Original scientific paper

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

B. Karakaş
H. Gündoğan

Abstract

$T{S}^2$ is a differentiable manifold of dimension 4. For every $X\in T{S}^2$, if we set $X=(p,x)$ we have $<\overrightarrow{p},\overrightarrow{x}>=0$ since $\overrightarrow{p}$ is orthogonal to $T_p{S}^2$, therefore $\parallel \overrightarrow{p}\parallel =1$. Those there could exist a one-to-one correspondence between $T{S}^2$ and $D{S}^2$. In this paper we gave and studied a one-to-one correspondence among $T{S}^2$, $D{S}^2$ 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 $$\alpha (t,\lambda )=\alpha (\overrightarrow{t})+A(\alpha (t))+\lambda \alpha (\overrightarrow{t})$$
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 ${\mathcal{R}}^3$.

Keywords

dual unit sphere; non-cylindrical ruled surface; spherical curve; anti-symmetric linear vector field; tangent bundle

Hrčak ID:

736

URI

https://hrcak.srce.hr/736

Publication date:

20.6.2003.

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