 30
 1
 Summary

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.