I. 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 particular fractional calculus. Thus, we begin with some definitions, for the details we refer to $[1,15,8]$.

The Gamma function of a positive integer $\eta$ is again a positive integer, while the gamma function $\mathrm{\Gamma }(-\eta )$ of a negative integer changes to infinities. The Gamma function any positive $\eta$ value is defined as follows:

The Gamma function $\mathrm{\Gamma }(\eta )$ is considered as a generalization of the factorial and $\mathrm{\Gamma }(\eta )$ is defined for by the integral

In the classical sense since $\mathrm{\Gamma }(0)=\frac{\mathrm{\Gamma }(1)}{0}$, then it follows that $\mathrm{\Gamma }(\eta )$ is not defined for integers $\eta \le 0$. However, the extension formula gives finite values for $\mathrm{\Gamma }(\eta )$, for $\mathrm{Re}(\eta )\le 0$ since $\mathrm{\Gamma }(\eta )$ is analytic everywhere except at $\eta =0,-1,-2,\dots$, and the residue at $\eta =k$ is given by

Now, if , then

Equation (1.1) can be used to define $\mathrm{\Gamma }(\eta )$ for and $\eta \ne -1,-2,\dots$ 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]

Ref

In the classical sense, $\mathrm{\Gamma }(\eta )$ 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:

for $\eta \ne 0,-1,-2,\dots$, and this function can be defined by neutral limit such as

It was also proven in [20] the existence of $r$ the derivative of the Gamma function and defined it by equation

Ref

for $r\in N_0$ and $n\in N$. Also,

for $r=1,2,\dots$, where

Thus, the definition can be extended to the whole real line where,

where $\gamma$ denotes Euler's constant, see [22].

For a function $f:V\subset \mathbb{R}\to \mathbb{R}$ with $0\in V$, the fractional derivative of order $\alpha$ is defined by:

where $n$ is the first integer greater than or equal to $\alpha$.

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].

4. $\frac{d^{\alpha }}{d{t}^{\alpha }}(Cf(t))=C\frac{d^{\alpha }}{d{t}^{\alpha }}f(t)$, where $C$ is a constant,

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 techniques along with fast implementation methods are significantly essential and still active research areas.

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.

II. FRACTIONAL DIFFERENTIAL CALCULUS ON MANIFOLDS

Assume that $N$ be an $m$ -dimensional differential manifold $(V,x_i)$ a local coordinate system on $N$ and $V_0=\{x\in V:0\le x_i\le b_i,i=1,2,...,m\}$ [5].

For a function $f:V_0\to \mathbb{R}$, the fractional derivative with respect to $x_i$:

Ref where ${\partial }_{x_i}^n=\frac{\partial }{\partial x_i}\circ \frac{\partial }{\partial x_i}\circ \dots \circ \frac{\partial }{\partial x_i}$ (n times, $i$ is fixed, $\alpha \ge 0$ ).

For

A fractional vector field $V\subset N$ is an object of the form $X^{\alpha }=X_i^{\alpha }{\partial }_i^{\alpha }$, where $X_i^{\alpha }\in {\mathfrak{S}}_V(N)$ $i=1,\dots ,m$.

The fractional vector fields on $V$ and ${\chi }_V^{\alpha }$ is generated by the operators ${\partial }_i^{\alpha },i=1,2,\dots ,m$ are denoted by ${\chi }_V^{\alpha }$. If $c:x=x(t),t\in I$ is a parameterized curve in $U$ then the fractional tangent vector field of $c$ is given by

A fractional covariant derivative is given by

where $X^{\alpha },Y^{\alpha }\in {\chi }_U^{\alpha }$ and ${\tilde{I}}_{ik}^j$ the functions defining the coefficients of a fractional linear connection on $N$. They are determined by the relations

Since it is essential to study fractional vector fields on a differentiable manifold N. For $R^n$, there is an obvious way to do this. Recall that ${\chi }^{\alpha }({\mathbb{R}}^n)$ denotes the space of fractional differentiable vector fields defined on R. Examples are the fractional vector fields $\frac{{\partial }^{\alpha }}{\partial u_1^{\alpha }},\dots ,\frac{{\partial }^{\alpha }}{\partial u_n^{\alpha }}$ determined by the natural coordinate functions $u_1,\dots ,u_n$.

Definition 2.1. Fractional Riemannian metric F on m-dimensional manifold N defines for every point $p\in N$, the scalar product of fractional tangent vectors in the fractional tangent space $T_p^{\alpha }N$ depending on the point p.

Let $A^{\alpha }=A_i^{\alpha }{\partial }_i^{\alpha }$ and $B^{\alpha }=B_j^{\alpha }{\partial }_j^{\alpha }$ any two fractional vectors tangent to the manifold N at the point p with coordinates $x=(x_1,\dots ,x_m)$ ( $A^{\alpha },B^{\alpha }\in T_p^{\alpha }N$ ) the scalar product is equal to

1. $F(A^{\alpha },B^{\alpha })=F(B^{\alpha },A^{\alpha })$, i.e., ${\tilde{g}}_{ij}={\tilde{g}}_{ji}$ (symmetricity condition).

2. if $A^{\alpha }\ne 0$, i.e. ${\tilde{g}}_{ij}{u}_i^{\alpha }{u}_j^{\alpha }\ge 0$, ${\tilde{g}}_{ij}{u}_i^{\alpha }{u}_j^{\alpha }=0$ iff $u_1^{\alpha }=\dots =u_n^{\alpha }=$

0 (positive definiteness).

3. $F(A^{\alpha },B^{\alpha }){|}_{p=x}$, i.e. ${\tilde{g}}_{ij}(x)$ are smooth function where .

Components of tensor field F in coordinate system are matrix valued functions ${\tilde{g}}_{ij}(x)$

III. RULE OF TRANSFORMATION FOR ENTRIES OF THE

Notes

${\tilde{g}}_{ij}(x)$ - entries of the matrix $\Vert {\tilde{g}}_{ij}\Vert$ are components of tensor field $F$ in a given coordinate system.

How do these components transform under transformation of coordinates $\{x_i\}\to \{x_{i^{\prime }}\}$?

Hence,

Example 2.2. Consider ${\mathbb{R}}^2$ with fractional polar coordinates in the domain , $X=(r^{\alpha }co{s}^{\alpha }\phi ,r^{\alpha }si{n}^{\alpha }\phi )$, then

We have that