Let be $\gamma:I\to \mathbb{R}^{3}$ a curve parameterized by arc length, $\{\overline{t},\overline{n}, \overline{b}\}$ the frame of Frenet, $k$ his curvature and $a>0$. Consider also the function $\mathcal{X}:I\times[0, 2\pi]\to \mathbb{R}^{3}$, given by $\mathcal{X}(s, \theta)=\gamma(s)+a(\overline{n}\cos(\theta)+\overline{b}\sin(\theta))$.
Prove that, if $\forall s\in I:k(s)< \frac{1}{a}$ then $S:=\mathcal{X}[I\times [0, 2\pi]]$ must be a regular surface.
The definition of regular surface that I'm following is Do Carmo's whone in his book of differential geometry. I've alredy prove almost everything, the problem that I have is with the injectivity, how to prove that $\mathcal{X}$ is injective?
Seems not to be able by definition, someone told me to do it using the inverse function theorem, but I only studied it with functions with the same dimension in domain and codomain. Any sugestions?