Let $k1$ and $k2$ be the principal curvatures of an oriented surface S and N be the field of unit normal vectors to S. Let $F_i : S \mapsto \Bbb R^3$ be the map defined as follows: $F_i(p) = p + \frac 1{k_i}N(p)$.
a) Prove that if $p_0$ is a non-umbilical point and the direction derivative of $k_i, i = 1, 2$ at $p_0$ in the corresponding principal direction does not vanish, then there exists a neighborhood $U$ of $p_0$ of $S$ such that $F_1(U)$ and $F_2(U)$ are regular surfaces.
For this one, I want to use the fact that $k1$ and $k2$ are nowhere zero to prove.
b) Prove that if $\alpha(t)$ is a line of curvature on $U$, which is tangent to the principal direction corresponding to the principal curvature $k1$, then after an appropriate reparametrization the curve $F_1(\alpha(t))$ is a geodesic on the surface $F_1(U)$.
If we assume that I'm able to correctly prove the initial component (that $F_1$ is a regular srface), how can I use the fact that $\alpha(t)$ is a line of curvature on $U$ to prove that, when plugged into the regular surface $F_1(U)$, the curve is geodesic? I'm thinking I can use this information to prove that the field of tangents is parallel along the curve $\alpha(t)$.