I just started a differential geometry course and I am very far removed from a lot of fundamental relevant background material (such as basic calculus and analysis) so please bear with me.

True or False: For any spherical curve, $\displaystyle c(s) $, the center of the osculating sphere, denoted $\displaystyle m(s) $, is collinear with the normal vector.

First things first, I'm pretty sure this is true. On a spherical curve, the osculating sphere is constant for all $\displaystyle s $. So, in particular, the center is constant. I think that the normal vector, N(s), points directly at the center of the osculating sphere (in fact, that the terminal point of the normal vector IS the center, at any point along the curve)...well either that or -N(s) does.

Does this sound like a decent approach?