EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 3, Article Number 6204 ISSN 1307-5543 – ejpam.com Published by New York Business Global General Deformations of Sprays on Finsler Manifolds S. G. Elgendi1,∗, A. Soleiman2 1 Department of Mathematics, Faculty of Science, Islamic University of Madinah, Madinah, Saudi Arabia 2 Department of Mathematics, College of Science, Jouf University, Skaka, Saudi Arabia Abstract. In this paper, we investigate the concept of general deformations of a spray S on a manifoldM . We then focus on a specific case, which we call a projective-like deformation. This type of deformation extends the notion of projective deformation but, unlike projective deformation, it does not necessarily preserve geodesics. We derive an explicit formula for the Jacobi endomorphism under projective-like deformations and analyze the conditions under which it remains invariant. As applications, we consider (α, β)-metrics and spherically symmetric metrics. We find a necessary and sufficient condition for an (α, β)-metric and the Riemannian metric α to be projectively related. Additionally, we provide and examine several explicit examples. 2020 Mathematics Subject Classifications: 53C60, 53B40, 58B20. Key Words and Phrases: Sprays, projective deformation, projective-like deformation, Jacobi endomorphism 1. Introduction In Finsler geometry, the concepts of sprays and its projective deformation play a crucial role in understanding how the geodesic structure of a manifold changes under transfor- mations that preserve the projective class of paths. A projective deformation of a spray refers to a modification of the spray such that the new spray generates the same set of unparameterized geodesics as the original one. This concept is closely related to projec- tive geometry, where only the direction of geodesics matters rather than their specific parameterization. For example, see [1–11]. A projective deformation [5, 7, 10, 12] of S results in a new spray S given by: S = S − 2P(x, y)C, (1) ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i3.6204 Email addresses: salah.ali@fsc.bu.edu.eg, salahelgendi@yahoo.com (S. G. Elgendi), asoliman@ju.edu.sa, amrsoleiman@yahoo.com (A. Soleiman) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 2 of 18 where P(x, y) is a scalar function known as the projective factor, and C is the canonical Liouville vector field. This transformation ensures that the integral curves of S and S remain the same up to reparameterization, meaning the geodesics of the manifold remain unchanged as point sets. Projective deformations [1, 7] are particularly useful in Finsler geometry for studying conditions under which a given Finsler metric can be related to another one via projectively equivalent sprays. These transformations are fundamental in various areas, including the study of curvature properties, and projective flatness. The theory of projective changes in Riemannian geometry has been deeply studied (locally and intrinsically) by many authors. As regards to Finsler geometry, a complete local theory of projective changes has been established ([5, 7, 8, 13, 14]). Moreover, an intrinsic theory of projective changes (resp. semi-projective changes) has been investigated in [10, 15]) following the KG-approach. In this paper, considering two sprays S1 and S2 on a manifold M , and using the fact that that the difference S1 − S2 is always vertical, we introduce the general deformation of a spray S as follows: S̃ = S − 2ζ, where ζ is a vertical vector field ζ ∈ Xv(TM). Then, we focus our attentions to an interesting special case. That is, we introduce what we called projective-like deformation. Precisely, let S be a spray on a manifold M , and consider any vertical vector field ξ ∈ Xv(TM). The projective-like deformation of S is given by S̃ = S − 2P(x, y)ξ, where P(x, y) ∈ C∞(T M) is a smooth, and positively homogeneous function of degree 1 in y, and called the deformation factor. We consider two special classes of Finsler metrics, namely, the class of (α, β)-metrics and the class of spherically symmetric metrics. For these classes, we figure out the relation between the background spray and the new one showing the explicit formula of the vertical vector which identifies the difference between the two sprays. Moreover, As by-product, we discuss when the background spray and new one are projective. For example, for an (α, β)-metric F = αϕ(s), then the associated sprays with F and the Riemannian metric α are projectively related if and only if 2αϕ′si0 + r00ϕ ′′bi − r00ϕ ′′ β α2 yi = 0. Under the projective-like deformation, we establish the relation between the two associ- ated Jacobi endomorphisms. Moreover, we discuss the invariance property of the jacobi endomorphism. Some explicit examples are studied. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 3 of 18 2. Preliminaries Let M be an n-dimensional differentiable manifold with its tangent bundle represented by (TM, πM ,M), and let (T M,π,M) denote the subbundle comprising nonzero tangent vectors. The local coordinates onM are expressed as (xi), while the associated coordinates on TM are (xi, yi), where yi are the components of the tangent vectors. The tangent bundle TM is naturally equipped with an almost-tangent structure J , locally given by J = ∂ ∂yi ⊗ dxi. Let’s recall some basics and properties of the Klein-Grifone approach to Finsler geometry. The canonical (or Liouville) vector field C on TM is a vertical vector field defined as: C = yi ∂ ∂yi . (2) More details can be found in [1, 9, 16]. A spray S is a vector field S ∈ X(T M) that satisfies JS = C and [C, S] = S. Locally, it has the expression: S = yi ∂ ∂xi − 2Gi ∂ ∂yi , (3) whereGi = Gi(x, y), known as the spray coefficients, are functions that exhibit 2-homogeneity in y. A nonlinear connection is an n-dimensional distribution H(T M) that serves as a com- plement to the vertical distribution V (T M) := kerπ∗. This implies that, for each z ∈ T M , the tangent space at z decomposes as: Tz(T M) = Hz(T M)⊕ Vz(T M). (4) Each spray S naturally induces a nonlinear connection Γ = [J, S] (see [16]) characterized by horizontal and vertical projectors: h = 1 2 (Id + [J, S]), v = 1 2 (Id− [J, S]). (5) Locally, these projectors take the forms: h = δ δxi ⊗ dxi, v = ∂ ∂yi ⊗ δyi, (6) where: δ δxi = ∂ ∂xi −N j i (x, y) ∂ ∂yj , δyi = dyi +N j i (x, y)dx i, N j i (x, y) = ∂Gj ∂yi . (7) For a vector k-form K on M , the graded derivations iK and dK on the Grassmann algebra of M are given by [17]: iKf = 0, iKdf = df ◦K, (8) S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 4 of 18 where f ∈ C∞(M) and df represents the exterior derivative of f . Furthermore, the derivation dK is defined as: dK := [iK , d] = iK ◦ d− (−1)k−1diK . (9) The Jacobi endomorphism (or Riemann curvature, as seen in [9]) Φ is formulated as: Φ = v ◦ [S, h] = Ri j ∂ ∂yi ⊗ dxj , (10) where: Ri j = 2 ∂Gi ∂xj − S(N i j)−N i kN k j . (11) The curvature tensor R of a spray S is given by: R = −1 2 [h, h] = Rℓ ij ∂ ∂yℓ ⊗ dxi ⊗ dxj , with: Rℓ ij = δGℓ i δxj − δGℓ j δxi . (12) The two curvature tensors are related as follows (see, for example, [3]): 3R = [J,Φ], Φ = iSR. (13) Definition 1. A Finsler manifold (M,F ) consists of an n-dimensional manifold M along with a function F : TM → R that meets the following conditions: a) F is smooth and strictly positive on T M . b) F is positively homogeneous of degree 1 in y: LCF = F . c) The Hessian matrix gij = ∂2E ∂yi∂yj has full rank n on T M , where E = 1 2F 2. The geodesic spray S associated with F is determined uniquely by the Euler-Lagrange equation: iSddJE = −dE. For simplicity, we use the notations δi := δ δxi , ∂i := ∂ ∂xi , ∂̇i := ∂ ∂yi . For a Finsler manifold (M,F ), the coefficients Gi of the geodesic spray of F are given by Gi = 1 4 gih(yr∂r∂̇hF 2 − ∂hF 2), (14) where gij are the components of the inverse metric tensor. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 5 of 18 2.1. Projective deformations of sprays The concepts of sprays and its projective deformation play a crucial role in under- standing how the geodesic structure of a manifold changes under transformations that preserve the projective class of paths. A complete local theory of projective changes has been established ([5, 7, 8, 13, 14]). A regular curve is a curve γ : I → M with the tangent lift γ′ : I → TM . A regular curve γ on M is a geodesic of a spray S if it satisfies the condition S ◦ γ′ = γ′′. Locally, if γ(t) = (xi(t)), then the geodesic equation is given by: d2xi dt2 + 2Gi ( x, dx dt ) = 0. (15) In other words, the curve γ(t) is a geodesic of S if its velocity yi = dxi dt satisfies the above equation. Two sprays, S and S, are said to be projectively related if their geodesics remain unchanged, up to an orientation-preserving reparameterization. In this case, one spray can be regarded as a projective deformation of the other. Locally, (1) modifies the spray coefficients as follows: G i = Gi + Pyi. To see why the geodesics remain unchanged under this deformation, consider the new geodesic equation: d2xi dt2 + 2G i ( x, dx dt ) = 0. Expanding G i , we obtain: d2xi dt2 + 2Gi ( x, dx dt ) + 2P ( x, dx dt ) dxi dt = 0. (16) Now, consider a reparameterization t̃ = t̃(t) with dt̃ dt > 0, then (15) takes the form d2xi dt̃2 + 2Gi ( x, dx dt̃ ) + d2 t̃ dt2( dt̃ dt )2 dxi dt̃ = 0. (17) This equation suggests that the new geodesic parameter t̃ is related to the original pa- rameter t by a reparameterization. Specifically, the function P ( x, dxdt ) induces a reparam- eterization of the geodesic equation, but the actual path traced by the geodesics remains unchanged. To see this explicitly, comparing (16) and (17), then the reparameterization t̃ = t̃(t) satisfies d2t̃ dt2 = 2P ( x, dx dt ) dt̃ dt . S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 6 of 18 The key observation is that a projective deformation modifies the acceleration along the geodesics but does not alter their direction. The new spray S still determines the same set of geodesics in terms of their geometric trajectories, though their parameterization may differ. 3. General deformations of sprays For any two sprays S1 and S2 on a manifold M , one can see that the difference S1−S2 is always vertical. Indeed, J(S1 − S2) = JS1 − JS2 = C − C = 0. Using the fact that a vector field X on TM is vertical if and only if JX = 0, we conclude that S1 − S2 is vertical. Now, using the fact that the difference of two sprays is vertical, we can define the general deformation of a spray S as follows. Definition 2. Let S be a spray on a manifold M , and ζ be a vertical vector field ζ ∈ Xv(TM). The general deformation of S is given by S̃ = S − 2ζ. (18) Moreover, the coefficients of the two sprays are related by G̃i = Gi + ζi. (19) By making use of (16) and (19), we have the following property: The general defor- mation (18) of S is projective, i.e., preserves geodesics, if and only if the vertical vector ζ is proportional to the Liouville vector field C (or equivalently, ζi is proportional to yi). For a given Finsler manifold (M,F ), any deforamtion of F implies a deformation of the geodesic spray. As an example, let’s consider the deformation of a a geodesic spray of a Finsler manifold (M,F ) due to a conformal change of the Finsler metric F . Example 1. Consider the conformal transformation of a Finsler manifold (M,F ) with the geodesic spray S F̃ = eσ(x)F. So the geodesic spray S is deformed as follows: S̃ = S − 2ζ. By [18], the spray coefficients are related by G̃i = Gi + ζi = Gi + 1 2 F 2σi − σ0y i, where σr := ∂rσ, and σ0 := σry r. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 7 of 18 Now, using the facts that C = yi∂̇i and ℓi = yi F , we have ζ = ζi∂̇i = F 2 2 ( gir − 2ℓiℓr ) σr(x)∂̇i = F 2 2 girσr(x)∂̇i − σ0(x)C = 1 2 F 2σi∂̇i − σ0C, where ζi = F 2 2 ( gir − 2ℓiℓr ) σr(x) = 1 2 F 2σi − σ0y i. Moreover, the conformal change is projective, i.e., preserves the geodesics of S, if and only if ζ = C. This occurs if and only if σi(x) = 0. That is, σ is constant. 3.1. Projective-like deformation One the special and important deformation of a spray is the projective deformation, that is, S = S − 2P(x, y)C. Under the projective deformation, the geodesics of a spray are preserved. So, in this subsection we consider another point of view of a more gen- eral deformation than the projective deformation but still a special case of the general deformation of a spray S. In this section, we introduce the porjective-like deformation as follows: Definition 3. Let S be a spray on a manifold M , and consider any vertical vector field ξ ∈ Xv(TM). The projective-like deformation of S is given by S̃ = S − 2P(x, y)ξ. (20) where P(x, y) ∈ C∞(T M) is a smooth, and positively homogeneous function of degree 1 in y, and called the deformation factor. Moreover, the coefficients of the two sprays are related by G̃i = Gi + Pξi. (21) In a similar manner to what we discuss in Property 3, if ξ := C, then the porjective- like spray deformation (20) reduces to the projective deformation, and P(x, y) is called projective factor. It is clear that, as S̃ is a spray, then ξ is homogenous of degree one, that is [C, ξ] = 0. (22) Taking the following vector one form µ := [J, ξ], we have: Lemma 1. The endomorphism µ has the following properties µJ = 0, Jµ = 0, µ2 = 0, Γµ+ µΓ = 0. Proof. Since J Γ̃ = JΓ = J , then Γ̃ = J(Γ + µ) = J + Jµ = J =⇒ Jµ = 0. Similarly, using the fact that Γ̃J = ΓJ = −J , we get µJ = 0. Therefore, we conclude that µ2 = 0. Since Γ̃2 = I and Γ2 = I, then we obtain that Γµ+ µΓ = 0. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 8 of 18 Proposition 1. Under the projective-like deformation (20) of a spray S with deformation factor P(x, y), the Barthel connection Γ̃ has the form Γ̃ = Γ− 2(Pµ+ dJP ⊗ ξ). (23) Consequently, the horizontal and vertical projections h̃ and ṽ becomes h̃ = h− L, ṽ = v + L, where, L is the vector 1-form defined by L := Pµ+ dJP ⊗ ξ. (24) Proof. Using the formula [17] [J, fX] = f [J,X] + dJf ⊗X − df ⊗ JX, and the fact that ξ is a vertical vector field, we obtain Γ̃ = [J, S̃] = [J, S − 2Pξ] = [J, S]− 2[J,Pξ] = Γ− 2(Pµ+ dJP ⊗ ξ). Hence, using the facts h̃ = 1 2(I + Γ̃), and ṽ = 1 2(I − Γ̃), the proof completes. Proposition 2. Under the projective-like deformation (20) of a spray S with deformation factor P(x, y). The Jacobi endomorphism Φ̃ is given by Φ̃ = Φ− (P2µ[S, µ] + S(P)µ) + (2dhP−PdµP−∇dJP)⊗ ξ −P v[S, µ]− dJP ⊗ v[S, ξ]− PdµP ⊗ ξ − 2dξP dJP ⊗ ξ −2P(v[ξ, h]− v[ξ,L]) + P{µ[S, h]− dJP ⊗ µ[S, ξ]− 2Pµ[ξ, h]} −{PdJ [S,µ]P + dJPdJ [S,ξ]P + 2PdJ [ξ,h]P} ⊗ ξ . Proof. Under the spray deformation (20), with deformation factor P(x, y), taking into account Proposition 1, one can show that [S̃, h̃] = [S − 2Pξ, h− L] = [S, h]− [S,L]− 2[Pξ, h] + 2[Pξ,L]. (25) On the other hand, taking into account the expression of L given by (24), and using the following relations: ξ ∈ Xv(T M), hξ = Lξ = 0, together with the facts that, for f, g ∈ C∞(T M),K ∈ Ψ1(T M), ω ∈ Λ1(T M): [X, fK] = (dXf)K + f [X,K], [X,ω ⊗ Y ] = [X,ω]⊗ Y + ω ⊗ [X,Y ], S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 9 of 18 dfK+ω⊗Y g = fdKg + ω ⊗ dY g. It follows that [S,L] = [S,Pµ+ dJP ⊗ ξ] = [S,Pµ] + [S, dJP ⊗ ξ] = S(P)µ+ P[S, µ] + [S, dJP]⊗ ξ + dJP ⊗ [S, ξ] = S(P)µ+ P[S, µ] + LSdJP ⊗ ξ + dJP ⊗ [S, ξ]. [Pξ, h] = dP ⊗ hξ − dhP ⊗ ξ + P[ξ, h] = P[ξ, h]− dhP ⊗ ξ. [Pξ,L] = dP ⊗ Lξ − dLP ⊗ ξ + P[ξ,L] = P[ξ,L]− dLP ⊗ ξ = P[ξ,L]− PdµP ⊗ ξ − dξP dJP ⊗ ξ. In view of the above relations, Equation (25) becomes [S̃, h̃] = [S, h]− S(P)µ− P[S, µ]− LSdJP ⊗ ξ − dJP ⊗ [S, ξ] + 2dhP ⊗ ξ − 2PdµP ⊗ ξ − 2dξP dJP ⊗ ξ − 2P([ξ, h]− [ξ,L]). Now, using the definition of the Jacobi endomorphism Φ̃ := ṽ ◦ [S̃, h̃] and we compose to the left both terms in the above formula by ṽ, taking into account Proposition 1, one can show that: Φ̃ := ṽ ◦ [S̃, h̃] = (v + L)[S̃, h̃] = v[S̃, h̃] + L[S̃, h̃]. (26) On the other hand, using the facts that v ◦ µ = vµ = µ, vξ = ξ, µ ◦ µ = Jµ = 0, µξ = 0, Lξ = Jξ = 0, v = J [S, h], we get v[S̃, h̃] = Φ− S(P)µ− P v[S, µ]− LSdJP ⊗ ξ − dJP ⊗ v[S, ξ] +2dhP ⊗ ξ − 2PdµP ⊗ ξ − 2dξP dJP ⊗ ξ − 2P(v[ξ, h]− v[ξ,L]). L[S̃, h̃] = Pµ[S̃, h̃] + dJP([S̃, h̃])⊗ ξ = P{µ[S, h]− Pµ[S, µ] + dJP ⊗ µ[S, ξ] + 2Pµ[ξ, h]} +{d(J [S,h]−PJ [S,µ]−dJP⊗J [S,ξ]−2PJ [ξ,h])P} ⊗ ξ = P{µ[S, h] + Pµ[S, µ] + dJP ⊗ µ[S, ξ] + 2Pµ[ξ, h]} +{dvP − PdJ [S,µ]P − dJPdJ [S,ξ]P − 2PdJ [ξ,h]P} ⊗ ξ In view of, the action of the dynamical covariant derivative ∇ on the semi-basic 1-form dJP is given by ∇dJP = LSdJP − dvP. Now, by substituting (26), the required formula of the Jacobi endomorphism Φ̃ is obtained, and this completes the proof. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 10 of 18 Theorem 1. The Barthel curvature tensor R̃ associated with the spray deformation (20), with deformation factor P(x, y), is determined by R̃ = R+ [h,L]−NL, where L defined by (24), and NL := 1 2 [L,L] is the Nijenhuis torsion of L. Proof. The proof follows from Proposition 23, together with the fact that R̃ = −1 2 [h̃, h̃], and taking into account the properties of the Frölicher-Nijenhuis bracket [17]. Corollary 1. The Berwald connection D̃◦ associated with the spray deformation (20), with deformation factor P(x, y), is determined by D̃◦ JXJY = D◦ JXJY D̃◦ h̃X JY = D◦ hXJY − [LX, JY ] + L[hX, JY ]. Proof. The proof follows from Proposition 23, together with the facts that (for example, see [19]) D◦ JXJY = J [JX, Y ], D◦ hXJY = v[hX, JY ]. This completes the proof. Now, we retrieve some important results of projective deformation. In the deformation (20), if ξ = C, then the deformation reduces to the projective deformation S = S − 2P(x, y) C (27) Corollary 2. [3] Under the projective deformation S = S − 2P(x, y) C, we have Γ = Γ− 2(PJ + dJP ⊗ C), h = h− PJ − dJP ⊗ C, v = v + PJ + dJP ⊗ C, Φ = Φ+ (P2−LSP)J + (2dhP−PdJP−∇dJP)⊗ C, R = R+ dJdhP ⊗ C + (PdJP − dhP) ∧ J, Proof. The proof follows from the above results, taking into account the fact that, under projective deformation, the vertical vector field ξ = C, the vector form µ = [J, C] = J , and hence the vector 1-form L = P J + dJP ⊗ C, together with the following relations: C(P) = P, [C, S] = S, [C, h] = 0, [C, J ] = −J, [C,L] = 0, hC = LC = 0, [S, J ] = −Γ, vC = C, vS = 0, vΓ = −v. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 11 of 18 4. Some special cases Many of the known special Finsler metrics yield a deformation for the background geodesic spray. As examples, in this section, we consider two rich classes of Finsler metrics. Namely, we consider the class of (α, β)-metrics and the class of spherically symmetric Finsler metrics. 4.1. (α, β)-metrics The geodesic spray S̃ of an (α, β)-metric F is given by [4, 20] G̃i = Gi + αQsi0 +Θ {−2αQs0 + r00} { yi α + Q′ Q− sQ′ b i } , (28) where G̃i (resp. Gi) are the coefficients of the geodesic spray of F (resp. α), and rij = 1 2 (bi|j + bj|i), sij = 1 2 (bi|j − bj|i), r00 := rijy iyj , si0 := sijy j , sij := shja ih, sj := sijb i, bi := bja ij Q := ϕ′ ϕ− sϕ′ , (29) Θ := Q− sQ′ 2(1 + sQ+ (b2 − s2)Q′) , (30) the symbol | refers to the covariant derivative with respect to the Levi-Civita connection of α, and Q′ (resp. ϕ′) mean the derivative of Q (resp. ϕ) with respect to s. The two sprays S̃ (resp. S) of the metrics F (resp. α) are related by S̃ = S − 2Pξ, where the vertical vector ξ is given by Pξ = Pξi∂̇i = ( αQsi0 +Θ {−2αQs0 + r00} { yi α + Q′ Q− sQ′ b i }) ∂̇i Let’s study when the two sprays are protectively related (that is, ξ = C) as shown in the following theorem. Theorem 2. Let F = αϕ(s) be an (α, β)-metric, then the associated sprays with F and the Riemannian metric α are projectively related if and only if 2αϕ′si0 + r00ϕ ′′bi − r00ϕ ′′ β α2 yi = 0. (31) In this case, we have S = Sα − 2PC, P = 1 2 r00ϕ ′ αϕ . S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 12 of 18 Proof. In [21], C. Shibata introduced the β-change of a Riemannian metric α, that is, F = f(α, β) where f is homogeneous function of degree 1 in α and β. He characterized when this change is projective change, that is, the sprays of F and α are projectively related. Namely, the two sprays are projectively related if and only if 2qsi0 + q0r00 ( bi − β α2 yi ) = 0, where q := f∂βf and q0 := f∂2 βf . Now, since f is homogeneous function of degree 1 in α and β, then we can write F = αf ( 1, β α ) = αf(1, s) = αϕ(s), where we set ϕ(s) := f(1, s). That is, q = f∂βf = αϕϕ′, q0 = f∂2 βf = ϕϕ′′. Moreover, contracting (31) by bi, we get 2αϕ′s0 = −r00ϕ ′′ ( b2 − β2 α2 ) . By substituting from the above equation and (31) into (28), we get Gi = Gi α + 1 2 r00ϕ ′ αϕ yi. Corollary 3. Let F = αϕ(s) be an (α, β)-metric and β be a closed 1-form, then the associated sprays with F and the Riemannian metric α are projectively related if and only if F is of Randers type or β is parallel with respect to the Levi-Civita connection of α. Proof. Assume that F = αϕ(s) is an (α, β)-metric which is projectively related to the Riemannian metric α as well as that β is a closed 1-form. Then sij = 0 and hence si0 = 0. That is (31) reduces to r00ϕ ′′ ( bi − β α2 yi ) = 0. Using [22, Lemma 3.2], bi − β α2 y i ̸= 0. Then, we have r00 = 0 or ϕ′′ = 0. The later implies that ϕ = c1s+ c2 which means that F is of Randers type. The choice r00 = 0 implies r00 = rijy iyj = 0. Taking the derivative twice with respect to yh and yk respectively together with using the fact that β is closed, we get rhk = 1 2 (bh|k + bk|h) = bh|k = 0. That is, β is parallel and this complete the proof. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 13 of 18 4.2. Spherically symmetric metrics A spherically symmetric Finsler metric F on Bn(r0) ⊂ Rn is given by F (x, y) = u φ (r, s) , where r = |x|, u = |y|, s = ⟨x, y⟩ |y| , | · | denotes the standard Euclidean norm and ⟨·, ·⟩ is the standard inner product on Rn, for more details, we refer for example to [23–25]. Moreover, the geodesic spray coefficients Gi of F are given by Gi = uPyi + u2Qxi, (32) where the functions P and Q have the following formulae Q := 1 2r −φr + sφrs + rφss φ− sφs + (r2 − s2)φss , P := −Q φ (sφ+ (r2 − s2)φs) + 1 2rφ (sφr + rφs). (33) The spray S of a spherically symmetric metric F is given by S = S0 − 2Pξ, where S0 is the spray of the Euclidean metric and Pξ = uPC + u2Qxi∂̇i. From which we get the following known corollary. Corollary 4. The spray of a spherically symmetric metric F = uφ(r, s) is projectively flat if and only if Q = 0. 5. Invariant Jacobi endomorphism and curvature In this section, under the projective-like deformation (18), we study some properties of the Jacobi endomorphism, Barthel connection, and the curvature of Barthel connection [26, 27]. Using the property (13) 3R = [J,Φ], Φ = iSR, we have the following theorem. Theorem 3. Under the general deformation (18), the Jacobi endomorphism Φ is invariant if and only if the curvature R is invariant. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 14 of 18 Proof. Let S be a spray on a manifold M with the Jacobi endomorphism Φ and curvature R. Now, consider the general deformation (18) of S. Assume that Φ is invarinat under the deforamtion (18), that is, Φ̃ = Φ. Then, we have 3R̃ = [J, Φ̃] = [J,Φ] = 3R. That is, the curvature R is invariant. Conversely, let R be invariant, then we get Φ̃ = i S̃ R̃ = iS−2ξR = iSR− 2iζR = Φ− 2iξR. But using the fact that R is semi-basic, we have iζR = 0. Consequently, Φ̃ = Φ, i.e., Φ is invariant. Proposition 3. Under the projective-like deformation (18), with deformation factor P(x, y), the curvature Barthel connection R is invariant if and only if the vertical one form L sat- isfies the following relation [h,L] = 1 2 [L,L], where h is the corresponding horizontal projection of the spray S. Proof. The proof is clear and we omit it. From which together with Theorem 3, we conclude that Theorem 4. Under the projective-like deformation (18), with deformation factor P(x, y), the following assertions are equivalent (i) the vertical one form L satisfies [h,L] = 1 2 [L,L]. (ii) the Jacobi endomorphism Φ is invariant. (iii) the curvature Barthel connection R is invariant. Proposition 4. The projective deformation (27) preserves the curvature the projective factor P is a Funk function, that is, dhP = PdJP. Proof. It is easy to see that the curvature of a spray S is preserved under any defor- mation of S if and only if R̃ = R, that is dJdhP ⊗ C + (PdJP − dhP) ∧ J = 0. The above equation is satisfied if and only if PdJP − dhP = 0. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 15 of 18 In the following two examples, for simplicity, we consider S̃ = S − 2Pξ = S − 2ζ, where we set ζ = Pξ. For calculations, we use the Finsler package [28], moreover, PDF and Maple files of the calculation are posted on https://github.com/salahelgendi/Calculations-for-the-paper-Deformation-of-sprays.git Example 2. Consider M = {(x1, x2) ∈ R2 : x1 > 0 with the Finsler function F F = √ x1((y1)2 + (y2)2), where (x1, x2; y1, y2) ∈ TR2 = R4. The coefficients Gi of the geodesic spray are given by G1 = 1 4 (y1)2 − (y2)2 x1 , G2 = 1 4 y1y2 x1 . The components of the Jacobi endomorphism are given by R1 1 = −1 2 (y2)2)2 (x1)2 , R1 2 = 1 2 y1y2)2 (x1)2 , R2 1 = 1 2 y1y2)2 (x1)2 , R2 2 = −1 2 (y1)2 (x1)2 . Now, consider the conformal transformation F̃ = e− 1 2 ((x1)2−(x2)2+x1+x2))F. Then the spray coefficients of F̃ are given by G̃1 = G1 − 1 4 2(x1)2(y1)2 − 2(x1)2(y2)2 − 4x1x2y1y2 + x1(y1)2 + 2x1y1y2 − x1(y2)2 x1 . G̃2 = G2 − 1 4 4(x1)2y1y2 + 2x1x2(y1)2 − 2x1x2(y2)2 − x1(y1)2 + 2x1y1y2 + x1(y2)2 x1 . We can write S̃ = S − 2Pξ = S − 2ζ, then the coefficients G̃i as follows: G̃1 = G1 + ζ1, G̃2 = G2 + ζ2. Hence the vector ζ is given by ζ = ζ1∂̇1 + ζ2∂̇2 where ζ1 = −1 4 2(x1)2(y1)2 − 2(x1)2(y2)2 − 4x1x2y1y2 + x1(y1)2 + 2x1y1y2 − x1(y2)2 x1 ζ2 = −1 4 4(x1)2y1y2 + 2x1x2(y1)2 − 2x1x2(y2)2 − x1(y1)2 + 2x1y1y2 + x1(y2)2 x1 S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 16 of 18 The components of Ri j of the Jacobi endomorphism are equal since the Jacobi endomor- phism is preserved under this transformation, moreover the components of R̃i j are given by R̃1 1 = R1 1 = 1 2 (y2)2 (x1)2 , R̃1 2 = R1 2 = −1 2 y1y2 (x1)2 , R̃1 2 = R2 1 = −1 2 y1y2 (x1)2 , R̃1 2 = R2 2 = 1 2 (y1)2 (x1)2 . The components of Ri jk of the curvature is preserved under this transformation, more- over the components of R̃i jk are given by R̃1 12 = R1 12 = −1 2 y2 (x1)2 , R̃1 21 = R1 21 = 1 2 y2 (x1)2 , R̃2 12 = R2 12 = 1 2 y1 (x1)2 , R̃2 21 = R2 21 = −1 2 y1 (x1)2 . Example 3. Consider M = R3 with the Finsler function F F = 4 √ (y1)4 + (y2)4 + (y3)4, where (x1, x2, x3; y1, y2, y3) ∈ TR3 = R6. The coefficients Gi of the geodesic spray are given by Gi = 0. The components of the Jacobi endomorphism are given by Ri j = 0, Ri jk = 0. Now, consider the deformation of F as follows F̃ = 4 √ f1(x1)(y1)4 + f2(x2)(y2)4 + f3(x3)(y3)4. Then the spray coefficients of F̃ are given by G̃1 = (y1)2 8f1(x1) df1(x 1) dx1 , G̃2 = (y2)2 8f2(x2) df2(x 2) dx2 , G̃3 = (y3)2 8f3(x3) df3(x 3) dx3 . Consider S̃ = S − 2Pξ = S − 2ζ, then the coefficients G̃i as follows: G̃1 = G1 + ζ1, G̃2 = G2 + ζ2, G̃3 = G3 + ζ3. Hence the vector ζ is given by ζ = ζ1∂̇1 + ζ2∂̇2 + ζ3∂̇3 where ζ1 = (y1)2 8f1(x1) df1(x 1) dx1 , ζ2 = (y2)2 8f2(x2) df2(x 2) dx2 , ζ3 = (y3)2 8f3(x3) df3(x 3) dx3 . The components of Ri j of the Jacobi endomorphism are equal since the Jacobi endomor- phism is preserved under this transformation. That is, we have R̃i j = Ri j = 0, R̃i jk = Ri jk = 0. S. G. Elgendi, A. Soleiman / Eur. J. Pure Appl. Math, 18 (3) (2025), 6204 17 of 18 Acknowledgements The authors sincerely thank the referee for their constructive comments and helpful suggestions, which have enhanced the quality of the paper. References [1] P. L. Antonelli, R. Ingarden, and M. Matsumoto. The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology. Kluwer Academic Publishers, Netherlands, 1993. [2] I. Bucataru and Z. Muzsnay. Projective metrizability and formal integrability. SIGMA, 7, 2011. [3] I. Bucataru and Z. Muzsnay. 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