Suppose you have some surface $S$ with a curve $c$ in it that allows for a Frenet triad. If we know that for every point along the curve $c(t)$ that $T(t) \in T_pS$ (always the case) but also that $B(t)\in T_pS$ where $B$ is the binormal vector and we further are told that every point along the curve in umbilical, then why is it that $c$ must be a planar curve?
I have tried to figure this out to no avail. I managed to prove that the Gaussian map at any point on the curve is either -$N(t)$ or $N(t)$ i.e. the normal to the curve is either parallel or anti-parallel to the normal to the surface.
I know that if every point in a surface is umbilical then it must be a sphere or a plane as its Gaussian curvature is constant. The proof of this however uses parameterisations and notions of connectedness which cannot be applied to the curve, so although I think the Gaussian curvature along the curve is constant I cannot prove it. I also know that the tangent and binormal together form a basis for the tangent plane, and that the Weingarten map is now just a constant $k$ and that every direction is a principal direction, but I cannot put these all together to show that the torsion must be 0 and therefore that the curve is planar.
Any help or hints welcome :)