Below is a problem from Do Carmo:
If the geodesic equations (i) and (ii) are parametrized by arc length, then (i) implies (ii), except in the case of coordinate curves.
$(i): u''+\Gamma_{11}^1(u')^2+2\Gamma_{12}^1u'v'+\Gamma_{22}^1(v')^2=0\\(ii): v''+\Gamma_{11}^2(u')^2+2\Gamma_{12}^2u'v'+\Gamma_{22}^2(v')^2=0$
If we take the the following approach to this problem:
Suppose the geodesic is parametrized by $\gamma=X(u(s),v(s)).$By construction of the geodesic equation, we know $\gamma''=(i)X_u+(ii)X_v+(*)N$, and also $\gamma'=u'X_u+v'X_v$. So $0=\langle \gamma'',\gamma'\rangle=(i)u'E+((i)v'+(ii)u')F+(ii)v'G$ and if $(i)=0$ we have $(ii)(u'F+v'G)=0$.
Now if we can show $u'F+v'G$ is nonzero, we are done. But the question is: is it possible to do so?