Let $\alpha:I\to\Bbb R^3$ be a curve parametrized by arc length $s$, with nonzero curvature and torsion. Consider the parametrized surface $$\textbf{x}(s,v) = \alpha(s)+vb(s),\ s\in I,-\epsilon<v<\epsilon,\epsilon>0$$where $b$ is the binormal vector of $\alpha$. Prove that if $\epsilon$ is small, $\textbf{x}(I\times (-\epsilon,\epsilon)) = S$ is a regular surface over which $\alpha(I)$ is geodesic.
The above is an exercise 4.4.17 in Do Carmo's differential geometry of curves and surfaces textbook.
I assumed $I$ is compact. $\textbf{x}_s = \alpha'(s)+vb'(s) = t(s)-\tau(s)n(s), \textbf{x}_v = b(s)$ by Frenet formula where $n$ is a normal vector and $t$ is tangent vector. Hence, $|\textbf{x}_s\times \textbf{x}_v|^2 = |n(s)+\tau(s)t(s)|^2 = 1+\tau^2(s)>0$. How can I go to the next step?
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