I wonder if some one can help me with the following problem:
Let $\gamma: \mathbb{R} \to \mathbb{R}^3$ be a regular curve parametrized by arclength. Show that if the tangent curve $\dot{\gamma}: \mathbb{R} \to \mathbb{R}^3$ is a non-constant geodesic on the unit sphere $S^2$, then the curvature $\kappa$ and torsion $\tau$ are constant.
If $\dot{\gamma}$ is a non-constant geodesic on $S^2$, then its second derivative $\gamma^{(3)}$, satisfies $\lVert \gamma^{(3)} \rVert^{\text{tan}}=0$ (it's tangential part is 0). Then it must be parallel to the normal (which on the unit sphere is just the coordinate) so $\gamma^{(3)}=r (x,y,z)$ for some $r$. Is this correct? If yes, how can I proceed from here?