Abstract
In this paper, we study general helices in the Sol³. We characterize the general helices in terms of their curvature and torsion. Finally, we find out their explicit parametric equations in the Sol³.
Highlights
Sol space, one of Thurston’s eight 3-dimensional geometries, can be viewed as R3 provided with Riemannian metric gSol3 = e2zdx2 + e−2zdy2 + dz[2]
The tangent vector can be written in the following form
We have dz = cos P, ds Integrating both sides, we have z (s) = cos Ps + C3, where C3 is constant of integration
Summary
Note that the Sol metric can be written as: gSol3 = The orthonormal basis dual to the 1-forms is e1 The connected component of the identity is generated by the following three families of isometries: (x, y, z) → (x + c, y, z) , (x, y, z) → (x, y + c, z) , (x, y, z) → e−cx, ecy, z + c . The Frenet frame satisfies the following Frenet–Serret equations:
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