The fundamental theory of differential geometry states that:
If there is a given curvature $\bar{\kappa}(s)>0$ and torsion $\bar{\tau}(s)$ which both of them are differentiable and continuous in $(a,b)$, then:
1.there exists an arc length parametrized curve $C:r=r(s)$ such that its curvature $\kappa(s)=\bar{\kappa}(s)$ and its torsion $\tau(s)=\bar{\tau}(s)$
2.the above curve $C$ is the only curve in Euclidean space (up to rigid transformation)
The question I want to ask is that why it emphasizes that $\bar{\kappa}(s)>0 $ in $(a,b)$ ? Isn't that $\bar{\kappa}(s)=0 $ means the curve is a straight line? If it is a straight line, the curve must be the only curve in the space.