>>> > All geodesics do obey Fermat's principle of Least Time.  That is the
>>> > reason why Snell's law exists.  Any object moves from one point in
>>> > space to another through the path with the least accumulated amount of
>>> > time.
> 
> Before making a general statement, why not see if it is true for
> a few simple cases. Here's a case: you throw a ball straight up
> into the air, and it comes back to you. What is the height h as
> a function of time t? Does the path minimize the total time?
>
> Well, if we treat the acceleration of gravity as approximately
> constant, call it g, then the time required for the ball to
> return to its starting place is
>
>      T = 2 v0/g
>
> where v0 is the initial upwards speed.
>
> If v0 = 25 meters/second, g = 10 meters/second^2, then T = 5 seconds.
> Is that the smallest possible time? Obviously not. Suppose that the
> ball travels straight up at speed 25 meters/second for 1 second, turns
> around, and comes back down at speed 25 meters/second for 1 second.
> Then the whole trip would only be 2 seconds. What if the ball travels
> at 25 meters per second for 1 second, and then turns around and travels
> 50 meters per second for 1/2 second. Then the trip would only be 1.5
> seconds. 5 seconds is *not* the smallest possible time.

You have two different metrics.  One involves with (g = 10m/sec^2), and
the other one has (g = 25m/sec^2).  Each one obeys the principle of
Least Time.  You need to study the calculus of Variations.  Learn how
the geodesic equations are derived from this very Langrangian method.