A relation among DS^{2}, TS^{2} and non-cylindrical ruled surfaces

Abstract

$TS^{2}$ is a differentiable manifold of dimension 4. For every $Xin TS^{2}$, if we set $X=(p,x)$ we have $=0$ since $vec{p}$ is orthogonal to $T_{p}S^{2}$, therefore $parallel vec{p}parallel =1$. Those there could exist a one-to-one correspondence between $TS^{2}$ and $DS^{2}$. In this paper we gave and studied a one-to-one correspondence among $TS^{2}$, $DS^{2}$ and a non cylindrical ruled surface. We showed that for a restriction of an anti-symmetric linear vector field A along a spherical curve $alpha (t)$ there exists a non-cylindrical ruled surface which corresponds to $alpha (t)$ and has the following prametrization [alpha (t,lambda )=alpha (vec{t})+A(alpha (t))+lambda alpha (vec{t})] So it is possible to study non-cylindrical ruled surfaces as the set of $(alpha (t),A(alpha (t)))$, where $alpha (t)in S^{2}$ and $A$ is an anti-symmetric linear vector field in ${cal R}^{3}$