Effect of local fractional derivatives on Riemann curvature tensor

Abstract

In this short note, we investigate the effect 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 $\mathcal{V}-$fractional derivatives. Then, according to this general operator, a particular Riemannian metric tensor field on the real affine space ${\mathbb{R}}^n$ that is different than Euclidean one is defined. In conlusion, we obtain that the Riemann curvature tensor of ${\mathbb{R}}^n$ endowed with this particular metric is identically $0$, namely, locally isometric to Euclidean space.

7 pages. Comments are welcome