In this paper, we have a tendency to investigate a particular Weingarten and linear Weingarten varieties of canal surfaces according to Bishop frame in Euclidean 3-space E 3 satisfying some fascinating and necessary equations in terms of the Gaussian curvature, the mean curvature, and therefore the second Gaussian curvature. On the premise of those equations, some canal surfaces are introduced.

This research aims to study Darboux sweeping surface in isotropic space I1 3 . We went through the geometric characteristics of sweeping surfaces in I1 3 . The first and second fundamental forms of the sweeping surface were evaluated. Furthermore, we investigate the mean and Gaussian curvature of the sweeping surface. We also show that the parametric curves on these surfaces are non-geodesic and non-asymptotic. Then, we derive the necessary and sufficient conditions for the sweeping surface to become a developable sweeping surface, minimal sweeping surface, and Weingarten surface. Finally, an example to illustrate the application of the results is introduced.

In this paper, we introduce the equiform-Bishop frame of a spacelike curve r lying fully on S 2 1 in Minkowski 3-space R 3 1. By using this frame, we investigate the equiform-Bishop Frenet invariants of special spacelike equiform-Bishop Smarandache curves of a spacelike base curve in R 3 1. Furthermore, we study the geometric properties of these curves when the spacelike base curve r is specially contained in a plane. Finally, we give a computational example to illustrate these curves.

We investigate the equiform Hasimoto surfaces P(σ, ℓ) in Minkowski 3-space in this paper. In E 3 1 , in different three cases, the geometric properties of equiform Hasimoto surfaces are discussed. For each case, the equiform Gaussian and mean curvatures of the equiform Hasimoto surface are determined. Then, in E 3 1 , we characterize the parameter equiform Bishop curves of equiform Hasimoto surfaces. Keywords— Hasimoto surface, Minkowski geometry, equiform Bishop frame, smoke ring equation.

In this paper, we study inextensible flows of spacelike curves lying fully on a spacelike surface Ω according to equiform frame in 4-dimensional Minkowski space ℝ1 4 . We give necessary and sufficient conditions for this inextensible flows which are expressed as a partial differential equation involving the equiform curvature functions in 4-dimensional Minkowski space ℝ1 4 . Finally we give an application of inextensible flows of spacelike curves in ℝ1 4 .

In this work, we show a local study of a cyclic surface in produced by equiform motions of a cycloid. We definepartially such surfaces with some conjecture on its curvature. In general, we shall consider the condition that scalar curvature is locally constant. We will conclude cases of scalar curvature of following equations accounts and will put theories to explain the results of research and example.