Ruled Surfaces with Bishop vectors via Smarandache geometry

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This paper introduces three distinct types of ruled surfaces, namely, the quasi-tangent surfaces , the quasi-normal surfaces, and the quasi-binormal surfaces. These types are determined by the orientation of their direction curves tangent, normal, and binormal to the base curve, respectively. This paper does not only introduce these surfaces but also determine their fundamental properties, including the first, the second, and the third fundamental forms, as well as the Gaussian and the mean curvatures. Also, the geodesic curvature, the normal curvature, and the geodesic torsion associated with the base curve for each type of surface are investigated. Furthermore, the conditions for the base curve to be as a geodesic, an asymptotic line, and a principal line for each type of surface are provided. Also, the conditions for these curves to be considered developable and minimal surfaces are introduced. Moreover, two illustrative examples are introduced to obtain our results

We investigate ruled surfaces in 3d Riemannian manifolds, i.e., surfaces foliated by geodesics. In 3d space forms, we find the striction curve, distribution parameter, and the first and second fundamental forms, from which we obtain the Gaussian and mean curvatures. We also provide model-independent proof for the known fact that extrinsically flat surfaces in space forms are ruled. This proof allows us to identify the necessary and sufficient condition the curvature tensor must satisfy for an extrinsically flat surface in a generic 3d manifold to be ruled. Further, we show that if a 3d manifold has an extrinsically flat surface tangent to any 2d plane and if they are all ruled surfaces, then the manifold is a space form. As an application, we prove that there must exist extrinsically flat surfaces in the Riemannian product of the hyperbolic plane, or sphere, with the reals, and that do not make a constant angle with the real direction.

A basic goal of this paper is to study the tube surface with a null Cartan curve with respect to the Bishop frame in Minkowski 3-space $E_{1}^{3}$ and to compute some geometric features for this kind of the tube surface such as, the first and the second fundamental forms, the mean and the Gaussian curvatures. Furthermore, using the mean curvature $H$ and the Gaussian curvature $K$, we investigate $\Omega(K,H)-$ surface condition of the tube surface with the null Cartan curve due to the Bishop frame in Minkowski 3-space $E_{1}^{3}$.

Attempts have been made to introduce ruled surfaces generated from any vector X, Bishop Darboux vector and Bishop vectors. Some properties with respect to developable and minimal and the relations between the integral invariants of these surfaces are discussed. Also the fundamental forms, geodesic curvatures, normal curvatures and geodesic torsions are calculated, and some results are obtained.

In this paper, a study of time-like ruled surfaces in Minkowski 3-space is investigated by strictly connected time-like straight line moving with Darboux's frame along a differentiable space-like curve. By using the striction curve and the distribution parameter of time-like ruled surfaces, some theorems related to the geodesic curvature and the second fundamental form tensor are obtained.