# Normal curvature

Jeff Lee in his book on Differential Geometry writes on page 159 that for a unit-speed curve $c$, which is the normal section, the normal curvature $k(c'(0))$ in the direction of its velocity vector at the given point $c(0)=p$ is positive if this curve bends away from the normal $N$ at $p$.
I wonder why it is necessarily positive. It surely must depend on the choice of the normal at the given point, mustn't it? For instance, on a sphere the great circles are normal sections with principal normals pointing inwards towards the centre, so that the angle is either 0 or 180 depending on the choice of the normal field on the sphere. This is actually clear from $k(c'(0))=\kappa(0)\cos\theta(0)$. We then get negative sign in case we choose outer normal.