Definition of curves.

Question

To define a regular smooth curve in $\mathbb R^3$ one can take some parametrization $(x_1(t), x_2(t), x_3(t))$ with the velocity vector $\frac{dx}{dt}$ is not equal to zero (it is the regularity condition). It is well known that locally every regular smooth curve is a solution for equation system $F_1 = F_2 = 0$ of maximal rank. It has obvious generalization to an higher dimensional submanifolds, e.g. for surfaces in $\mathbb R^4$. And it also well known that in general embedded submanifold can not be represented as a solution of equations(with maximal rank condition) $globally$, e.g. $\mathbb{RP}^2$ embedded in $\mathbb R^4$. The question is simple. Is there any regular closed smooth curve $(x_1(t), x_2(t), x_3(t))$ with the velocity vector $\frac{dx}{dt}$ is not equal to zero such that it can not be globally represented as a solution of regular equation system: $F_1 = F_2 = 0$, ${\frac{\partial F_i}{\partial x_j}}$ has rank = 2.

Answer 1

First, consider the case of a trivial vector bundle over the circle. You realize zero section of the bundle as the regular zero level set of a compactly supported map on the total space of the bundle. Next, the normal bundle of any smoothly embedded circle in the Euclidean space is trivial. Lastly use the fact that a tubular neighborhood if such circle is diffeomorphic to the total space of the normal bundle .