Smarandache Tn Curves Of General Helices In The Sol<Sup>3</Sup> Space

Source abstract

In this paper, they study Smarandache TN curves of general helices in the Sol . They characterize Smarandache TN curves of general helices in terms of their curvature and torsion. Finally, found that the explicit parametric equations. 3 1. Riemannian Structure of Sol Space Sol 3 Sol space, one of Thurston's eight 3-dimensional geometries, can be viewed as R provided with Riemannian metric (2) 3 where (x, y, z) are the standard coordinates in R . Note that the Sol metric can also be written as: (3) where ( 4 ) and the orthonormal basis dual to the 1-forms is (5) Proposition 2.1. For the covariant derivatives of the Levi-Civita connection of the left-invariant metric g 3, defined above the Sol following is true: (6) where the (i, j )-element in the table above equals for our basis Lie brackets can be easily computed as: