Consider the following problem from Chapter 3 of Curves and Surfaces, 2nd edition, by Montiel and Ros:
Now, for me it is easy to see that
$$
(dF_r)_p(v) = v + r (dN)_p(v), \quad v \in T_pS.
$$
If $v = e_i$ is a principal direction,
$$
(dF_r)_p(e_i) = (1 + rk_i)e_i.
$$
This differs from what the books asks us to show. What am I missing? Is it really a typo?
Best Answer
We directly have that ${\rm d}(F_r)_p(v) = v+r\,{\rm d}N_p(v)$. So if ${\rm d}N_p(e_i) = -k_i(p)e_i$, we have that $${\rm d}(F_r)_p(e_i) = e_i - rk_i(p)e_i = (1-rk_i(p))e_i.$$The problem here is that the principal directions are the eigenvectors of the shape operator, and this is minus the differential of the Gauss map.