This is a question I encountered at DoCarmo's Differential Geometry of Curves and Surfaces p258.
I do not know this sentence just below the second equation:
"(Of course the geodesic may be tangent to a parallel which is not a geodesic and then v'(s)=0. However, Clairaut's relation shows that this happens only at isolated points)"
My question is that how to apply Clairaut's relation and attain this result.
On page 259, they have Figure 4-20. The point $p_1$ is such a point. Note that they do not draw the parallel that passes through $p_1.$
Clairaut says that, since $\cos 0 = 1,$ and for nonzero $\theta$ we have $\cos \theta < 1,$ it follows that $r > r_0,$ where $r_0$ is the value of $r$ at $p_1.$