Int. J. Anal. Appl. (2022), 20:26 Fractional Order Riemann Curvature Tensor in Differential Geometry Wedad Saleh∗ Department of Mathematics, Taibah University, Al- Medina, Saudi Arabia ∗Corresponding author: wed_10_777@hotmail.com Abstract. This study discussed some interesting aspects and features of fractional curvature in the differential manifold. In particular, Riemannian fractional curvature tensor, Livi-Civita fractional con- nection and Bianchi fractional identity are presented. 1. Introduction In mathematics, several special functions appear in many applications such as the Gamma function that plays some significant roles in the theory of integral differential equations in particu- lar fractional calculus. Thus, we begin with some definitions, for the details we refer to ( [1], [15], [8]). The Gamma function of a positive integer η is again a positive integer, while the gamma function Γ(−η) of a negative integer changes to infinities. The Gamma function any positive η value is defined as follows: Γ(η) = ∫ ∞ 0 tη−1e−tdt. The Gamma function Γ(η) is considered as a generalization of the factorial and Γ(η) is defined for η > 0 by the integral Γ(η) = ∫ ∞ 0 tη−1e−t dt. In the classical sense since Γ(0) = Γ(1) 0 , then it follows that Γ(η) is not defined for integers η ≤ 0. However, the extension formula gives finite values for Γ(η), for <(η) ≤ 0 since Γ(η) is analytic Received: Mar. 7, 2022. 2010 Mathematics Subject Classification. 26A33, 58A05, 58D17. Key words and phrases. fractional geometrical object; fractional manifold; revised Riemann-Liouville fractional calculus on manifolds. https://doi.org/10.28924/2291-8639-20-2022-26 ISSN: 2291-8639 © 2022 the author(s). https://doi.org/10.28924/2291-8639-20-2022-26 2 Int. J. Anal. Appl. (2022), 20:26 everywhere except at η = 0,−1,−2, ..., and the residue at η = k is given by Resη=kΓ(η) = (−1)k k! . Now, if η > 0, then Γ(η + 1) = ηΓ(η). (1.1) Equation (1.1) can be used to define Γ(η) for η < 0 and η 6= −1,−2, . . . and further, this is one of the most important formulas that were satisfied by the Gamma function. Even though the Gamma function is defined as a locally summable function on the real line by [17] Γ (η) = ∫ ∞ 0 tη−1e−tdt, η > 0. (1.2) In the classical sense, Γ(η) function was not defined for the negative integer thus, there was an open problem to give a satisfactory definition. However, by using the neutral limit, it has been shown in [21] that the Gamma function (1.2) is defined as follows: Γ (η) = N − lim ε→0 ∫ ∞ ε tη−1e−tdt for η 6= 0,−1,−2, ..., and this function can be defined by neutral limit such as Γ (−n) = N − lim ε→0 ∫ ∞ ε t−n−1e−tdt = ∫ ∞ 1 t−n−1e−tdt + ∫ 1 0 t−n−1 [ e−t − n∑ i=0 (−1)i i! ti ] dt − n−1∑ i=0 (−1)i i!(n− i) ,n ∈N. It was also proven in [20] the existence of r the derivative of the Gamma function and defined it by equation Γ(r)(0) = N − lim ε→0 ∫ ∞ ε t−1 lnr te−tdt = ∫ ∞ 1 t−1 lnr te−tdt + ∫ 1 0 t−1 lnr t [ e−t − 1 ] dt Γ(r)(−n) = N − lim ε→0 ∫ ∞ ε t−n−1 lnr te−tdt = ∫ ∞ 1 t−n−1 lnr te−tdt + ∫ 1 0 t−n−1 lnr t [ e−t − n∑ i=0 (−1)i i! ti ] dt − n−1∑ i=0 (−1)i i! r!(n− i)−r−1 Int. J. Anal. Appl. (2022), 20:26 3 for r ∈N0 and n ∈N. Also, Γ(−r) = (−1)r r! ψ(r) − (−1)r r! γ for r = 1, 2, . . . , where ψ(r) = r∑ i=1 1 i . Thus, the definition can be extended to the whole real line where, Γ(0) = Γ ′ (1) = −γ, where γ denotes Euler’s constant, see [22]. For a function f : V ⊂R→R with 0 ∈ V , the fractional derivative of order α is defined by: dα dtα f (t) = 1 Γ(−α) ∫ t 0 f (s) − f (0) (t − s)1+α ds, α < 0 (1.3) dα dtα f (t) = 1 Γ(n−α) dn dtn ∫ t 0 f (s) − f (0) (t − s)α−n+1 ds, α > 0 (1.4) where n is the first integer greater than or equal to α. The relation (1.3) gives a fractional integral and (1.4) gives a fractional derivative. We express some of the operators of fractional derivatives, see for example, [4,7,9,10,12,16]. (1) dα dtα tγ = Γ(1 + γ) Γ(1 + γ + α) tγ−α, α ∈R or (α ∈C) and 1 + γ 6= 0,−1, ...,−n, (2) dn dtn dα dtα f (t) = dn+α dtn+α f (t), n ∈N, (3) dα dtα (f1(t) + f2(t)) = dα dtα f1(t) + dα dtα f2(t), (4) dα dtα (Cf (t)) = C dα dtα f (t), where C is a constant, (5) dα dtα f (βt) = βα dα [d(βt)]α f (βt). It is well known that fractional calculus is an essential and advantageous branch of mathematics, having a broad range of applications at almost every department of sciences. Techniques of fractional calculus have been employed in the modeling of many different phenomena in engineering, physics , and mathematics.The problem in fractional calculus is not only essential but also quite challenging ,which usually involves complicated mathematical solution techniques. However, a general solution theory for almost every issue in this area has yet to be established. Each application has developed its approaches and implementations. Consequently, a single standard method for the problems in fractional calculus has not emerged yet. Therefore, funding reliable and efficient solution tech- niques along with fast implementation methods are significantly essential and still active research areas. 4 Int. J. Anal. Appl. (2022), 20:26 Further, it is also realized that the operators of fractional integration and derivation have physical and geometric interpretations, which streamline along with their utilization for related issues in various fields of science( see [2], [8], [10], [11], [12], [14], [18], [19]). Moreover, the fractional differential calculus on a differential manifold is studied in( [2], [3], [4], [6], [13]). Even though fractional calculus is a handy and important topic, however, the research on geometric interpretation and applications are limited ,and not many in current literature. Thus, in this study, we focus on the Riemannian curvature tensor, Livi-Civita connection and Bianchi’s identity on fractional differentiable manifolds and discuss some related properties. We also give some examples. 2. Fractional Differential Calculus on Manifolds Assume that N be an m-dimensional differential manifold (V,xi ) a local coordinate system on N and V0 = {x ∈ V : 0 ≤ xi ≤ bi, i = 1, 2, ...,m} [5]. For a function f : V0 →R, the fractional derivative with respect to xi : ∂ α i f (x)= 1 Γ(n−α) ∂ n xi ∫ xi 0 f (x1, ...,xi−1,s,xi+1, ...,xm)− f (x1, . . . ,xi−1,0,xi+1, . . . ,xm) (xi − s)α−n+1 ds, where ∂nxi = ∂ ∂xi ◦ ∂ ∂xi ◦ ...◦ ∂ ∂xi (n times, i is fixed, α ≥ 0). For α ∈ (0,1),γ > −1, ∂ α i (xi) γ = Γ(1+γ) Γ(1+γ −α) ;∂ α i = δ j i . A fractional vector field V ⊂ N is an object of the form Xα = Xαi ∂ α i , where X α i ∈=V (N) i =1, ...,m. The fractional vector fields on V and χαV is generated by the operators ∂ α i , i = 1,2, ...,m are denoted by χ α V . If c : x = x(t),t ∈ I is a parameterized curve in U then the fractional tangent vector field of c is given by x α (t)= 1 Γ(1+α) ∂ α t xi(t)∂ α i . A fractional covariant derivative is given by 5αXαY α = X α i (∂ α i Y α j + Γ̃ j ikY α k )∂ α j where Xα,Y α ∈ χαU and Γ̃ j ik the functions defining the coefficients of a fractional linear connection on N. They are determined by the relations 5α∂α i ∂ α k = Γ̃ j ik∂ α j . Since it is essential to study fractional vector fields on a differentiable manifold N. For Rn, there is an obvious way to do this. Recall that χα(Rn) denotes the space of fractional differentiable vector fields defined on R. Examples are the fractional vector fields ∂α ∂uα1 , ..., ∂α ∂uαn determined by the natural coordinate functions u1, ...,un. Definition 2.1. Fractional Riemannian metric F on m-dimensional manifold N defines for every point p ∈ N, the scalar product of fractional tangent vectors in the fractional tangent space Tαp N depending on the point p. Int. J. Anal. Appl. (2022), 20:26 5 Let Aα = Aαi ∂ α i and B α = Bαj ∂ α j any two fractional vectors tangent to the manifold N at the point p with coordinates x =(x1, ...,xm) (A α,Bα ∈ Tαp N) the scalar product is equal to 〈Aα,Bα〉F |p = A α i (x)g̃ij(x)B α j (x) = (A α 1 , ...,A α n)   g̃11 ... g̃1n . ... . . ... . . ... . g̃n1 ... g̃nn     Bα1 . . . Bαn   where (1) F(Aα,Bα)= F(Bα,Aα), i.e., g̃ij = g̃ji (symmetricity condition). (2) F(Aα,Aα) > 0 if Aα 6=0, i.e. g̃ijuαi u α j ≥ 0, g̃iju α i u α j =0 iff u α 1 = ... = u α n =0 (positive definiteness). (3) F(Aα,Bα) |p=x, i.e. g̃ij(x) are smooth function where 0 < α < 1. Components of tensor field F in coordinate system are matrix valued functions g̃ij(x) F = g̃ij(x)d α xi ⊗dαxj. Rule of Transformation for Entries of the Matrix g̃ij(x) g̃ij(x)- entries of the matrix ‖ g̃ij ‖ are components of tensor field F in a given coordinate system. How do these components transform under transformation of coordinates {xi}→{xi′} ? F = g̃ijdx α i ⊗dx α j = g̃ij ( ∂xαi ∂xα i′ dx α i′ ) ⊗ ( ∂xαj ∂xα j′ dx α j′ ) = ∂xαi ∂xα i′ g̃ij ∂xαj ∂xα j′ dx α i′ ⊗dx α j′ = g̃i′j′dx α i′ ⊗dx α j′ . Hence, g̃i′j′ = ∂xαi ∂xα i′ g̃ij ∂xαj ∂xα j′ . Example 2.1. Consider R2 with fractional polar coordinates in the domain y > 0, X =(rαcosαϕ,rαsinαϕ), then ∂αX ∂rα = (α!cos α ϕ,α!sin α ϕ) . ∂αX ∂ϕα = ( α!e iαπ r α sin α ϕ,α!r α cos α ϕ ) . g̃ij = ( (α!)2 [ cos2α ϕ+sin2α ϕ ] (α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ (α!)2rα [ eαπi +1 ] sinα ϕcosα ϕ (α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] ) We have that F =(α!) 2 [ cos 2α ϕ+sin 2α ϕ ] (dr α ) 2 +2(α!) 2 r α [[ e iαπ +1 ] sin α ϕcos α ϕ ] dr α dϕ α +(α!) 2 r 2α [ e 2iαπ sin 2α ϕ+cos 2α ϕ ] (dϕ α ) 2}. 6 Int. J. Anal. Appl. (2022), 20:26 Notice that , as expected, when α =1, one recovers the classical formula. F =(dr) 2 + r 2 (dϕ) 2 . α = 0.1 0.2 g̃rr 0.905[C 0.2 + S0.2] 0.843[C0.4 + S0.4] g̃rϕ = g̃ϕr 0.905r 0.1[i + 1]S0.2C0.2 0 g̃ϕϕ 0.905r0.2[−S0.2 −C0.2] 0.843r0.4[−S0.4 + C0.4] F 0.905[C0.2 + S0.2]dr0.2 0.843[C0.4 + S0.4]dr0.4 +1.8101r0.1[i + 1]S0.2C0.2(dr).1(dϕ).1 + +0.905r0.2[−S0.2 −C0.2](dϕ).2 0.843r0.4[−S0.4 + C0.4](dϕ)0.4 Table 1. C = cos ϕ,S = sin ϕ α = 0.3 0.4 g̃rr 0.805[C 0.6 + S0.6] 0.787[C0.8 + S0.8] g̃rϕ = g̃ϕr 0.805r 0.3[[i + 1]S0.3C0.3] 1.574r0.4S0.3C0.3 g̃ϕϕ 0.805r0.6[−S0.6 + C0.6] 0.787r0.8[S0.8 + C0.8] F 0.805[C0.6 + S0.6](dr)0.6 0.787[C0.8 + S0.8](dr)0.8 +1.61r0.3[[i + 1]S0.3C0.3](dr).3(dϕ).3 +3.148r0.4S0.4C0.4dr .4dϕ.4 +0.805r0.6[−S0.6 + C0.6](dϕ).6 +0.787r0.8[S0.8 + C0.8](dϕ).8 Table 2. C = cos ϕ,S = sin ϕ α = 0.5 0.6 g̃rr 0.785[C + S] 0.798[C 1.2 + S1.2] g̃rϕ = g̃ϕr 0.785r 0.5[i + 1]S0.5C0.5 0 g̃ϕϕ 0.785r[−S + C] 0.798r1.2[S1.2 + C1.2] F 0.785[C + S]r(dr) 0.798[C1.2 + S1.2](dr)1.2 +1.57r0.5[i + 1]S0.5C0.5(dr)0.5(dϕ)0.5 + 0.785r[−S + C]dϕ +0.798r1.2[S1.2 + C1.2](dϕ)1.2 Table 3. C = cos ϕ,S = sin ϕ Int. J. Anal. Appl. (2022), 20:26 7 α = 0.7 0.8 g̃rr 0.826[C 1.4 + S1.4] 0.867[C1.6 + S1.6] g̃rϕ = g̃ϕr 0.826r 0.7[i + 1]S0.7C0.7 1.734r0.8S0.8C0.8 g̃ϕϕ 0.826r1.4[−S1.4 + C1.4] 0.867r1.6[S1.6 + C1.6] F 0.826[C1.4 + S1.4](dr)1.4 0.867[C1.6 + S1.6](dr)1.6 +1.652r0.7[i + 1]S0.7C0.7(dr)0.7(dϕ)0.7 +3.468r0.8S0.8C0.8dr0.8dϕ0.8 +0.826r1.4[−S1.4 + C1.4](dϕ)1.4 +0.867r1.6[S1.6 + C1.6](dϕ)1.6 Table 4. C = cos ϕ,S = sin ϕ α = 0.9 1 g̃rr 0.925[C 1.8 + S1.8] 1 g̃rϕ = g̃ϕr 0.925r 0.9[i + 1]S0.9C0.9 0 g̃ϕϕ 0.925r1.8[−S1.8 + C1.8] r2 F 0.925[C1.8 + S1.8](dr)1.8 (dr)2 +1.85r0.9[i + 1]S0.9C0.9(dr)0.9(dϕ)0.9 + +0.925r1.8[−S1.8 + C1.8](dϕ)1.8 r2 (dϕ)2 Table 5. C = cos ϕ,S = sin ϕ Remark 2.1. Let N is an m-dimensional Riemannian manifold with fractional metric tensor g̃,then we shall denote the fractional derivatives of the elements of tensor g̃ as follows: g̃ij,k = ∂α ∂xαk g̃ij, and g̃ij,kl = ∂α ∂xαl ∂α ∂xαk g̃ij = ∂2α ∂xαl ∂x α k g̃ij, i, j,k, l =1, ...,n. Definition 2.2. Asymmetric fractional connection is called Levi-Civita fractional connection if it is compatible with metric, i.e., if it preserves the scalar product: ∂ α Xα 〈Y α ,Z α〉= 〈5αXαY α ,Z α〉+ 〈Y α,5αXαZ α〉 for arbitrary fractional vector fields Xα,Y α, and Zα. In local coordinates Christoffel symbols of Levi-Civita fractional connection are given by: Γ̃ k ij = 1 2 g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij). Proof. Since ∂ α j ei = Γ̃ m ij em (2.1) Γ̃ m ij emel =(∂ α j ei)el (2.2) 8 Int. J. Anal. Appl. (2022), 20:26 Γ̃ m ij gml = ∂ α j (ei.el)−ei(∂ α j el) = ∂ α j gil − Γ̃ m lj emei = ∂ α j gil − Γ̃ m lj gmi, (2.3) then Γ̃ m ij gml + Γ̃ m lj gmi = ∂ α j gil, (2.4) which implies that Γ̃ m ij g̃ml + Γ̃ m lj g̃mi = ∂ α j g̃il. (2.5) In this equation, the index m is a dummy, so only the indices i,j ,and l are specified. We can cyclically permute these indices to generate two more equations: Γ̃ m jl g̃mi + Γ̃ m il g̃mj = ∂ α l g̃ji (2.6) Γ̃ m li g̃mj + Γ̃ m ji g̃ml = ∂ α i g̃lj (2.7) since Γ̃mij = Γ̃ m ji , then Γ̃ m lj g̃mi + Γ̃ m il g̃mj = ∂ α l g̃ij (2.8) Γ̃ m il g̃mj + Γ̃ m ij g̃ml = ∂ α i g̃lj. (2.9) We can now add (2.5)to (2.9) and subtract (2.8)to get 2Γ̃ m ij g̃ml = ∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij 2Γ̃ m ij g̃mlg̃ kl = g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij) since g̃mlg̃ kl = δkm, then Γ̃ k ij = 1 2 g̃ kl (∂ α j g̃il +∂ α i g̃lj −∂ α l g̃ij), we can write Γ̃ k ij = 1 2 g̃ kl (g̃il,j + g̃lj,i − g̃ij,l). � Example 2.2. For 2-dimential polar coordinates X =(rαcosαϕ,rαsinαϕ). The metric tensor and its inverse here are: g̃ij = ( (α!)2 [ cos2α ϕ+sin2α ϕ ] (α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ (α!)2rα [ eαπi +1 ] sinα ϕcosα ϕ (α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] ) g̃i j = ( A−1(α!)2r2α [ e2απi sin2α ϕ+cos2α ϕ ] −A−1(α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ −A−1(α!)2rα [ e2απ +1 ] sinα ϕcosα ϕ A−1(α!)2 [ cos2α ϕ+sin2α ϕ ] ) . where A =(α!) 4 r 2α {[ cos 2α ϕ+sin 2α ϕ ][ e 2απi sin 2α ϕ+cos 2α ϕ ] − [ e απi +1 ] sin 2α ϕcos 2α ϕ } Therefore, ∂ α r g̃ij = ( 0 (α!)3 [ eαπi +1 ] sinα ϕcosα ϕ (α!)3 [ eαπi +1 ] sinα ϕcosα ϕ (α!)(2α)!rα [ e2απi sin2α ϕ+cos2α ϕ ] ) ∂ α ϕg̃ij = ( (α!)(2α)! [ eαπi +1 ] cosαϕsinαϕ (α!)3rα [ eαπi +1 ][ eαπi sin2α ϕ+cos2α ϕ ] (α!)3rα [ eαπi +1 ][ eαπi sin2α ϕ+cos2α ϕ ] (α!)(2α)!r2α [ eαπi +1 ] sinα ϕcosα ϕ ) . Int. J. Anal. Appl. (2022), 20:26 9 Then , Γ̃ r rr = 1 2 g̃ rl (∂ α r g̃rl +∂ α r g̃lr −∂ α l g̃rr) = −A−1(α!)5rα [ e απi +1 ]2 sin 2α ϕcos 2α ϕ, Γ̃ r rϕ = 1 2 g̃ rl (∂ α ϕg̃rl +∂ α r g̃lϕ −∂ α l g̃rϕ) = 0 = Γ̃ r ϕr, Γ̃ r ϕϕ = 1 2 g̃ rl (∂ α ϕg̃ϕl +∂ α ϕg̃lϕ −∂ α l g̃ϕϕ) = (α!) 2 r α (2A) −1 {[ e 2απi sin 2α ϕ+cos 2α ϕ ][ 2(α!) 3 r α [ e απi +1 ][ e απi sin 2α ϕ+cos 2α ϕ ] − (α!)(2α)!rα [ e 2απi sin 2α ϕ+sin 2α ϕ ]] − (α!)(2α)!rα [ e απi +1 ] sin 2α ϕcos 2α ϕ } , Γ̃ ϕ rr = 1 2 g̃ ϕl (∂ α ϕg̃rl +∂ α r g̃lr −∂ α l g̃rr) = −(α!)2(2A)−1 { (α!)(2α)!r α [ e απi +1 ]2 sin 2α ϕcos 2α ϕ+ [ cos 2α ϕ+sin 2α ϕ ][ e απi +1 ] × [ (α!) 3 r α [ e απi sin 2α ϕ+cos 2α ϕ ] +(α!) 3 sin α ϕcos α ϕ− (α!)(2α)!cosα ϕsinα ϕ ]} , Γ̃ ϕ rϕ = 1 2 g̃ ϕl (∂ α ϕg̃rl +∂ α r g̃lϕ −∂ α l g̃rϕ) = (α!) 2 (2A) −1 { −(α!)(2α)!2rα [ e απi +1 ] cos 2α ϕsin 2α ϕ +(α!)(2α)!r α [ cos 2α ϕ+sin 2α ϕ ][ e 2απi sin 2α ϕ+cos 2α ϕ ]} = Γ̃ ϕ ϕr, Γ̃ ϕ ϕϕ = 1 2 g̃ ϕl (∂ α ϕg̃ϕl +∂ α ϕg̃lϕ −∂ α l g̃ϕϕ) = (α!) 2 r 2α (2A) −1 { − [ e απi +1 ] sin α ϕcos α ϕ [ 2(α!) 3 [ e απi +1 ][ e απi sin 2α ϕ+cos 2α ϕ ] −(α!)(2α)! [ e 2απi sin 2α ϕ+cos 2α ϕ ]] +(α!)(2α)! [ e απi +1 ][ sin 2α ϕ+cos 2α ϕ ] sin α ϕcos α ϕ } . 3. Fractional Curvature Definition 3.1. The fractional curvature R̃ of order α of a Riemannian manifold N is a correspondence that associates to every pair Xα,Y α ∈ χα a mapping R̃(Xα,Y α): χα(N)×χα(N)→ χα(N) given by R̃(X α ,Y α )Z α =5αXα 5 α Y α Z α −5αY α 5 α Xα Z α −5α[Xα,Y α]Z α , where Zα ∈ χα and 5α is the fractional Riemannian connection. Remark 3.1. R̃(X α ,Y α )Z α = 5αXα 5 α Y α Z α −5αY α 5 α Xα Z α −5α[Xα,Y α]Z α = −(5αY α 5 α Xα Z α −5αXα 5 α Y α Z α −5α[Y α,Xα]Z α ) = −R̃(Y α,Xα)Zα. Proposition 3.1. The fractional curvature R̃ of a Riemannian manifold has the following properties: 10 Int. J. Anal. Appl. (2022), 20:26 (1) R̃ is bilinear in χα(N)×χα(N), that is, R̃(f X α +gY α ,Z α )W α = f R̃(X α ,Z α )W α +gR̃(Y α ,Z α )W α , R̃(X α , f Y α +gZ α )W α = f R̃(X α ,Y α )W α +gR̃(X α ,Z α )W α , where f ,g ∈=(M),Xα,Y α,Zα,Wα ∈ χα(N) (2) For any Xα,Y α ∈ χα(N), R̃(Xα,Y α) is linear R̃(X α ,Y α )(Z α +W α )= R̃(X α ,Y α )Z α + R̃(X α ,Y α )W α , R̃(X α ,Y α )(f Z α )= f R̃(X α ,Y α )Z α , where Zα,Wα ∈ χα(N) Proof. (1) R̃(f X α +gY α ,Z α )W α =5αf Xα+gY α 5 α Zα W α −5αZα 5 α f Xα+gY α W α −5α[f Xα+gY α,Zα]W α =(f 5αXα +g5 α Y α)5 α Zα W α −5αZα(f 5 α Xα W α +g 5αY α W α ) −5αf [Xα,Zα]+g[Y α,Zα]−(Zαf )Xα−(Zαg)Y α W α = f 5αXα 5 α ZαW α +g 5αY α 5 α ZαW α − (Zαf )5αXα W α −f 5αZα 5 α XαW α − (Zαg)5αY α W α −g 5αZα 5 α Y αW α − f 5α[Xα,Zα] W α −g 5α[Y α,Zα] W α +(Z α f )5αXα W α +(Z α g)5αY α W α = f (5αXα 5 α Zα W α −5αZα 5 α Xα W α −5α[Xα,Zα]W α ) +g(5αY α 5 α Zα W α −5αZα 5 α Y α W α −5α[Y α,Zα]W α ) = f R̃(X α ,Z α )W α +gf R̃(Y α ,Z α )W α . Also, R̃(X α , f Y α +gZ α )W α = −R̃(f Y α +gZα,Xα) = −f R̃(Y α,Xα)Wα −gR̃(Zα,Xα)Wα = f R̃(X α ,Y α )W α +gR̃(X α ,Z α )W α . (2) R̃(X α ,Y α )(Z α +W α ) =5αXα 5 α Y α (Z α +W α )−5αY α 5 α Xα (Z α +W α )−5α[Xα,Y α](Z α +W α ) =5αXα 5 α Y α Z α +5αXα 5 α Y α W α −5αY α 5 α Xα Z α −5αY α 5 α XαW α −5α[Xα,Y α]Z α −5α[Xα,Y α]W α =(5αXα 5 α Y α Z α −5αY α 5 α Xα Z α −5α[Xα,Y α]Z α ) +(5αXα 5 α Y α W α −5αY α 5 α Xα W α −5α[Xα,Y α]W α ) = R̃(X α ,Y α )Z α + R̃(X α ,Y α )W α . Int. J. Anal. Appl. (2022), 20:26 11 Also, R̃(X α ,Y α )(f Z α ) =5αXα 5 α Y α (f Z α )−5αY α 5 α Xα (f Z α )−5α[Xα,Y α](f Z α ) =5αXα((Y α f )Z α + f 5αY α Z α )−5αY α((X α f )Z α + f 5αXα Z α ) −(([Xα,Y α]f )Zα + f 5α[Xα,Y α] Z α ) = X α (Y α f )Z α +(Y α f )5αXα Z α +(X α f )5αY α Z α + f 5αXα 5 α Y αZ α −Y α(Xαf )Zα +(Xαf )5αY α Z α +(Y α f )5αXα Z α + f 5αY α 5 α XαZ α −([Xα,Y α]f )Zα − f 5α[Xα,Y α] Z α =([X α ,Y α ]f )Z α + f (5αXα 5 α Y α Z α −5αY α 5 α Xα Z α −5α[Xα,Y α]Z α )− ([Xα,Y α]f )Zα = f R̃(X α ,Y α )Z α . � Proposition 3.2 (Bianchi Fractional Identity). R̃(X α ,Y α )Z α + R̃(Y α ,Z α )X α + R̃(Z α ,X α )Y α =0. Proof. R̃(X α ,Y α )Z α + R̃(Y α ,Z α )X α + R̃(Z α ,X α )Y α =5αXα 5 α Y α Z α −5αY α 5 α Xα Z α −5α[Xα,Y α]Z α +5αY α 5 α ZαX α −5αZα 5 α Y α X α −5α[Y α,Zα]X α +5αZα 5 α XαY α −5αXα 5 α Zα Y α −5α[Zα,Xα]Y α =5αXα[Y α ,Z α ]+5αY α[Z α ,X α ]+5αZα[X α ,Y α ] −5α[Xα,Y α] Z α −5α[Y α,Zα]X α −5α[Zα,Xα]Y α = [X α , [Y α ,Z α ]]+ [Y α , [Z α ,X α ]]+ [Z α , [X α ,Y α ]] = 0. � In local coordinates R̃(∂ α i ,∂ α j )∂ α k = R̃ l ijk∂ α l , and R̃ijkm = 〈 R̃(∂ α i ,∂ α j )∂ α k ,∂ α m 〉 = 〈 R̃ l ijk∂ α l ,∂ α m 〉 = R̃ l ijk 〈∂ α l ,∂ α m〉 = R̃ l ijkg̃ml. The fractional Riemannian curvature tensor acts on fractional vector fields as follows: R̃(X α ,Y α ,Z α ,W α )= 〈 R̃(X α ,Y α )Z α ,W α 〉 . Proposition 3.3. (1) R̃ijkl + R̃jkil + R̃kijl =0. (2) R̃ijkl =−R̃jikl. (3) R̃ijkl =−R̃ijlk. (4) R̃ijkl = R̃klij. 12 Int. J. Anal. Appl. (2022), 20:26 Proof. (1) is just the Bianchi fractional identity again. (2) R̃ijkl = 〈 R̃(∂ α i ,∂ α j )∂ α k ,∂ α l 〉 = 〈 −R̃(∂αj ,∂ α i )∂ α k ,∂ α l 〉 = − 〈 R̃(∂ α j ,∂ α i )∂ α k ,∂ α l 〉 = −R̃jikl. (3) is equivalent to R̃ijkk =0, whose proof follows: R̃ijkk = 〈 R̃(∂ α i ,∂ α j )∂ α k ,∂ α k 〉 = 〈 5α∂α i 5α∂α j ∂ α k −5 α ∂α j 5α∂α i ∂ α k −5 α [∂α i ,∂α j ]∂ α k ,∂ α k 〉 , but 〈 5α∂α j 5α∂α i ∂ α k ,∂ α k 〉 = ∂ α j 〈 5α∂α i ∂ α k ,∂ α k 〉 − 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α k 〉 , and 〈 5α[∂α i ,∂α j ]∂ α k ,∂ α k 〉 = 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 , then R̃ijkk = ∂ α j 〈 5α∂α i ∂ α k ,∂ α k 〉 −∂αi 〈 5α∂α j ∂ α k ,∂ α k 〉 + 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 = 1 2 ∂ α j (∂ α i 〈∂ α k ,∂ α k 〉)− 1 2 ∂ α i ( ∂ α j 〈∂ α k ,∂ α k 〉 ) + 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉 = − 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉+ 1 2 [ ∂ α i ,∂ α j ] 〈∂αk ,∂ α k 〉=0. (4) By Bianchi fractional identity we have R̃ijkl + R̃jkil + R̃kijl =0 R̃jkli + R̃klji + R̃ljki =0 R̃klij + R̃likj + R̃iklj =0 R̃lijk + R̃ijlk + R̃jlik =0 summing the equations above, we obtain 2R̃kijl +2R̃ljki =0, then R̃kijl =−R̃ljki = R̃jlki. � Proposition 3.4. The following expression holds 2R̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2Γ̃ rjkΓ̃ s img̃rs +2Γ̃ r ikΓ̃ s jmg̃rs. Proof. From the definition of the Christoffel symbols, 5α∂α i ∂αj = Γ̃ k ij ∂ α k , 2 〈 5α∂α i ∂ α j ,∂ α m 〉 = 2 〈 Γ̃ k ij ∂ α k ,∂ α m 〉 = 2Γ̃ k ij g̃mk = g̃im,j + g̃jm,i − g̃ij,m, Int. J. Anal. Appl. (2022), 20:26 13 an appropriate rearrangement of the indices yields the following expression: 2 〈 5α∂α j ∂ α k ,∂ α m 〉 = 2 〈 Γ̃ i jk∂ α i ,∂ α m 〉 = 2Γ̃ i jkg̃im = g̃jm,k + g̃km,j − g̃jk,m. (3.1) ∂ α i < 5 α ∂α j ∂ α k ,∂ α m >=< 5 α ∂α i 5α∂α j ∂ α k ,∂ α m > + < 5 α ∂α j ∂ α k ,5 α ∂α i ∂ α m > whence, by (3.1) 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 +2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 = 2∂ α i 〈 5α∂α j ∂ α k ,∂ α m 〉 = ∂ α i ( g̃img il (∂ α k gjl +∂ α j gkl −∂ α l gjk) ) = g̃jm,ki + g̃km,ji − g̃jk,mi. (3.2) By switching i and j we also have that 2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 = g̃im,kj + g̃km,ij − g̃ik,mj. (3.3) Combining (3.2) and (3.3) yields 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 −2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 . By definition R̃ ( ∂ α i ,∂ α j ) ∂ α k =5 α ∂α i 5α∂α j ∂ α k −5 α ∂α j 5α∂α i ∂ α k whence 2R̃ijkm = 2 〈 R̃(∂ α i ,∂ α j )∂ α k ,∂ α m 〉 = 2 〈 5α∂α i 5α∂α j ∂ α k ,∂ α m 〉 −2 〈 5α∂α j 5α∂α i ∂ α k ,∂ α m 〉 , so we have proven that 2R̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 +2 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 . By the definition of the Christoffels, 〈 5α∂α i ∂ α k ,5 α ∂α j ∂ α m 〉 = 〈 Γ̃ r ik∂ α r , Γ̃ s jm∂ α s 〉 = Γ̃ r ikΓ̃ s jm 〈∂ α r ,∂ α s 〉 = Γ̃ r ikΓ̃ s jmg̃rs, 〈 5α∂α j ∂ α k ,5 α ∂α i ∂ α m 〉 = 〈 Γ̃ r jk∂ α r , Γ̃ s im∂ α s 〉 = Γ̃ r jkΓ̃ s im 〈∂ α r ,∂ α s 〉 = Γ̃ r jkΓ̃ s img̃rs, 14 Int. J. Anal. Appl. (2022), 20:26 then 2R̃ijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2Γ̃ rjkΓ̃ s img̃rs +2Γ̃ r ikΓ̃ s jmg̃rs. � Remark 3.2. If α =1,then 2Rijkm = gjm,ki +gkm,ji −gjk,mi −gim,kj −gkm,ij +gik,mj −2Γ rjkΓ s imgrs +2Γ r ikΓ s jmgrs. Since gkm,ji = gkm,ij, then 2Rijkm = gjm,ki −gjk,mi −gim,kj +gik,mj −2Γ rjkΓ s imgrs +2Γ r ikΓ s jmgrs, then 2Rijkm = g̃jm,ki + g̃km,ji − g̃jk,mi − g̃im,kj − g̃km,ij + g̃ik,mj −2Γ rjkΓ s imgrs +2Γ r ikΓ s jmgrs. Remark 3.3. For any pair of fractional tangent vectors Xα,Y α ∈ Tαp N we shall denote with Γ̃(Xα,Y α) the following fractional vector in Tαp N: Γ̃(X α ,Y α )= Γ̃ k ij X α i Y α j ∂ α k . Proposition 3.5. The following expressions hold for any pair Xα,Y α ∈ Tαp N: 2R̃(X α ,Y α ,Y α ,X α ) = ∂ α i ( g̃img il )( ∂ α k gjl +∂ α j gkl −∂ α l gjk ) + g̃img il ( ∂ α i ∂ α k gjl +∂ α i ∂ α j gkl −∂ α i ∂ α l gjk ) −∂αj ( g̃jmg jl ) (∂ α k gil +∂ α i gkl −∂ α l gik)− g̃jmg jl ( ∂ α j ∂ α k gil +∂ α j ∂ α i gkl −∂ α j ∂ α l gik ) +2 ‖ Γ̃ (Xα,Y α) ‖2 −2 〈 Γ̃ (X α ,X α ) , Γ̃ (Y α ,Y α ) 〉 . Proof. Since g̃rsX α i Y α j Y α k X α mΓ̃ r ikΓ̃ s jm = 〈 X α i Y α k Γ̃ r ik∂ α r ,Y α j X α mΓ̃ s jm∂ α s 〉 = 〈 Γ̃(X α ,Y α ), Γ̃(X α ,Y α ) 〉 =‖ Γ̃(Xα,Y α) ‖2, and g̃rsX α i Y α j Y α k X α mΓ̃ r jkΓ̃ s im = 〈 X α i X α mΓ̃ s im∂ α s ,Y α j Y α k Γ̃ r jk∂ α r 〉 = 〈 Γ̃(X α ,X α ), Γ̃(Y α ,Y α ) 〉 , This completes the proof. � Remark 3.4. If α =1, then 2R(X,Y,Y,X)= gjm,ki −gjk,mi −gim,kj +gik,mj +2 ‖ Γ(X,Y ) ‖2 −2〈Γ(X,X),Γ(Y,Y )〉 . Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper. Int. J. Anal. Appl. (2022), 20:26 15 References [1] I.D. Albu, M. Neamtu, D. Opris, The Geometry of Fractional Osculator Bundle of Higher Order and Applications, in: Processing of International Conference on Differential Geometry Lagrange and Hamilton Spaces, Dedicated to Acad. Radu Miron at eighty, September 3-8, 2007, Iasi, Romania https://doi.org/10.48550/arXiv.0709.2000. [2] M. Axtell, M.E. 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Fractional Differential Calculus on Manifolds Rule of Transformation for Entries of the Matrix ij(x) 3. Fractional Curvature References