i'm studying diferential geomtry and i'm in the part of geodesics, my professor always define a curve that can define the tangent field but for calculating the geodeiscs and normal curvatures at each point $\alpha(t)$ of the curve he define the frame $D=\{\alpha^{'}(t),\alpha(t)\times N(\alpha(t)),N(\alpha(t))\}$ where $N$ is the Gauss aplication of the surface, so we get

$\alpha^{''}=<\alpha^{''},\alpha\times N(\alpha)>(\alpha\times N(\alpha)) + <\alpha^{''},\ N(\alpha)>N(\alpha)$

where this coeficients are the geodesic and normal curvatures of $\alpha$, my question is, whats the relation between $D$ and the Frenet-Serret $F=\{\alpha^{'}(t),n(t),b(t)\}$ where $n$ and $b$ are the normal and binormal of the curve, there is a relation between $n$ and $\alpha\times N(\alpha)$, or $b$ and $N(\alpha)$?

i'm having several dificulties about exercises involving this because i don't know how to compare results of curves, like torsion with the Darboux vectors


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