:19) ( z j g in − z i g jn )+ 12 z z h (cid:2) − C hmj g in + C hmi g jn + C hin δ mj − C hjn δ mi (cid:3) . (C.4)Thus we are able to evaluate the curvature tensor e R nmij := ∂Z mni ∂y j − ∂Z mnj ∂y i + Z hni Z mhj − Z hnj Z mhi . By lowering the index e R nmij := u mt e R ntij , we obtain the representation91 z ( c ( x )) F a ( x ) e R nmij = 1 F ( a − a )( h nj h mi − h ni h mj ) + 1 F S nmij + 12 (cid:16) (ln z ) nj g mi − (ln z ) ni g mj − (ln z ) mj g ni + (ln z ) mi g nj (cid:17) − aF z (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni ) (cid:17) + 12 z (cid:18) − aF l n + 12 z z n (cid:19) ( z i g mj − z j g mi ) − z (cid:18) − aF l m + 12 z z m (cid:19) ( z i g jn − z j g in )+ aF z ( l s z s )( g in g mj − g jn g mi ) − z (cid:0) z h g hs z s (cid:1) ( g in g mj − g jn g mi )+ 12 z z h (cid:16) C hmj g in − C hmi g jn − C hnj g mi + C hin g mj (cid:17) , (C.5)where S nmij = (cid:18) ∂C mni ∂y j − ∂C mnj ∂y i + C hni C mhj − C hnj C mhi (cid:19) F and S nmij = g mt S ntij . Henceforth we assume the zero-degree homogeneity of the function z ( x, y ) with re-spect to the argument y , having the identities(ln z ) ni l i = − F (ln z ) n , (ln z ) i l i = 0 . (C.6)By performing the contraction in (C.5), we get1 z ( c ( x )) F a ( x ) e R nmij l m l j = 12 F (cid:16) − (ln z ) n l i − (ln z ) ni − (ln z ) i l n (cid:17) + 12 z (cid:18) − aF l n + 12 z z n (cid:19) z i + 12 z aF z i l n − z (cid:0) z h g hs z s (cid:1) h in + 12 z z h C hin , or02 1 z ( c ( x )) F a ( x ) e R nmij l m l j = − F (ln z ) n l i − (ln z ) ni − F (ln z ) i l n + 12 z z n z i − z (cid:0) z h g hs z s (cid:1) h in + 1 z z h C hin . (C.7)The vanishing e R nmij = 0 (C.8)holds when(ln z ) ni = − F (ln z ) n l i − F (ln z ) i l n + 12 z z n z i − z (cid:0) z h g hs z s (cid:1) h in + 1 z z h C hin , in which case from (C.5) we get1 F S nmij = − F ( a − a )( h nj h mi − h ni h mj )+ 12 (cid:16) F (ln z ) n l j + 1 F (ln z ) j l n − z z n z j + 12 z (cid:0) z h g hs z s (cid:1) h jn (cid:17) g mi − (cid:16) F (ln z ) n l i + 1 F (ln z ) i l n − z z n z i + 12 z (cid:0) z h g hs z s (cid:1) h in (cid:17) g mj − (cid:16) F (ln z ) m l j + 1 F (ln z ) j l m − z z m z j + 12 z (cid:0) z h g hs z s (cid:1) h jm (cid:17) g ni + 12 (cid:16) F (ln z ) m l i + 1 F (ln z ) i l m − z z m z i + 12 z (cid:0) z h g hs z s (cid:1) h im (cid:17) g nj + aF z (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni ) (cid:17) − z (cid:18) − aF l n + 12 z z n (cid:19) ( z i g mj − z j g mi ) + 12 z (cid:18) − aF l m + 12 z z m (cid:19) ( z i g jn − z j g in )+ 14 z (cid:0) z h g hs z s (cid:1) ( g in g mj − g jn g mi ) . Due simplifying yields11 F S nmij = − F ( a − a )( h nj h mi − h ni h mj )+ 12 F (cid:16) (ln z ) n l j + (ln z ) j l n (cid:17) g mi − F (cid:16) (ln z ) n l i + (ln z ) i l n (cid:17) g mj − F (cid:16) (ln z ) m l j + (ln z ) j l m (cid:17) g ni + 12 F (cid:16) (ln z ) m l i + (ln z ) i l m (cid:17) g nj + 14 z (cid:0) z h g hs z s (cid:1)h h jn g mi − h in g mj − h jm g ni + h im g nj i + aF z (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni ) (cid:17) + aF z h l n ( z i h mj − z j h mi ) − l m ( z i h jn − z j h in ) i + 14 z (cid:0) z h g hs z s (cid:1) ( g in g mj − g jn g mi )= − F ( a − a )( h nj h mi − h ni h mj )+ a − zF (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni ) + l n ( z i h mj − z j h mi ) − l m ( z i h jn − z j h in ) (cid:17) + 14 z (cid:0) z h g hs z s (cid:1)(cid:16) h jn g mi − h in g mj − h jm g ni + h im g nj + g in g mj − g jn g mi (cid:17) , which is S nmij = a (2 − a )( h nj h mi − h ni h mj ) + F z (cid:0) z h g hs z s (cid:1) ( h nj h mi − h ni h mj )+ a − z (cid:16) z n ( l i h mj − l j h mi ) − z m ( l i h nj − l j h ni )+ l n ( z i h mj − z j h mi ) − l m ( z i h jn − z j h in ) (cid:17) F. (C.9)Therefore, the known vanishing S nmij l i = 0 requires( a − (cid:16) (ln z ) n h mj − (ln z ) m h nj (cid:17) = 0 . (C.10)2Whenever a = 1, we should take z n = 0, which means that the function z is independentof y . Lastly, it is worth noting that the case a = 1 would mean S nmij = h nj h mi − h ni h mj . (C.11)In this case C Ind. = 0 (C.12)
Appendix D: Evaluation of the coefficients N kmni in the F N -space To evaluate the coefficients N kmni = ∂N kmn /∂y i , we use (II.3.32) and obtain N kmni = 1 F l i h kn ∂F∂x m + 1 F ( l k h ni + l n h ki ) ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m − l k ∂ (cid:18) F h ni (cid:19) ∂x m − ∂C kns ∂y i N sm − C kns N smi − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) (cid:18) F h ki h ns + 2 l k C nsi (cid:19) N sm + 1 F (1 − H ) l k l n h si N sm + 1 F (1 − H ) l k l s h ni N sm − (cid:18) y khp t pi + HF y kh l i (cid:19) F H (cid:18) ∂U hn ∂x m + L hms U sn (cid:19) − y kh F H (cid:18) ∂U hni ∂x m + L hms U sni (cid:19) , or N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kns ∂y i N sm − C kns N smi − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm
3+ 1 F (1 − H ) l k l n h si N sm − H F l k l s h ni N sm − (cid:18) y khp t pi + HF y kh l i (cid:19) F H (cid:0) − N sm U hns − N smn U hs (cid:1) − y kh F H (cid:18) ∂U hni ∂x m + L hms U sni (cid:19) . Reducing similar terms leads to N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kns ∂y i N sm − C kns N smi − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm + 1 F (1 − H ) l k l n h si N sm − H F l k l s h ni N sm − y kp t piv y vh F H N sm U hns − h vs y kp t piv N smn + HF y kh l i F H (cid:0) N sm U hns + N smn U hs (cid:1) − y kh F H (cid:18) ∂U hni ∂x m + L hms U sni (cid:19) . Applying here (II.3.30) and (II.3.31) yields N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kns ∂y i N sm − C kns N smi − C kis N smn − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm
4+ 1 F (1 − H ) l k l n h si N sm − H F l k l s h ni N sm − (cid:18) C kiv − (1 − H ) 1 F ( l v δ ki + l i δ kv − l k g iv ) (cid:19) y vh F H U hns N sm + (1 − H ) 1 F ( l i h ks − l k h is ) N smn + HF y kh l i F H (cid:0) N sm U hns + N smn U hs (cid:1) − y kh F H (cid:18) ∂U hni ∂x m + L hms U sni (cid:19) . From the representation U hn = 1 F H t hn − F HU h l n (see (II.3.5)) let us evaluate the coeffcients U hni = ∂U hn ∂y i . We get U hni = − HF l i F H t hn + 1 F H t hni + 1 F l i HU h l n − F HU hi l n − F HU h h ni = − HF ( l i U hn + l n U hi ) + 1 F H t hni + 1 F (1 − H ) HU h l n l i − F HU h h ni . Here, t hni = C sni t hs − (1 − H ) 1 F (cid:18) l i t hn + l n t hi − HF t h g ni (cid:19) (see (II.3.30)). From this it follows that U hni = − HF ( l i U hn + l n U hi ) + 1 F H (cid:20) C sni t hs − (1 − H ) 1 F (cid:0) l i t hn + l n t hi (cid:1)(cid:21) + 1 F F H (1 − H ) Ht h g ni − F HU h ( g ni − (2 − H ) l n l i ) . We have here U hni = − HF ( l i U hn + l n U hi ) + 1 F H (cid:20) C sni t hs − F (cid:0) l i t hn + l n t hi (cid:1)(cid:21) + 1 F H H F (cid:0) l i t hn + l n t hi (cid:1) − F F H H t h g ni + 2 − HF HU h l n l i = C sni t hs F H − F F H (cid:0) l i t hn + l n t hi (cid:1) − F H U h g ni + 2 + HF HU h l n l i . U hni = C sni t hs F H − F (cid:0) l i U hn + l n U hi (cid:1) − F H U h h ni . (D.1)We may readily deduce the contraction y vh F H U hns = C vsn − F ( l s h vn + l n h vs ) − F Hh ns l v . (D.2)Inserting yields N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kns ∂y i N sm − C kns N smi − C kis N smn − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm + 1 F (1 − H ) l k l n h si N sm − H F l k l s h ni N sm − (cid:18) C kiv − (1 − H ) 1 F ( l v δ ki + l i δ kv − l k g iv ) (cid:19) (cid:20) C vsn − F ( l s h vn + l n h vs ) − F Hh ns l v (cid:21) N sm +(1 − H ) 1 F ( l i h ks − l k h is ) N smn + HF l i (cid:20) C ksn − F (cid:0) l s h kn + l n h ks (cid:1) − HF h ns l k (cid:21) N sm + HF l i N smn h ks + y kh F H ∂ F (cid:0) l i U hn + l n U hi (cid:1) ∂x m + y kh F H ∂ F H U h h ni ∂x m − ∂C kni ∂x m − C sni y kh F H ∂ (cid:18) U hs + 1 F HU h l s (cid:19) ∂x m − y kh F H L hms U sni , or6 N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kni ∂x m − ∂C kns ∂y i N sm − C kns N smi − C kis N smn − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N smi + 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm + 1 F (1 − H ) l k l n h si N sm − H F l k l s h ni N sm − (cid:20) C vsn C kiv − F (cid:0) l s C kin + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F δ ki Hh ns N sm +(1 − H ) 1 F ( l i δ kv − l k g iv ) (cid:20) C vsn − F ( l s h vn + l n h vs ) (cid:21) N sm − (1 − H ) 1 F l k h is N smn + HF l i (cid:20) C ksn − F (cid:0) l s h kn + l n h ks (cid:1) − HF h ns l k (cid:21) N sm + 1 F l i N smn h ks + y kh F H ∂ F (cid:0) l i U hn + l n U hi (cid:1) ∂x m + y kh F H F H h ni ∂U h ∂x m + y k H ∂ F h ni ∂x m + 2 y k H m F h ni − C sni y kh F H ∂ (cid:18) F HU h l s (cid:19) ∂x m − y kh F H L hsm U sni − C vni y kh F H ∂U hv ∂x m . Finally we apply here (II.3.32), obtaining N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − F l k ∂h ni ∂x m − ∂C kni ∂x m − ∂C kns ∂y i N sm − C kns N smi − C kis N smn + N kms C sni − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F l n (1 − H ) l k h is N sm + 1 F h ks ( l n N smi + l i N smn ) − F (1 − H ) l k ( h ns N smi + h is N smn )+ 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − F (1 − H ) l k C nsi N sm − H F l k l s h ni N sm − (cid:20) C vsn C kiv − F (cid:0) l s C kin + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F δ ki Hh ns N sm +(1 − H ) 1 F l i (cid:20) C ksn − F (cid:0) l s h kn + l n h ks (cid:1)(cid:21) N sm − (1 − H ) 1 F l k (cid:20) C isn − F ( l s h in + l n h is ) (cid:21) N sm + HF l i (cid:20) C ksn − F (cid:0) l s h kn + l n h ks (cid:1) − HF h ns l k (cid:21) N sm + y kh F H ∂ F (cid:0) l i U hn + l n U hi (cid:1) ∂x m + y kh F H F H h ni ∂U h ∂x m + y k H ∂ F h ni ∂x m + 2 y k H m F h ni − C sni l k ∂l s ∂x m − y kh F H L hsm U sni − C vni (cid:20) − F h kv ∂F∂x m − l k ∂l v ∂x m − C kvs N sm − F (1 − H ) l k h vs N sm (cid:21) + C vni y kh F H L hsm U sv . Reducing similar terms yields now N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − (1 − H ) 1 F l k (cid:20) ∂h ni ∂x m + 2 C nsi N sm + h ns N smi + h is N smn (cid:21) − D m C kni − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F l n (1 − H ) l k h is N sm + 1 F h ks ( l n N smi + l i N smn )+ 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki (cid:1) N sm − F (1 − H ) h ki h ns N sm − H F l k l s h ni N sm − (cid:20) C vsn C kiv − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F h ki Hh ns N sm − (1 − H ) 1 F l k l i Hh ns N sm − F l i (cid:0) l s h kn + l n h ks (cid:1) N sm − (1 − H ) 1 F l k (cid:20) C isn − F ( l s h in + l n h is ) (cid:21) N sm − HF l i l k Hh ns N sm + y kh F H ∂ F (cid:0) l i U hn + l n U hi (cid:1) ∂x m + y kh F H F H h ni ∂U h ∂x m + y k Hh ni ∂ F ∂x m + 2 y k H m F h ni − y kh F H L hsm U sni − C vni (cid:20) − C kvs N sm − F (1 − H ) l k h vs N sm (cid:21) + C vni y kh F H L hsm U sv . The next step is to transform the representation to N kmni = 1 F l i h kn ∂F∂x m + 1 F l n h ki ∂F∂x m − F h kn ∂l i ∂x m − F h ki ∂l n ∂x m + 1 F l k h ni ∂F∂x m − (1 − H ) 1 F l k (cid:20) ∂h ni ∂x m + 2 C nsi N sm + h ns N smi + h is N smn (cid:21) − D m C kni − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F l n (1 − H ) l k h is N sm + 1 F h ks ( l n N smi + l i N smn )+ 1 F (cid:0) h ni h ks − l n l k h si − l n l s h ki − l i l s h kn (cid:1) N sm − F (1 − H ) h ki h ns N sm − H F l k l s h ni N sm − (cid:20) F S nksi − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F h ki Hh ns N sm − F l k l i Hh ns N sm − F l i l n h ks N sm + (1 − H ) 1 F l k l s h in N sm + (1 − H ) 1 F l k l n h is N sm + y kh F H F (cid:18) l i ∂U hn ∂x m + l n ∂U hi ∂x m (cid:19) + h kn ∂ F l i ∂x m + h ki ∂ F l n ∂x m + y kh F H F H h ni ∂U h ∂x m − F l k Hh ni ∂F∂x m + 2 1 F l k H m h ni − y kh F H L hsm U sni + C vni y kh F H L hsm U sv , where S nksi = F ( C vin C ksv − C vsn C kiv ) . We come to N kmni = − (1 − H ) 1 F l k (cid:20) ∂h ni ∂x m + 2 C nsi N sm + h ns N smi + h is N smn (cid:21) − D m C kni − F l i (cid:0) l n h ks − (1 − H ) l k h ns (cid:1) N sm + 1 F l n (1 − H ) l k h is N sm + 1 F h ks ( l n N smi + l i N smn ) + 1 F (cid:0) h ni h ks − l n l s h ki − l i l s h kn (cid:1) N sm − (cid:20) F S nksi − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F h ki h ns N sm − F l i l n h ks N sm − H F l k ( l i h ns + l n h is ) N sm + y kh F H F (cid:18) l i ∂U hn ∂x m + l n ∂U hi ∂x m (cid:19) + y kh F H F H h ni ∂U h ∂x m + 2 1 F l k H m h ni + y kh F H L hsm (cid:20) F ( l i U sn + l n U si ) + H F U s h ni (cid:21) = − (1 − H ) 1 F l k D m h ni − D m C kni − F l i l n h ks N sm + 1 F (cid:0) h ni h ks − l n l s h ki − l i l s h kn (cid:1) N sm − (cid:20) F S nksi − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F h ki h ns N sm − F l i l n h ks N sm − H F l k ( l i h ns + l n h is ) N sm + y kh F H F (cid:20) l i (cid:18) ∂U hn ∂x m + U hs N smn + L hsm U sn (cid:19) + l n (cid:18) ∂U hi ∂x m + U hs N smi + L hsm U si (cid:19)(cid:21) + y kh F H F H h ni (cid:18) ∂U h ∂x m + L hsm U sm (cid:19) + 2 1 F l k H m h ni . Use here the equality D m h ni = − H H m h ni (see (II.3.21)).The rest is N kmni = 2 H H m F l k h ni − D m C kni − F l i l n h ks N sm + 1 F (cid:0) h ni h ks − l n l s h ki − l i l s h kn (cid:1) N sm − (cid:20) F S nksi − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F h ki h ns N sm − F l i l n h ks N sm − H F l k ( l i h ns + l n h is ) N sm − y kh F H F (cid:0) l i U hns + l n U his (cid:1) N sm − y kh F H F H h ni U hs N sm = 2 H H m F l k h ni − D m C kni − F l i l n h ks N sm − F (cid:0) l n l s h ki + l i l s h kn (cid:1) N sm − (cid:20) F S nksi − F (cid:0) l i C kns + l n C kis (cid:1)(cid:21) N sm − (1 − H ) 1 F (cid:0) h ki h ns − h in h ks (cid:1) N sm − F l i l n h ks N sm − H F l k ( l i h ns + l n h is ) N sm − F (cid:20) l i (cid:18) C ksn − F (cid:0) l s h kn + l n h ks (cid:1) − HF h ns l k (cid:19) + l n (cid:18) C ksi − F (cid:0) l s h ki + l i h ks (cid:1) − HF h is l k (cid:19)(cid:21) N sm . Since the indicatrix curvature equals H , we have S nksi = − (1 − H ) (cid:0) h ki h ns − h in h ks (cid:1) . N kmni = 2 H H m F l k h ni − D m C kni . (D.3)Thus, Proposition II.3.5 is valid. Appendix E: Implications from angle
Below, the consideration refers to an arbitrary
Finsler space. No assumptions con-cerning the curvature of indicatrix are made. We use the angle α = α { x } ( y , y ) whichis the geodesic-arc distance on the indicatrix, in accordance with the initial definition(I.1.1).In terms of the function E := 12 α (E.1)the preservation equation d i α + (1 /H ) H i α = 0 proposed by (I.1.28) reads ∂E∂x i + N k i ∂E∂y k + N k i ∂E∂y k = − H H i E, (E.2)where N k i = N ki ( x, y ) , N k i = N ki ( x, y ) , and H i = ∂H/∂x i . Henceforth, H = H ( x ) . (E.3)We are aimed to extract the required coincidence limits from the function E , treatingthe indicatrix naturally to be a particular Riemannian space metricized by the help of themetric tensor induced by the Finsler metric tensor.Let a set of scalars u a = u a ( x, y ) be used to coordinatize the indicatrices; the indices a, b, ... will be specified over the range (1,2,...N-1). We shall use the derivative objects u am = ∂u a ∂y m , u amk = ∂u am ∂y k . The scalars are assumed to be positively homogeneous of degree zero with respect to thevariable y : u a ( x, ky ) = u a ( x, y ) , k > , ∀ y, (E.4)which directly entails the identities u am y m = 0 , u amk y k = − u am . (E.5)Using the parametrical representation l i = t i ( u a ) of the indicatrix, where l i are unitvectors (possessing the property F ( l ) = 1), we can construct the induced metric tensor i ab ( u c ) = g mn t ma t nb ≡ h mn t ma t nb (E.6)on the indicatrix by the help of the projection factors t ma = ∂t m /∂u a (the method wasdescribed in detail in Section 5.8) of [1]).The validity of the equalities F u bm t mc = δ bc , F u bm t kb = h km , F u ck = g km t ma i ac , t ne i ec t ic = h ni , F h mn = i ab u am u bn (E.7)2can readily be verified.From the identity l m t ma = 0 it follows that l m t mab = − i ab , (E.8)where t mab = ∂t ma /∂u b .From (E.6) we get i ab,c = 2 F C mnk t ma t nb t kc + g nm ( t mac t nb + t ma t nbc ) . With the coefficients i ac,b = ∂i ac /∂u b we obtain( i ac,b + i bc,a − i ab,c ) = 2 F C mnk t ma t nb t kc + 2 g nm t mab t nc , which entails ( i ae,b + i be,a − i ab,e ) i ec = 2 F C mnk t ma t nb t ke i ec + 2 g nm t mab t ne i ec , so that t ic (cid:16) i cab − F C mnk t ma t nb t ke i ec (cid:17) = h im t mab and t iab = t ic (cid:16) i cab − F C mnk t ma t nb t ke i ec (cid:17) − l i i ab (E.9)(this equation is equivalent to (5.8.8) of [1]).The indicatrix Christoffel symbols i cab = 12 i ce (cid:18) ∂i ea ∂u b + ∂i eb ∂u a − ∂i ab ∂u e (cid:19) and the indicatrix curvature tensor I aebd := ∂i eab ∂u d − ∂i ead ∂u b + i f ab i efd − i f ad i efb (E.10)will be used.Constructing the tensor S abcd = −
13 ( I acbd + I adbc ) , (E.11)where I acbd = I aebd i ec , we obtain the useful identity S abcd − S acbd = − I adbc . (E.12)It follows that i ab (cid:0) u bmj + i bcv u vm u cj (cid:1) = h pq (cid:0) t pa t qb u bmj + i bcv t pa t qb u vm u cj (cid:1) = h pq t pa ∂ F h qm ∂y j − u bm t qbc u cj + i bcv t qb u vm u cj . Taking t pab from (E.9), we get i ab (cid:0) u bmj + i bcv u vm u cj (cid:1) = h pq t pa ∂ F h qm ∂y j − u bm t qf (cid:16) i f bc − F C rsk t rb t sc t ke i ef (cid:17) u cj + i bcv t qb u vm u cj . t ke i ef = F u fh g hk , so that i ab (cid:0) u bmj + i bcv u vm u cj (cid:1) = h pq t pa ∂ F h qm ∂y j + 1 F C mjq , (E.13)from which it follows that u bmj + i bcv u vm u cj = F u bt ∂ F h tm ∂y j + 1 F C mj t (E.14)and u amk u bn i ab + u am u fk u bn i caf i cb = − F ( h nk l m + h nm l k ) + 1 F C kmn . (E.15)Now we consider the quantity (E.1) on the indicatrix: E = M ( x, u , u ) , (E.16)where M is a scalar.There arise the objects M a c def = ∂M a ∂u c , M a c d def = ∂M a c ∂u d − i f d c M a f , (E.17)together with M a c def = ∂M a ∂u c − i f a c M f , M a b c def = ∂M a b ∂u c = ∂ M∂u a ∂u b ∂u c − i f a b M f c (E.18) M a b c d def = ∂M a b c ∂u d − i f d c M a b f = ∂ M∂u a ∂u b ∂u c ∂u d − i f a b ∂M f c ∂u d − i f d c M a b f . (E.19)It follows that ∂ M∂u a ∂u b ∂u c ∂u d = M a b c d + i f a b ∂M f c ∂u d + i f d c M a b f . (E.20)In the limit u → u we have ∂M∂u a → , ∂M∂u a → , (E.21)and ∂ M∂u a ∂u b → i ab , ∂ M∂u a ∂u b → − i ab , ∂ M∂u a ∂u b → i ab , (E.22)4together with ∂ M∂u a ∂u b ∂u c → − i ebc i ea , ∂ M∂u a ∂u b ∂u c → − i f ab i fc (E.23)(see Section 3.2 in [12]).Also, M a b c d → S abcd (E.24)(see (3.2.69) in [12]). From (E.20) it follows that ∂ M∂u a ∂u b ∂u c ∂u d → S abcd − i f ab i edc i fe . (E.25)We find ∂E∂y k = u d k ∂M∂u d , ∂ E∂y k ∂y n = u d k n ∂M∂u d + u d k ∂ M∂u d ∂u t u t n (E.26)and ∂ E∂y k ∂y n ∂y m = u d k n ∂ M∂u d ∂u v u v m + u d k ∂ M∂u d ∂u t ∂u s u t n u s m . (E.27)Moreover, ∂ E∂y k ∂y n ∂y m ∂y j = u d k n ∂ M∂u d ∂u v ∂u s u v m u s j + u d k n ∂ M∂u d ∂u v u v m j + u d k ∂ M∂u d ∂u t ∂u s u t n u s m j + u d k ∂ M∂u d ∂u t ∂u s ∂u t u t n u s m u t j . (E.28)These observations entail the limits y → y : ∂E∂y m → , ∂E∂y m → , (E.29) ∂ E∂y m ∂y n → F h mn , ∂ E∂y m ∂y n → F h mn , ∂ E∂y m ∂y n → − F h mn , (E.30)and ∂ E∂y k ∂y m ∂y n → − (cid:0) u ank u bm i ab + u am u bk u cn i ebc i ea (cid:1) , (E.31)plus ∂ E∂y k ∂y m ∂y n → − (cid:0) u amk u bn i ab + u an u bk u cm i ebc i ea (cid:1) , (E.32)5together with ∂ E∂y k ∂y n ∂y m ∂y j → − u akn (cid:0) i ab u bmj + i f cv i af u vm u cj (cid:1) − u amj i f cv i af u vk u cn + (cid:0) S abcd − i f a b i edc i fe (cid:1) u ak u bn u cm u dj , or ∂ E∂y k ∂y n ∂y m ∂y j → − ( u akn + i acv u vk u cn ) i ab (cid:0) u bmj + i bcv u vm u cj (cid:1) + S abcd u ak u bn u cm u dj . (E.33)In this way we arrive at the reductions ∂ E∂y k ∂y m ∂y n → − (cid:0) − F ( h nm l k + h nm l k ) + 1 F C kmn (cid:1) (E.34)and ∂ E∂y k ∂y m ∂y n → − (cid:0) − F ( h nk l m + h mk l n ) + 1 F C kmn (cid:1) . (E.35)Taking into account (E.13), we can write ∂ E∂y k ∂y n ∂y m ∂y j → − ∂ F h tk ∂y n + 1 F C knt h tq ∂ F h qm ∂y j + 1 F C mj q + S abcd u ak u bn u cm u dj , which is F ∂ E∂y k ∂y n ∂y m ∂y j → (cid:0) l n h tk + l k h tn − F C knt (cid:1) ( − l j h tm − l m h tj + F C mjt )+ F S abcd u ak u bn u cm u dj , or F ∂ E∂y k ∂y n ∂y m ∂y j → l n ( − l j h km − l m h kj + F C mjk ) + l k ( − l j h nm − l m h nj + F C mjn )+ F C knm l j + F C knj l m − F C knt C mjt + F R abcd u ak u bn u cm u dj . (E.36)Now we differentiate the preservation law (E.2) with respect to y m and y n and make y → y , which yields ∂ i h mn − h mn F ∂ i F + N kim h kn + N kin h km + 1 F N ki (cid:16) − ( h nk l m + h mn l k ) + F C kmn (cid:17) + 1
F N ki (cid:16) − ( h km l m + h nm l k ) + F C kmn (cid:17) = − H H i h mn . D i h mn − F h mn d i F = − H H i h mn (E.37)with D i h mn = ∂ i h mn + N ki ∂h mn ∂y k + N kim h kn + N kin h km (E.38)and d i F = ∂ i F + N ki l k . (E.39)In the vanishing case ∂ i F + N ki l k = 0 (E.40)we obtain by differentiation the equalities ∂ i l n + 1 F N ki h kn + N kin l k = 0 (E.41)and ∂ i h mn + N ki ∂h kn ∂y m + N kim h kn + N kin h km − F N ki h kn l m + 1 F N ki l k h mn + F N kinm l k = 0 . So we can write D i h mn + F N kinm l k = 0 (E.42)together with F N kinm l k = 2 H H i h mn . (E.43) References [1] H. Rund,
The Differential Geometry of Finsler Spaces,
Springer, Berlin 1959.[2] D. Bao, S. S. Chern, and Z. Shen,
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Bull.Soc. Sci. Lett. Lodz Ser. Rech. Deform (2006), 147-152.[5] Z. L. Szab´o, All regular Landsberg metrics are Berwald, Ann Glob Anal Geom (2008), 381-386.[6] L. Tam´assy, Angle in Minkowski and Finsler spaces, Bull. Soc. Sci. Lett. LodzSer. Rech. Deform (2006), 7-14.[7] G. S. Asanov, Finsleroid gives rise to the angle-preserving connection, arXiv: arXiv: Publ. Math. Debrecen (2010), 245–259.[10] G. S. Asanov, Finsler connection preserving angle in dimensions N ≥ arXiv: N ≥ Publ.Math. Debrecen (2011), 181–209.[12] J. L. Synge,