In baby do Carmo's book Differential Geometry of Curves and Surfaces, definition 8a states that a regular connected curve $C$ in a regular surface $S$ is called a geodesic if, for each point $p \in S$, the parameterization $\alpha(s)$ of a coordinate of $p$ by the arc length $s$ is a parameterized geodesic (it is geodesic for all $t$ in the interval domain $I$ of the curve $C$).
My question is about a comment in the book, right after this definition:
From a point of view exterior of the surface $S$, Definition 8a is equivalent to saying that $\alpha''(s) = kn$ (principal normal vector) is normal to the tangent plane, that is, parallel to the normal of the surface."
A regular curve is a geodesic iff this happens for every point in the curve.
I don't get it. Isn't the principal normal vector at a point of a curve always parallel to the normal of the surface? When calculating the second fundamental form of $\alpha'(0)$, we get $<N,kn> = k<N,n> = k_n$. I believe my confusion comes from always thinking about spheres and cylinders and planes, and so we would have $<N,n> = 1$. Please enlighten me on this.