Is there any situation in which a geodesic maximize the path length between two points? Some people (even in here) claim that geodesics are, in general, stationary curves. Locally speaking, geodesics always minimize arc length (see Manfredo, for example). But I can't visualize a surface having geodesics that maximize the arc length in Riemannian manifolds. Is it possible?
Consider the sphere for instance (Figure 1). The geodesic represented as full line minimizes the path locally. What about the geodesic pictured as dashed line? Certainly, it's not maximize the arc length since there is a lot of other curves that have longer paths. Is this dashed geodesic also a locally minimum path? Or is it neither maximum nor minimum?

 A: No curve between two points on a Riemannian manifold ever maximizes arc length. Given any curve $\gamma$ from $p$ to $q$, you can always find a longer curve, e.g. by leaving $\gamma$ for a little while and taking a detour, and then coming back and rejoining $\gamma$ where you left off. 
The dashed curve in your drawing is a saddle point for the distance function. You can envision the space of all smooth curves with the same endpoints as a sort of infinite-dimensional manifold; in that manifold, this is a saddle with a 1-dimensional space of directions in which the distance function decreases, and an infinite-dimensional space of directions in which it increases. 
The Morse index theorem says that every geodesic segment has at most a finite-dimensional space of directions in which distance decreases (the dimension of this space is called the index of the geodesic), and the index is equal to the number of interior points along the geodesic that are conjugate to one endpoint, counted with multiplicity. (Roughly speaking, you can think of two points being conjugate along a geodesic if there's a 1-parameter family of geodesics between the same two points. This is not exactly right, but thinking of it that way will give you a good intuition about conjugate points.)  
On the sphere, two points are conjugate along a geodesic if and only if they are antipodal points. The dashed geodesic in your drawing contains one interior point that's conjugate to the starting point (namely the point antipodal to the starting point), and therefore it has index 1. 
