TITLE: A regular curve on regular surface in $ \mathbb R^3 $ being of constant tangent length
QUESTION [0 upvotes]: I was given this in diff. geometry on which I am stuck:

Let $ S \subset \mathbb R^3 $ be a regular orientable surface in $ \mathbb R^3 $ such that on it we define the regular curve $ \gamma : I \to S $ a regular curve on the surface S and we define the following positive basis for $ \mathbb R^3 $ as $ \{ T(t) = \gamma ' (t) \space \space ; \space \space N(t) = N(\gamma(t)) \space \space ; \space \space V(t) = N(t) \times T(t)  \} $ and we are to show there exists smooth functions on the curve a,b,c such that:
$ T' = aV+bN $
$ V' = -aT+cN $
$ N' = -bT-cV $

I know the approach is that all three basis elements are smooth and therefore differentiable and the vectors $ T' ; V' ; N' $ are in $\mathbb R^3 $ and therefore are linear combinations of the given basis and therefore I can write for the first one $ T' = aV+bN+dT $ taking inner product with the tangent T and using orthogonality we obtain $ \langle {T', T} \rangle = d \langle {T, T} \rangle $ and because of regularity I know the tangent never vanishes therefore for $ d=0 $ as desired I need to have $ \langle {T', T} \rangle = 0 $ which is equivalent to saying $ ||T|| = const $ which I am not sure if it holds with the given data I was given as I know I forgot nothing and typed it as it was given to me and you see where my block comes from, I certainly appreciate any helper's opinion on this, thanks

REPLY [2 votes]: Review how the Frenet frame works. Any time you have a smoothly-varying orthonormal basis $e_1,e_2,e_3$ at each point of the curve, you're going to have $e_i'$ orthogonal to $e_i$ (by the argument you gave) and $\langle e_i',e_j\rangle  = -\langle e_j',e_i\rangle$ (by the same argument).
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