= Solution
If $\kappa\equiv0$, then $T'=0$, so the unit tangent is constant and
$$
\gamma(s)=p+sT.
$$
Thus both curves are straight line segments. Equal arc length makes their parameter intervals equally long, and a rotation in $SO(3)$ followed by a translation maps one to the other. Hence they are related by a <proper Euclidean motion of Euclidean three-space>.
The conclusion fails for $\kappa\equiv1$, because curvature alone does not determine a space curve. A unit circle has curvature one and torsion zero. The <circular helix> with $a=b=1/2$,
$$
\eta(s)=\left(\frac12\cos(\sqrt2s),
\frac12\sin(\sqrt2s),
\frac{s}{\sqrt2}\right),
$$
is unit speed and has curvature one but torsion one. Proper Euclidean motions preserve torsion, so equal-length pieces of these two curves cannot be related by one.
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