I want to show $\gamma'' = \kappa_n N + \kappa_g N \times \gamma'$. We assume $\gamma$ is parametrized by arc lenght. $N$ is the normal vector and $k_n$, $k_g$ are, respectively, the normal and geodesic curvatures. I know the following facts:
- $\{N, \gamma', \gamma'\times N\}$ is a basis for $\mathbb{R}^3$.
- $\gamma ' \perp \gamma ''$
- we can write $\gamma ''$ =A$N$ +B$\gamma ' \times N$, A and B being real constants (i.e. the coefficient along $\gamma'$ is $0$).
- $\kappa_g=\gamma''\cdot(N\times \gamma')$
- $\kappa_n = \|\gamma''\cdot N\|$
I really don't know where to begin to prove it. Help would be appreciated.