A paradox between intuition and equations regarding the surface of revolution? On the surface of revolution $$\sigma(u,v)=(f(u)cosv,f(u)sinv, g(u))$$ by the geodesic equations (i.e. $(1)\ \ddot{u}=f(u) \dfrac{df}{du} \dot{v}^2$ and $(2)\ \dfrac{d}{dt}(f(u)^2\dot{v})=0 $) we can show that every meridian (i.e. v=constant) and every u=constant in case of only $df/du=0$ are geodesics. If $u=u_0$ and $df/du\ne 0$ according to Eq. $(1)$ the curve is not a geodesic but considering the geometry it must be; because the shortest distance between two point each on $u=u_0$ and $\frac{df}{du}\ne 0$ (both same $u=u_0$) is the very curve of that $u=u_0$. Is this a contradiction?
Thanks a lot.

PS - Equations and Pic. are taken from Section 8.3. of Elementary Differential Geometry by Pressley. 
 A: To give you an example, consider the sphere, which is a surface of revolution given by 
$$(\sin u \cos v, \sin u \sin v, \cos u),$$
that is, $f(u) = \sin u$ and $g(u) = \cos u$, $u\in (0, \pi)$. Note that $f'(u) = \cos u$ is $0$ if and only if $u = \pi/2$, which corresponds to the great circle. Note that if $u_0 \neq \pi/2$, the curve given by $u = u_0$ is not a great circle. 
Note that geodesic are curves that corresponds to critical points of the length functional. We can check that the curve $u=  u_0$ cannot be a critical point of the length functional if $f'(u_0) \neq 0$. To see this, let 
$$\gamma_{u_0} (v) = (f( u_0) \cos v, f( u_0) \sin v, g (u_0))$$
be a parametrization of the curve $u = u_0$. Then the length $L(\gamma_{u_0})$ of this curve is given by $L(\gamma_{u_0}) = 2\pi f(u_0)$. So if we varies this $u_0$, then 
$$\frac{d}{du} L(\gamma_u) = 2\pi f'(u),$$
thus there is a family of curves in the surface so that 
$$\frac{d}{du} L(\gamma_u)\bigg|_{u=u_0} = 2\pi f'(u_0) \neq 0.$$
Thus the curve $\gamma_{u_0}$ cannot be a geodesic. 
Geometrically, you might think of wrapping a rubber band on this surface of revolution (To represent the curve $u = u_0$). The rubber band would be stable only when $f'(u_0) =0$, if $f'(u_0)\neq 0 $, the rubber band would tend to move to reduce it's length. 
