The Characterizations of Parallel q-Equidistant Ruled Surfaces

Abstract

In this paper, parallel q-equidistant ruled surfaces are defined such that the binormal vectors of given two differentiable curves are parallel along the striction curves of their corresponding binormal ruled surfaces, and the distance between the asymptotic planes is constant at proper points, which is related to symmetry. The characterizations and some other useful relations are drawn for these surfaces as well. If the surfaces are considered to be closed, then the integral invariants such as the pitch, the angle of the pitch, and the drall of them are given. Finally, some examples are presented to indicate that the distance between the proper points on the corresponding asymptotic planes is always constant.

Evolution for First Eigenvalue of LT,f on an Evolving Riemannian Manifold

“…In addition, for a special case of the (1,1)-tensor T, on Bianchi classes, some bounds of the the first non-zero eigenvalue are derived under the normalized Ricci flow. To develop this area more in the future, one can consider the techniques of the Singularity theory and Submanifold theory presented in [21][22][23][24][25][26][27][28], and it may find some new and interesting results.…”

The Invariants of Dual Parallel Equidistant Ruled Surfaces

“…The concept of parallel equidistant ruled surfaces was first introduced by Valeontis [11]. The resources [12][13][14][15][16][17][18][19] can be examined for studies on these surfaces. Although dual numbers are a set of numbers defined by William Kingdon Clifford (1845-1879) [20], their first applications in geometry began with Eduard Study [21].…”

Another application of Smarandache based ruled surfaces with the Darboux vector according to Frenet frame in $E^{3}$

“…Therefore, it is one of the most used surfaces in architectural designs. There are numerous studies on surfaces, [5,16,17,22,24]. In surface theory, there are special surfaces of which one is named the ruled surface.…”