Effect of local fractional derivatives on Riemann curvature tensor

Abstract

In this paper, we give a main example indicating the ineffectiveness of the local fractional derivatives on the Riemann curvature tensor that is a common tool in calculating curvature of a Riemannian manifold. For this, first we introduce a general local fractional derivative operator that involves the mostly used ones in the literature as conformable, alternative, truncated $M-$ and $V-$fractional derivatives. Then, according to this general operator, a particular Riemannian metric on the real affine space $R^n$ that is different from the Euclidean one is defined. In conclusion, our main example states that the Riemann curvature tensor of $R^n$ endowed with this particular metric is identically 0, that is, one is locally isometric to Euclidean space.