I want to show that the normal curvature of any curve on a sphere of radius $r$ is $\pm \frac{1}{r}$.
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The normal curvature is $\kappa_n=\gamma '' \cdot \textbf{N}$, where $\gamma$ is a unit-speed curve and $\textbf{N}$ is the unit normal of the sphere.
I have done the following:
Let $\gamma (t)$ be a unit-speed curve of the sphere with center $\alpha$ and radius $r$.
Then $(\gamma -\alpha) \cdot (\gamma -\alpha )=r^2$.
Differentiating in respect to $t$, we get $$\gamma ' \cdot (\gamma -\alpha )\cdot \gamma '=0 \Rightarrow \gamma ' \cdot (\gamma -\alpha )=0$$
Differentiating again in respect to $t$, we get $$\gamma '' \cdot (\gamma -\alpha )+\gamma ' \cdot \gamma '=0 \Rightarrow \gamma '' \cdot (\gamma -\alpha )=-\gamma ' \cdot \gamma ' =-1$$
Is it correct so far?
How can we calculate the unit normal $\textbf{N}$ of the sphere?