State and prove necessary and sufficient conditions for a Frenet curve $\gamma :I \to \mathbb{R}^3$ to be contained in some $2$-sphere $S^2=\{x \in \mathbb{R}^3; ||x−c||=r\}$ of centre $c$ and radius $r>0$.
The only one I can think of is the torsion needs be equal to $0$ as only a plane curve can lie on the surface of a two sphere. Would one of the conditions also be that the inner product of the tangent and position vectors must be $0$ (i.e. they are orthogonal) as any bit of tangent vector in the direction of the position would push the curve off the surface?