# 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?

• what figure are u talking about? – Seyhmus Güngören Feb 26 '15 at 21:40
• click on the "1" in the "Figure 1" to see it – Mr. K Feb 26 '15 at 21:42
• There is no reason that there is such a maximal one. Indeed, any two points can be joined by a curves with arbitrary long length. – user99914 Feb 27 '15 at 4:14
• I agree with you. So what can we say about the dashed geodesic? Is it a local minimum? – Mr. K Feb 27 '15 at 11:11

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.