I'm taking a course on differential geometry, and up until now I'd always thought that the definition of a geodesic is (loosely speaking) a curve on a surface with the minimal length between its endpoints. My professor, taking his lead from do Carmo, however, defines it as any curve whose geodesic curvature $\kappa_g=0$. We showed that this is equivalent to satisfying the following pair of nonlinear ordinary differential equations:
$(\boldsymbol{E}u' + \boldsymbol{F}v')' = \frac12(\boldsymbol{E}_u(u')^2 + 2\boldsymbol{F}_uu'v' + \boldsymbol{G}_u(v')^2)$
$(\boldsymbol{F}u' + \boldsymbol{G}v')' = \frac12(\boldsymbol{E}_v(u')^2 + 2\boldsymbol{F}_vu'v' + \boldsymbol{G}_v(v')^2)$
We then went through an incredibly painful calculation on the length of the family of curves $\gamma_\lambda$ to show that geodesics (i.e, those curves satisfying the geodesic equations above) are critical points of the functional
$\displaystyle\mathcal{L}(\lambda) = \int_a^b{\left\|\frac{d\gamma_\lambda}{dt}\right\| dt},$
which is the length of the curve. Therefore, according to my professor's (and the textbook's) definition, geodesics are not necessarily length-minimizing, just critical points of $\mathcal{L}$. Therefore, on a sphere, two non-antipodal points have two geodesics: the obvious length-minimizing one, and the other one going the long way around the sphere (which is, in this case, a saddle point of $\mathcal{L}$). This is not just an oversight on my professor's part, he explicitly brought attention to this fact.
My question is, what are the advantages and disadvantages of these two conflicting definitions? I still see the length-minimizing one almost everywhere.
On a related note, the fact that a geodesic is only a critical point, not necessarily a minimum, leaves open the possibility of a geodesic actually being the longest path between two points. Are there any situations where this is actually possible? It seems you could always perturb a curve slightly to stay within the image of a chart while still increasing its length infinitesimally. Are there some weird spaces where this is not the case?