Original scientific paper
A relation among DS^{2}, TS^{2} and non-cylindrical ruled surfaces
B. Karakaş
H. Gündoğan
Abstract
$TS^{2}$ is a differentiable manifold of dimension 4. For every $%
X\in TS^{2}$, if we set $X=(p,x)$ we have $<\vec{p},\vec{x}>=0$ 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 ${\cal R}^{3}$.
Keywords
dual unit sphere; non-cylindrical ruled surface; spherical curve; anti-symmetric linear vector field; tangent bundle
Hrčak ID:
736
URI
Publication date:
20.6.2003.
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