Curve Inside a Sphere: Differentiating Alpha

[Celso]

TL;DR

Let $S^2 \subset R^3$ the sphere whose center is at the origin and has radius 1.There is a function $\alpha \colon I \rightarrow R^3$ parameterized by arc length, regular, such as $\alpha (I) \subset S^2$ with constant and positive curvature.

Proof that there exists a differentiable function $g \colon I \rightarrow R$ such as $\alpha (s) = - \frac{1}{k} \vec{n} + g(s) \vec{b}$.

Show that $(\frac{1}{k})^2 + (g(s))^2 = 1$, therefore $g(s)$ is constant

Honestly I don't know where to begin. I started differentiating alpha trying to show that its absolute value is constant, but the equation got complicated and didn't seem right.

 

[fresh_42]

I think the constant curvature of $\alpha$ is the key here. What does the curvature formula say?