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

$$\alpha''=\langle\alpha'',\alpha\times N(\alpha)\rangle\,\alpha\times N(\alpha) + \langle\alpha'',\ N(\alpha)\rangle \,N(\alpha)$$

where these coefficients are the geodesic and normal curvatures of $\alpha$.

My question is, what's 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, is there 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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