Classification results on surfaces in the isotropic 3-space

The isotropic 3-space I^3 which is one of the Cayley--Klein spaces is obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we give several classifications on the surfaces in I^3 with the constant relative curvature (analogue of the Gaussian curvature) and the constant isotropic mean curvature. In particular, we classify the helicoidal surfaces in I^3 with constant curvature and analyze some special curves on these.Comment: 12 pages, 2 figure

Effect of local fractional derivatives on Riemann curvature tensor

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.Comment: 7 pages. Comments are welcom

Translating solitons of translation and homothetical types

We prove that if a translating soliton can be expressed as the sum of two curves and one of these curves is planar, then the other curve is also planar and consequently the surface must be a plane or a grim reaper. We also investigate translating solitons that can be locally written as the product of two functions of one variable. We extend the results in Lorentz-Minkowski space.MTM2017-89677-P, MINECO/AEI/FEDER, U

On certain surfaces in the isotropic 4-space

The isotropic space is a special ambient space obtained from the Euclidean space by substituting the usual Euclidean distance with the isotropic distance. In the present paper, we establish a method to calculate the second fundamental form of surfaces in the isotropic 4-space. Further, we classify some surfaces (spherical product surfaces and Aminov surfaces) in the isotropic 4-space with vanishing curvatures