Jeff Lee in his book on Differential Geometry writes on page 159 that for a unit-speed curve , which is the normal section, the normal curvature in the direction of its velocity vector at the given point is positive if this curve bends away from the normal at .
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 . We then get negative sign in case we choose outer normal.
I'm just wondering…it should actually read "the normal curvature is…nonzero…if the curve bends away from the normal", shouldn't it? Or is there a misunderstanding on my side? Is it a mistake?